There is something about prime numbers… An air of mystery surrounds them, that makes them one of the most alluring (and most studied) objects in all of mathematics. Despite hundreds of years of prime number research, there is still so much we do not know about them. Of course, we know that there are infinitely many prime numbers, with a first **proof due to Euclid** and many, many **other equally fascinating proofs** that continue to be found. Nonetheless, many open problems about their distribution among the natural numbers remain wide open. The Riemann Hypothesis, for instance, is intimately intertwined with the **distribution of prime numbers**.

In addition to the mysterious nature of the prime numbers as a whole, certain individual primes have a special place in my heart, for various reasons. In this blog post, I will list a few of my favorite primes, together with the fascinating properties that make them special… to me! The reader and other mathematicians would certainly compose different lists of favorite primes.

Without further ado, the list begins with the very first of all prime numbers…

The number 2 is the first prime, the smallest prime, and the **pain of number theorists’ existence**. It is such an *odd* prime that there is no other quite like it (unless you look for prime ideals above 2 in other number fields other than Q!). All sorts of curious facts come back to the fact that 2 is the unique even prime. For example:

- If q=m
^{n}-1 is a prime number, for some natural number m, then either q=2 or m=2. Primes of the form q=2^{n}-1 are called Mersenne primes (which will make another guest appearance below). - See also the special role of 2 in the construction of Fermat numbers and Fermat primes below.
- For any n>1, a polygon with 2
^{n}sides can be constructed with a ruler and compass, but if you replace 2 by any other prime p, this is no longer true (we will come back to this point later on). - The
**Law of Quadratic Reciprocity**gives a beautiful relationship between pairs of primes, but the prime 2 is a complete outlier in this regard, and it does not behave at all like the rest of the primes. - The group (
**Z**/p^{n}**Z**)^{x}is cyclic for all primes p>2 and all n>0, but it is not cyclic for p=2 and n>2.

p=37 might be my all-time favorite prime, for silly reasons such as 37*3 = 111, 37*6 = 222,… , and also for deeper reasons such as the fact that 37 is the first **irregular prime**. The regular primes are those exponents for which Fermat’s last theorem has a “simple proof” (first discovered by Lamé, who proposed an erroneous proof of Fermat’s last theorem, which was later fixed by Kummer for regular primes). The irregular primes, 37, 59, 67, 101, 103, 131, 149,… are those for which Kummer’s proof doesn’t work. In particular, this means that the class group of the ring of integers of the 37th cyclotomic field is of order divisible by 37… and in this case it is exactly of order 37.

Another couple of reasons why I am fascinated by the number 37 come from the **theory of elliptic curves**. A map between two elliptic curves is called an isogeny, and it turns out that cyclic, rational isogenies are somewhat rare. The size of the kernel of the map is called the degree of the isogeny, and Barry Mazur showed that there are only finitely many primes that are degrees of isogenies of elliptic curves. As it turns out, p=37 is one of the degrees that can occur… but it only occurs for two (isomorphism classes of) elliptic curves (**1225.b1** and **1225.b2**), and these elliptic curves are** rather special**. The second reason will be explained below.

The prime number 163 is really nice for several reasons. For instance, e^{pi*Sqrt(163)} is really close to being an integer (it is 262537412640768743.99999999999925… so an integer to 12 decimal places) which has a very **interesting explanation** coming from elliptic curves with complex multiplication. Not completely unrelated to this the previous fact, Q(Sqrt(-163)) is the “last” of the imaginary quadratic fields of class number 1 (there are only nine such fields, and this is the one with largest discriminant in absolute value). And also in the same family of amazing facts: the values of the polynomial x^{2}-x+41 for x=0 up to x=40 are prime numbers! Finally, 163 is the largest possible degree of a cyclic, rational isogeny for an elliptic curve defined over Q.

Fermat’s little theorem says that if p is an odd prime, then p is a divisor of the number 2^{(p-1)} – 1. A **Wieferich prime** is a prime p such that p^{2} is a divisor of 2^{(p-1)} – 1. We only know two Wieferich primes: 1093 and 3511. The crazy thing is that we conjecture that there are infinitely many Wieferich primes… but we only know two of them! More concretely, we expect log(log(x)) Wieferich primes below x, and since log(log(x)) grows so slowly, we are not surprised we haven’t found any others yet. I became interested in Wieferich primes (in fact, Wieferich places) when they unexpectedly showed up in **some work of mine**.

The **twin prime conjecture** claims that there are infinitely many natural numbers n such that n and n+2 are both primes. Sometimes, it is useful to have a “large” pair of twin primes to compute with, and 4001 and 4003 are easy to remember, large enough for most purposes, and not too large at the same time. That’s it. They are stuck in my head, and I use them very often!

The set E(Q) of all rational points on an elliptic curve E defined over Q is a finitely generated abelian group (thanks to the **Mordell-Weil theorem**), so E(Q) has a finite torsion subgroup T(E/Q), and also R(E/Q) rational points of infinite order such that E(Q) is isomorphic to T(E/Q) + **Z**^{R(E/Q)}. No one knows how large the rank R(E/Q) of an elliptic curve over Q can be, or what values R(E/Q) can take for that matter. The largest known rank is 28 (an **example due to Noam Elkies**). So it is interesting to find the “simplest” elliptic curves with any given rank. We organize elliptic curves by their conductor, so it is interesting to find examples of elliptic curves with rank R(E/Q)=0, 1, 2, 3, 4,… with the smallest possible conductor. Here is the beginning of such a list, with curves given by their **LMFDB.org** label:

- R(E/Q) = 0, conductor 11, curve 11.a1.
- R(E/Q) = 1, conductor 37, curve 37.a1.
- R(E/Q) = 2, conductor 389, curve 389.a1.
- R(E/Q) = 3, conductor 5077, curve 5077.a1.
- R(E/Q) = 4, conductor 234446 = 2*117223, curve 234446a1.
- R(E/Q) = 5, conductor 19047851, curve 19047851.a1.

The curves of rank 3 and conductor 5077 have a special place in the history of number theory, and 5077a1 is called the “Gauss curve” (see the paragraph at the **bottom of this LMFDB page**). As far as I know, there is an **elliptic curve of rank 6** and conductor 5187563742=2*3*2777*311341 but it is not proven to be the smallest such conductor!

Even though we have a proof that there are infinitely many prime numbers, finding very large prime numbers is a very difficult task. Thus, it would be of great interest if there was a simple formula or function that produced prime numbers. One famous such “formula” was proposed by Fermat, who famously claimed that the numbers of the form F_{n} = 2^{2^n}+1, known as **Fermat numbers**, are always prime. The first few Fermat numbers F_{0} = 3, F_{1} = 5, F_{2} = 17, F_{3} = 257, and F_{4} = 65537 are, indeed, prime numbers. However, Fermat’s claim has been proven to be fantastically wrong, since every single other Fermat number that we have been able to factor has turned out to be a composite number. For instance, Euler proved in 1732 that F_{5} = 4294967297 = 641*6700417.

Fermat primes, if you can find them, are really cool, because of the **Gauss-Wantzel theorem** which says that a regular polygon with n sides can be constructed with a compass and ruler (straightedge, no markings) if and only if n is the product of a power of 2 and any number of distinct Fermat primes. So, in particular, there is a construction of a polygon with 65537 using just a compass and a ruler!

It should be obvious why I love this one! One can ask if there are palindromic numbers, with digits in order, that are prime. The sequence that I have in mind is 1, 121, 12321, 1234321, etc., and none of these numbers are prime, until you reach

12345678910987654321

which is prime! Coincidentally, 1234567891010987654321 is also prime. If you continue the pattern… it turns out that the next (probable!) prime is the 17350-digit number 1234567…244524462445…7654321 according to **OEIS.org**.

As we mentioned above in the entry for p=2, if q=m^{n}-1 is a prime number, for some natural number m, then either q=2 or m=2. Moreover, if q=2^{n}-1 is prime, then n is prime (and if so, q is called a Mersenne prime). Unfortunately, this is not a necessary and sufficient criterion and some prime values of n do not yield a Mersenne number q (for instance, 2^{11}-1 = 23*89 is composite). The largest known prime (as of the writing of this post) is a Mersenne prime (the 51st Mersenne prime that we have been able to find), namely the prime number M_{51} = 2^{82,589,933} − 1. It is worth noting the mind-blowing fact that M_{51} has 24,862,048 digits.

A really cool fact about Mersenne primes is their relationship to even **perfect numbers**: if 2^{p}-1 is prime, then 2^{p-1}(2^{p}-1) is a perfect number (proved by Euclid!) and, viceversa, if n is an even perfect number, then it is of this form (proved by Euler!). So the largest even perfect number we are aware of is 2^{82,589,932} * (2^{82,589,933} − 1) … a perfect number with 49,724,095 digits!

In each era of the history of mathematics, there have been open problems and conjectures that mathematicians have paid particular attention to, maybe because of the intrinsic beauty of the problem, its perceived importance within an area of study, or simply put, because of the fame that a solution would bestow on the solver. At several points in time, lists of such problems have been compiled and advertised for various reasons. Such lists, as historical artifacts, serve as a snapshot of the state-of-the-art of mathematics, and the challenges themselves give us insight into the types of problems that were teasing the curious minds of the mathematicians of a given time period.

One of the first documented examples of a list of mathematical problems dates back to the year 1220 (CE). It was composed as a list of challenges to be solved by the mathematician Leonardo Pisano, who is better known nowadays by one of his nicknames: *Fibonacci* (see Devlin’s “The Man of Numbers” for an account of Fibonacci’s life and works). As the story goes, in 1202 Fibonacci authored *Liber Abaci* (Book of the Abacus), which is credited as a key text in the introduction of the Hindu-Arabic numerals to European mathematics. The book and techniques that Pisano detailed in his volume were quickly understood as a monumental advance in math and science. Within a few years, *Liber Abaci* had been widely praised, copied, and distributed, and the scientific advisors to the Holy Roman Emperor Frederick II became well aware of the impact of Fibonacci’s book and of the rumored unparalleled mathematical skills of its author. Thus it was high time to invite Fibonacci to join the emperor’s court. As a way to introduce Pisano, one of the scholars of the court, Johannes of Palermo, compiled a list of mathematical challenges that were presented to Leonardo, to be solved as a demonstration to the Emperor of his sophisticated mathematical knowledge.

The full list that Palermo put together is not known, but we know three of the featured problems because Fibonacci described their solutions in two of his books, *Flos* (Flower) and *Liber Quadratorum* (Book of Squares). The three challenges read as follows:

- To find a rational number such that, when 5 is added to its square, the result is the square of another rational number, and when 5 is subtracted from its square, the answer is also the square of a rational number.
- Find a number such that if it be raised to the third power, and the result added to twice the same number raised to the second power, and if that result be then increased by ten times the number, the answer is twenty.
- Three men owned a store of money, their shares being 1/2, 1/3, and 1/6. But each took some money at random until none was left. Then the first man returned 1/2 of what he had taken, the second 1/3, the third 1/6. When the money now in the pile was divided equally among the men, each possessed what he was entitled to. How much money was in the original store, and how much did each man take?

Fibonacci’s solution of the first of Palermo’s problems was 41/12. Note that

(41/12)^2 – 5 = (31/12)^2 and (41/12)^2+5 = (49/12)^2,

as required. The result is that the three squares (31/12)^2, (41/12)^2, and (49/12)^2 are in an arithmetic progression with difference 5, and we say that the three squares are congruent modulo 5. We also refer to 5 as a *congruent number* because such arithmetic progression of squares exists with common difference 5. One may ask (and Fibonacci indeed asked this question in his Book of Squares) what natural numbers are *congruent*. In other words, suppose n>0 is a natural number. Are there three square numbers a^2, b^2, and c^2 such that b^2-n=a^2 and b^2+n = c^2? For instance, n=6 is also a congruent number because (1/2)^2, (5/2)^2, and (7/2)^2 are three squares with common difference 6. Indeed, we have

(5/2)^2-6 = (1/2)^2 and (5/2)^2+6 = (7/2)^2.

The quest to characterize the set of all congruent numbers, known as the * congruent number problem*, is still ongoing to this day, and it has generated a large body of research, with a long list of partial and conditional results (most notably Tunnell’s criterion).

Palermo’s second problem asks for a solution of the equation x^3+2x^2+10x=20. Leonardo found an approximate solution of the equation that is correct to nine decimal places (namely, x=1.3688081075…), and expressed it in sexagesimal notation, as it was the custom at the time in precise astronomical calculations. The problem of finding a solution (or rather, an approximate solution) of a cubic polynomial equation was a problem that appeared in several Arab texts of the time. This equation in particular first appeared in Omar Khayyam’s “On proofs for problems concerning Algebra,” a text that contains the first systematic approach to solving cubic equations. The study of cubic equations would continue to be a hot topic in mathematics for a few hundred years, until the sixteenth century, when the Italian Renaissance mathematicians Cardano, Del Ferro, and Tartaglia would describe exact algebraic solutions of cubic equations.

While the third of Palermo’s problems seems to be the easiest of the three, as it only involves linear equations, it was nonetheless an interesting challenge, because there was no symbolic notation at the time and such problems were solved in a narrative form. The problem in question was very similar to other problems that Fibonacci described solutions for in his book *Liber Abaci*, so one suspects that this particular challenge was an opportunity for Leonardo to showcase the problem solving skills that had made him well-known in the scientific community. In modern notation, the problem asks for the following unknowns. Suppose T is the original sum of money. Let x, y, z be the amounts that each man takes from the pile, and let e be the equal amount of money that is given to each man at the end. Then 3e = x/2 + y/3 + z/6 or, equivalently, 18e = 3x+2y+z, and T = x+2e = 2y+3e = 5z+6e. After some clever manipulation, Fibonacci arrives at the smallest possible solution of this system of equation, which is T=47 and e=7, with x=33, y=13, and z=1.

One might say that the current *New Golden Age of Mathematics* kicked off in the year 1900, during the International Congress of Mathematicians (ICM) that was held in Paris in August of that same year. One particular lecture captivated the audience at the time, and several generations of mathematicians afterwards: David Hilbert’s lecture on “**Mathematical Problems**.” The lecture began as follows:

*Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?* (translated from the German by Dr. Mary Winston Newson).

Hilbert discussed 23 unsolved problems that he considered of “deep significance,” and which we will enumerate below. Undoubtedly, Hilbert’s list had a remarkable impact in the direction of mathematical research in the 20th century and, to this day, those problems in the list that are unresolved are still at the front and center of mathematical research. Certainly, many of these problems were already well-known and attractive before the year 1900, but when Hilbert called the attention to these particular questions, they became magnets for the scrutiny of mathematicians all around the world. Some of the problems were solved almost immediately. For instance, the third problem was solved by Max Dehn, a student of Hilbert, in the year 1900, with a negative answer (in fact, unbeknown to Dehn or Hilbert, the problem had been **solved in 1884 by Birkenmajer**!). However, many of the problems, such as the 8th problem, remain wide open and their allure still generates much research.

Here is the list of Hilbert’s 23 problems, together with a quick parenthetical remark about their status.

- The continuum hypothesis: there is no set whose cardinality is strictly between that of the integers and that of the real numbers. (Resolved in 1963.)
- Prove that the axioms of arithmetic are consistent. (Resolved in 1936.)
- Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second? (Resolved in 1884 and 1900.)
- Construct all metrics where lines are geodesics. (Partial progress depending on the interpretation of the problem.)
- Are continuous groups automatically differential groups? (Resolved in 1953, but an interpretation of this problem, the Hilbert-Smith conjecture, is still open.)
- Mathematical treatment of the axioms of physics. (Partially resolved.)
- Is a^b transcendental, for an algebraic number a =/= 0,1, and an irrational algebraic number b? (Resolved in 1934.)
- Problems on Prime Numbers. These include the Riemann hypothesis, Goldbach’s conjecture, and the twin prime conjecture. (All three are unresolved.)
- Find the most general law of the reciprocity theorem in any algebraic number field. (Partially resolved.)
- Find an algorithm to determine whether a given polynomial diophantine equation with integer coefficients has an integer solution. (Resolved in 1970.)
- Solving quadratic forms with algebraic numerical coefficients. (Resolved in 1924.)
- Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field. (Unresolved.)
- Solve 7th degree equations using algebraic (variant: continuous) functions of two parameters. (Unresolved.)
- Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? (Resolved in 1959.)
- Rigorous foundation of Schubert’s enumerative calculus. (Partially resolved.)
- Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. (Unresolved.)
- Express a non-negative rational function as quotient of sums of squares. (Resolved in 1927.)
- (a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions? (Resolved in 1928.), and

(b) What is the densest sphere packing? (Resolved in 1998.) - Are the solutions of regular problems in the calculus of variations always necessarily analytic? (Resolved in 1957.)
- Do all variational problems with certain boundary conditions have solutions? (Resolved during the course of the 20th century.)
- Proof of the existence of linear differential equations having a prescribed monodromy group. (Partially resolved.)
- Uniformization of analytic relations by means of automorphic functions. (Partially resolved.)
- Further development of the calculus of variations. (Progress.)

What were Hilbert’s criteria to select these specific problems? Certainly, there are very famous problems conspicuously missing from Hilbert’s list. For instance, Fermat’s last “theorem” (which would not be a proven theorem until much later in the 20th century) is missing. Hilbert himself addresses this issue to some extent in his essay. First, of course, not every problem could make it in one list:

*The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved numerous others come forth in its place. Permit me in the following, tentatively as it were, to mention particular definite problems, drawn from various branches of mathematics, from the discussion of which an advancement of science may be expected.*

And second, some problems are special cases of broader mathematical programs. Such is the case of Fermat’s last theorem, which is an example of a diophantine equation, and therefore it may be considered as a special case of the challenge proposed by Hilbert’s 10th problem (notice, though, that Fermat’s equation always has trivial solutions). In addition, Hilbert gives some indication of what types of problems he was looking for when composing a list of challenges:

*Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.*

After Hilbert, other mathematicians followed suit and created lists of their own, such as Edmund Landau, who proposed his own list in 1912.

During the ICM of 1912, following Hilbert’s example, Edmund Landau discussed progress in our understanding of the Riemann zeta function, and then presented a list of four open problems in mathematics. In particular, his lecture concentrated on four questions that pertain to the prime numbers, two of which were already mentioned by Hilbert under his 8th challenge.

- Goldbach’s conjecture: can every even integer greater than 2 be written as the sum of two primes? (Unresolved.)
- Twin prime conjecture: are there infinitely many primes p such that p + 2 is prime? (Unresolved.)
- Legendre’s conjecture: does there always exist at least one prime between consecutive perfect squares? (Unresolved.)
- Are there infinitely many primes p such that p - 1 is a perfect square? In other words: are there infinitely many primes of the form n^2 + 1? (Unresolved.)

Landau characterized the problems in his list as “unattackable at the present state of mathematics” (see Pintz’s “**Landau’s problems on primes**” for a great discussion). While all four problems are still unresolved more than a hundred years after Landau’s lecture, we do have several partial results towards these questions, and a deeper understanding of the conjectures. Landau’s 2nd problem was vastly generalized by Hardy and Littlewood in 1923 in what is now known as the *first* Hardy–Littlewood conjecture, which quantifies the number of twin primes (and other types of prime tuples) up to a certain bound in a concrete, yet conjectural form, akin to the statement of the prime number theorem.

In 1962, Bateman and Horn stated a much broader conjecture that generalizes both Landau’s problems 2 and 4 into a single problem, subsumes the Hardy-Littlewood conjecture, and quantifies the number of primes up to a given bound that are of a specified by a definition in terms of polynomials. For example, Landau’s 4th problem asks for primes of the polynomial form n^2+1, and the twin prime conjecture asks for numbers n such that n and n+2 are both prime.

The most surprising and substantial progress towards the twin prime conjecture was a result proved in 2013 by Yitang Zhang, who proved the existence of infinitely many primes within a fixed distance of each other. The twin prime conjecture says that there are infinitely many primes that are 2 units apart, and Zhang showed that there are infinitely many pairs of primes that are less than 70 million units apart. After Zhang’s splashy result, combined efforts by a team of mathematicians (the so-called Polymath Project), and results of Maynard, showed that there are infinitely many pairs of primes that are at most 246 units apart. While this is still a far cry from the twin prime conjecture, it is quite an impressive result!

With respect to Landau’s 3rd problem, known as Legendre’s conjecture, Ingham showed in 1937 that there is a certain lower bound N such that there is at least a prime between consecutive cubes larger than N. More recently, in 2001, Baker, Harman, and Pintz showed that, for numbers n larger than a certain lower bound, there is a prime between n^2 and approximately n^2+n^(21/20), which is a bit larger than (n+1)^2=n^2+n+1, which Legendre’s conjecture predicts.

Finally, there is also significant progress towards the 1st of Landau’s problems: the Goldbach conjecture. Building on work of Vinogradov, in 1938 results of Chudakov, Van der Corput, and Estermann showed that “almost all” even numbers are the sum of two primes. Their result says, a bit more precisely, that the density of even numbers that satisfy Goldbach is 100% or, equivalently, that the counterexamples to Goldbach are very sparse within the natural numbers. Unfortunately, since their result is a density argument, one cannot rule out the existence of isolated counterexamples to the conjecture.

There are many other partial technical results toward the Goldbach conjecture, which we will not go over here, but it is worth highlighting that in 2013, Helfgott proved the so-called weak Goldbach conjecture: every odd number larger than 5 can be written as the sum of three prime numbers (which in turn implies that every even number can be written as the sum of at most four primes).

In 1949, André Weil proposed four conjectures that, over the course of the next two decades, would wholly revolutionize the area of algebraic geometry. The conjectures describe rather technical properties of zeta functions attached to algebraic varieties over finite fields. In particular, the four conjectures say that:

- Zeta functions are rational.
- Zeta functions satisfy a functional equation and Poincaré duality.
- Zeta functions satisfy an analog of the Riemann hypothesis.
- The degrees of the factors of a zeta function are given by Betti numbers.

Though first stated by Weil, these conjectures were a long time in the making. The first known results that are directly related to the Weil conjectures date back to Gauss (1801) and his work on what we now call Gauss sums. Much later, in 1924, the conjectures had started to take form, and an early version was stated by Emil Artin in the special case of curves (which were proved by Weil himself). Finally, Weil stated the conjectures in full generality in 1949. Although the statements naturally reside in the realm of algebraic geometry, the interest in the conjectures grew immediately because of the implied connection to a different area of mathematics (algebraic topology) via Betti numbers. The conjectural connection between areas predicted the existence of a new cohomological theory that could connect them and explain the presence of Betti numbers in the factorization of zeta functions of algebraic varieties. A flurry of mathematical activity ensued, which culminated in the discovery of *étale cohomology* by Artin and Grothendieck, with the purpose of attacking the conjectures. The first conjecture (rationality) was shown by Dwork in 1960, the second and forth by Grothendieck and his collaborators in 1965, and the most difficult one, the third Weil conjecture, was shown by Deligne in 1974.

As the 20th century and a millennium closed to an end, and surely inspired by Hilbert’s highly influential list of problems, in 1998 the vice-president of the International Mathematical Union, V. I. Arnold, in 1998 wrote to a number of mathematicians with a request to collect a list of “great problems for the 21st century.” One of the recipients of the request was Steve Smale (known for his research in topology, dynamical systems and mathematical economics, and a recipient of the Fields medal in 1966), who composed a list of 18 problems for a lecture on the occasion of Arnold’s 60th birthday, and which appeared in print in his paper “**Mathematical Problems for the Next Century**.” In this paper, Smale explains his criteria in choosing problems:

- Simple statement. Also preferably mathematically precise, and best even with a yes or no answer.
- Personal acquaintance with the problem.
- A belief that the question, its solution, partial results or even attempts at its solution are likely to have great importance for mathematics and its development in the next century.

The list of problems was as follows:

- The Riemann hypothesis. (Unresolved.)
- The Poincaré conjecture. (Resolved in 2003.)
- P versus NP. (Unresolved.)
- Shub-Smale tau-conjecture on the integer zeros of a polynomial of one variable. (Unresolved.)
- Can one decide if a diophantine equation f(x,y) = 0 has an integer solution in exponential time? (Unresolved.)
- Is the number of relative equilibria (central configurations) finite, in the n-body problem of celestial mechanics, for any choice of positive real numbers m_1, … , m_n as the masses? (Partially resolved in 2012 for “almost all” systems of five bodies.)
- The Thomson problem on minimizing the distribution of N points on a 2-sphere. (Unresolved.)
- Extend the mathematical model of general equilibrium theory to include price adjustments. (Unresolved.)
- The linear programming problem. (Unresolved.)
- Pugh’s closing lemma. (Partially resolved in 2016.)
- Is one-dimensional dynamics generally hyperbolic? (This problem had two parts, the first part is unresolved, and the second part is resolved.)
- In other words, is the subset of all diffeomorphisms whose centralizers are trivial dense in Diff^r(M)? (Partially resolved in the C^1 topology in 2009.)
- Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane (Hilbert’s 16th problem). (Unresolved.)
- Do the properties of the Lorenz attractor exhibit that of a strange attractor? (Resolved in 2002.)
- Navier-Stokes existence and smoothness. (Unresolved.)
- The jacobian conjecture. (Unresolved.)
- Solving polynomial equations in polynomial time in the average case. (Resolved in 2016.)
- Limits of intelligence regarding the fundamental problems of intelligence and learning, both from the human and machine side. (Unresolved.)

In his write-up of the mathematical problems, Smale includes three additional problems as an addenda which he describes as “a few problems that don’t seem important enough to merit a place on our main list, but it would still be nice to solve them.” The problems in question are (19) a mean value problem in complex variables, (20) is the three-sphere a minimal set?, and (21) is an Anosov diffeomorphism of a compact manifold topologically the same as the Lie group model of John Franks?

Shortly after Smale’s problems were published, in 2000 the Clay Mathematics Institute of Cambridge, Massachusetts (CMI), established a list of seven problems to celebrate mathematics in the new millennium. In the **words of the institute**:

*The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude.*

Known as the Millennium Prize Problems, and with a $1 million prize allocated for the solution of each problem, the seven challenges are as follows.

- The Birch and Swinnerton-Dyer conjecture. (Unresolved.)
- The Hodge conjecture. (Unresolved.)
- Navier-Stokes existence and smoothness. (Unresolved.)
- The P versus NP problem. (Unresolved.)
- The Poincaré conjecture. (Resolved in 2003.)
- The Riemann hypothesis. (Unresolved.)
- The Yang–Mills existence and mass gap. (Unresolved.)

Each problem is accompanied by a **beautiful expository paper** by an expert in the field, namely (1) Andrew Wiles, (2) Pierre Deligne, (3) Charles Fefferman, (4) Stephen Cook, (5) John Milnor, (6) Peter Sarnak, and (7) Michael Douglas.

The only Millennium problem that has been resolved to date is the Poincaré conjecture. Building on work of Hamilton, Grigori Perelman gave a proof in 2003, and after a thorough review of the correctness of the proof, Perelman was poised to receive both the Clay Math $1 million award, and a Fields medal. However, he rejected both awards, alleging that the prize was unfair, as he considered his contributions to be no greater than Hamilton’s.

As many other young couples, when my wife and I looked for our first home, we were obsessed with the idea of an “open concept” home. After having religiously logged an uncountable number of HGTV-hours, we were made to believe that an open concept was the only way we would achieve a truly happy home, so we looked for the ideal open concept home for what must have been an eternity. We swiftly discarded any listings that did not feature the two magic words. Our realtor was ready to drive her car off of a bridge, with us inside of it, when thankfully an open concept home became available on the market, in the neighborhood of our choice, in the school district of our dreams, and we purchased it. Finally, our very own open concept home! Our own chance to the pursuit happiness, the start of our own family, all of it in a glorious open format living arrangement.

How do I wish now, with 20-20 vision, for a closed concept home or, perhaps, a middle ground: a clopen concept.

Fast-forward a few years, we have two school-aged kids under 10 years old, we are still living in the same open concept home, and a pandemic is raging outside. There is one door in our entire first floor (the door to a half bathroom) and, on the second floor, three doors that lead to three bedrooms (plus bathrooms). One of the bedrooms is a guest room that now doubles as a home office, and it is the sole refuge from the daily unstable chaos our home life has become. When we are all home (meaning, 100% of the time), there is one spot for one of us to hide away, while the other adult has to share the open concept first floor with the kids. My wife and I take turns hiding away at the paradisiacal home office, which typically means that my wife is in the home office 99 turns for each one of my turns. To be fair, my wife works in industry, so she has daily critical meetings, while my teaching and research meetings are not of such a high profile. Maybe if I had daily briefings with the Dean, then I would get a turn a bit more often.

On a typical day, by 9am, we have finished breakfast and everyone positions themselves at their battle stations. The open concept first floor features three areas carefully separated by noise-superconductor air: the kitchen/dining area, the living room, and a third space that serves as a smaller cozy living room, library, and my office. We bought a small desk for the older kid to work at a corner of the living room, and the younger kid typically works on her homework at the dining room table.

My wife nods “good luck,” and walks upstairs, with a perceptible spring in her step. The kids fire up their devices (their work is on Seesaw and similar educational platforms), and I open my laptop, while praying that everything goes smoothly for once.

Narrator: *“It will not go smoothly.*“

“Dad, have you seen my charger? My iPad is almost out of batteries,” Julia yells from the kitchen area.

We spend a few minutes trying to locate the charger and, by the time we find it (mysteriously, it walked to the basement), the iPad’s battery is dead. We plug it in, restart the machine, but Julia is now logged out of her online account. Her password is a QR code printed on a piece of paper, and Julia cannot remember the last time she has seen it. Once again, we walk in circles around the open concept first floor, looking for the little piece of paper that will permit my work-day to begin, my ticket to do math if you will. Eventually, Julia remembers the QR code is safely tucked inside the iPad protective cover. All is well, she can begin her work.

I fill up my coffee mug, and return to my desk. The laptop awakes, and the inbox displays a long list of unread email messages that patiently await for my attention…

“Dad, I need help with math,” Natalie calls from her desk in the living room.

Her math homework consists of a number of haphazardly phrased word problems, which make me fume inside, but I do not want to vent my frustration in front of my already frazzled daughter. Her questions are not about math, but about how to express the answer. I suggest the most logical way to represent the answer, but she insists that her teacher does not want the answers in such a (otherwise logical) format. I insist it would certainly be ok to write it the way I propose, and hint that maybe she might have misunderstood the teacher’s instructions. My daughter proceeds to break into tears.

“THAT’S NOT THE WAY MY TEACHER HAS TOLD US TO DO IT, DAD!”

“Please, can you be quiet,” Julia pleads from the adjacent kitchen area.

“What is going on?” my wife asks from the second floor, stepping out of her meeting for a moment.

“We have it under control, we will figure it out,” I reply in an attempt to pacify the audience on the bleachers. Natalie and I take a deep breath and go over the way she is required to answer her math riddles. We find a middle ground and she writes the solution in two ways. All is well, and she returns to her desk.

One of my first meetings of the day begins over Zoom, with two collaborators. As soon as I speak and they hear my voice, Cotton and Hyper start to loudly squeak supplicating for food. The two demanding voices belong to our Guinea pigs, whose home resides in the same living area where my desk is. (In fact, this area is now known as the Piggies’ room.) We forgot to feed them in the morning, so they are rightfully yelling at me to provide the hay and vegetables they deserve. I request a minute from my colleagues, turn the camera and mic off, and quickly throw some hay pellets and spinach into the piggy enclosure. Now that they feel heard and appreciated, they will let me continue my research conversation.

However, I forgot that I had hired a teenager in the neighborhood to clean my yard, and now there are insanely loud leaf-blower noise-waves invading and resonating and amplifying throughout our open concept home.

“WHAT IS THAT NOISE!” Julia and Natalie yell in unison.

“That’s Andrew cleaning the yard!” I reply, but it is unclear whether they heard me.

I text chat with my collaborators for the time being because turning my mic on would surely be a Zoom session killer move.

By the time my meeting ends, it is lunch time. My kids demand Mac and Cheese, and I have no strength to fight for a healthier alternative, so I resignedly boil some water. Mesmerized, staring into the bubbling water, I wonder when this nightmare will end, if ever. My wife’s meeting ends, almost magically, precisely when the kids food is already served on the table. We have a quiet lunch, we share some of the technological issues we faced in the morning, for the millionth time, and we rejoice in any of our small accomplishments during the morning session.

The kids are done with their work, so they run to their play area in the basement. My wife runs to her next meeting on the second floor, and the open concept first floor becomes an eerily quiet area. Even the Guinea pigs are quiet, taking a nap. It is a mirage, and it will not last, so I quickly go back to my laptop to finish taking care of my inbox.

As predicted, the fleeting peace comes to an abrupt halt when my kids run upstairs once again to prepare for their ballet classes. In order to have sufficient space to perform, each one needs a different room and device where to connect from for their same Zoom-ballet lesson. Natalie will dance in the living room, and Julia in the Piggies’ room, which means that I have to evacuate the desk and move to the dinning table.

My mug is full of coffee once again, and I try to concentrate to no avail. A cacophony of dance instructions and terrible iPad-speaker quality classical music blare from the other two spaces in the open concept home. Not even my best headphones can cancel the pandemonium that permeates every corner of the house. I decide to use this time to take care of menial tasks, hoping that I will have the energy and concentration to tackle research when the ballet lessons conclude.

It is 4pm and my kids have put away their devices, and have been kicked out of the house, to play outside, ride their bikes, or pick up sticks if they must “but get the heck out of here.” I have one more meeting with a graduate student, so I connect to Zoom once again. We are chatting when the connection dies and the screen freezes. After a few seconds of confusion, I realize the problem: my wife is heating up water for a tea cup in the microwave. Why would that be a problem, you ask? Well, it turns out that our microwave and our Wi-Fi signal are somehow and inexplicably intertwined and incompatible, and whenever someone turns the microwave on, the Wi-Fi signal cannot reach the corner where my desk is located. It seems that the open concept home was perfectly designed so that waves amplify and collapse throughout the house in the most frustrating ways.

“PLEASE stop the microwave! I am in a meeting!” I beg to my wife. “Please use the kettle when I am in Zoom calls!” How many puzzled students and colleagues have heard me yelling at my family complaining about the microwave? We shall never know.

By 5pm I am mentally exhausted, and angrily pace around the open concept first floor, thinking of alternative configurations that would have made this house a closed concept. I daydream of walls and soundproof doors. Perhaps we should cover the walls with egg crates to minimize noise travel. Maybe there is a way to transform our open concept space into a clopen concept with some sort of removable tarps or barriers to block the large entryways between the three areas… Hopefully, this nightmare will end soon, and there will be no need for drastic measures.

Damn you, HGTV.

I went to a real math conference today, an in-person conference, not one of the many virtual seminars and conferences that are so prevalent in these pandemic days. I miss travel so much that I just had to lay down on my bed, close my eyes for a few minutes, and imagine that I was about to go far, far away, at least one or two plane rides away, away from home for a few days, dressed in actual clothes, no sweat pants in my suitcase, no day-time pajamas for a change.

If I am going to do this, I want to do it right. The conference of my choice is, of course, on my favorite topics, with plenary talks by some of my favorite mathematicians, and hosted by a university located in a mildly exotic destination that I have never been to. Nowhere too fancy, I don’t need a paradise: just a city I’ve never visited, or a State or country that is in my list of places to go to, or some old famous Institute that I want to visit some day. I plan my trip, and book a flight and a modest room at a hotel near the university that is hosting the event. Excitedly, I pretend to text a few math friends, to see if any of them are going as well. Great news: many of them would not miss such a great conference and will be there too.

During one of my flights to my dream conference, I write the notes for my talk, and rehearse what I will say. Planning a talk is one of my favorite exercises, as it forces me to step back and consider a broad bird’s-eye view of the project. What would others think? What would they find interesting? What parts should I highlight? What examples are best suited for the exposition?

“Oh wow, are you a mathematician?” asks a fellow passenger sitting next to me in the plane. “Yes, I am a university professor and a mathematician,” I answer. Now I wait for the typical cringe-inducing response from a stranger (the mildest one being “oh, I am not a math person”) but, instead, the stranger says “oh wow I loved mathematics in school, I wish I learned more!” And we spend the rest of the flight pleasantly chatting about a variety of cool math topics that satisfy the curiosity of my impromptu flight companion. Even the third person in our same row seems to be curiously listening into our conversation. Ahhh, the power of the imagination is limitless.

When I arrive at my destination, I get on an Uber, and start texting math friends to see if they have arrived yet. After some characteristic indecisiveness, one of us forms some dinner plans, and we enjoy together some delicious foreign cuisine, while we catch up and talk about our teaching, our departments, some math gossip, our students’ shenanigans, and a bit about our latest research projects. Then we move it to a bar where we run into other math people, and we all enjoy a few drinks and laughs, and the conversation is liberally peppered with the nerdiest of math puns that we all enjoy without shame.

Back at the hotel, still feeling the buzz effect of the wine and beer, I review the notes for my lecture one more time, watch a show (whatever I want, no compromising!) and go to sleep. In the morning, I head to the conference building, and enjoy my walk through campus, admiring some of the beautiful architecture (and frowning at some of the more questionable design choices), enjoying the crisp morning air. When I arrive at the hall, the greatest display of fresh-brewed coffee and breakfast items (the good stuff) awaits the conference participants. I proudly display my name batch and begin to mingle, first approaching a group of people where someone I know is talking to others. I introduce myself to the small group of faculty and grad students, and they politely introduce themselves in return.

The organizers do a slow clap to catch our attention: the first lecture is about to begin. It is time to leave everything behind, all my obligations, my parental duties, my long work/home to-do lists, my worries, my goals for this term. Everything stays outside of the lecture hall except the hot cup of coffee that I carry with me (I pretend not to see the “NO FOOD OR DRINKS INSIDE AUDITORIUM” sign, even though it is unavoidable on the way in). I sit, I sip coffee, and enjoy the show. A math friend is sitting next to me, and we exchange brief comments or looks (surprise, puzzlement, horror, amazement, meh) during the talks, and these hints of comradery put me at ease knowing that I am surrounded by my kind of math people. Some talks are enjoyable, some talks are fascinating, some talks are awful, and one or two talks are extremely relevant to my current interests. I love them all, equally so.

At lunch, I join a mixed group of postdocs, students, and faculty, and I slowly work my way through the group, meeting each one of them, paying particular attention to the grad students. I ask them who their advisor is, “oh yes I know them, please say hi,” what are they working on, “oh that’s an interesting question!” how is the research going, have they thought about this other similar question, and are they aware of this paper, it might be helpful. Are they speaking at the conference, if so when, so I make sure I attend their talk.

The conference resumes with more talks, including mine. It is an imaginary conference and an imaginary talk with an imaginary audience, but I am a bit nervous all the same, so I am glad when it is well-received and it is over. The event continues, more math, more socializing, more networking. I am as always amazed at the ingenuity and depth of knowledge of the mathematicians around me, and I try to absorb as much as the amazing math energy they irradiate until I am now once again motivated to go back home, and do more math and prove new results.

I open my eyes. I am still on my bed, in my day-time sweat pants and hoodie, looking at the ceiling. The naive mental exercise has worked at least to some extent, and some motivation is crawling back to me. Enough to go back in front of my computer, open a current LaTeX draft, and think a bit more about a project. I might even have an idea or two of how to, maybe, make some progress.

After my one year at Colby College, I traveled from Waterville, ME, to Ithaca, NY, to work as a postdoc at Cornell University for three years. The job was all that a mathematician can dream of a postdoc but the next stages of my life and career were always on my mind, so (like many postdocs are advised to do) I applied to a few jobs during year 2 of my postdoc to maximize my chances of landing the wildly desired tenure-track job of my dreams. My records say I applied to 7 jobs. Unfortunately, I did not hear back from any of those institutions that I applied to. The search was as uneventful as it was unsuccessful. Nonetheless, just like any other search, it was nerve-wracking, but nothing came out of it, so let us never mention it again.

Flash forward to year 3 of my postdoc, and the stakes had never been higher. The goal of Job Search Round #3 (ehem… #4) was to find a tenure-track position, so that I would not have to go through any further searches (hopefully for a while). The trick, though, was that this time my search was rather limited in geographical terms. When I first started looking for jobs while at BU, my wife and I made a deal: she would move with me wherever my postdoc took me, for three years (which actually ended up being 4 years total), and then we would move back to New England, as close to the Boston area as possible. After the failed search on postdoc year 2 (remember? the one I said I would not mention again?), all the eggs were in the one basket of my search on year 3. The pressure was on to find a tenure-track job in New England. The fact that I did not hear back at all from any of the schools I applied to during my second year at Cornell did not bode well.

Let the search begin! As a first step, once again I updated all my materials (CV, research and teaching statement), and assembled my poker hand of letter writers: Berger, Gouvêa, Ramakrishna, Rohrlich, and Stevens. (My heartfelt thanks to all my letter writers over the years!)

And then the daily search for jobs began. As in previous years, I mined every tool at my disposal to find all the openings at schools in the New England area, within a 3-hour drive from Boston. My records show that I applied to 15 jobs that year… which was a scary low number, given the large number of jobs I had applied to during previous (successful) searches.

The applications started to go out sometime in October, and then the long dreadful wait started.

Luckily, I had some good news early on. By late December, I had heard from three institutions: UConn, and two small colleges in New England, which will remain anonymous for reasons that will become clear later on. All three, as a first step, wanted to set up interviews with me at the Joint Math Meetings that would take place in San Diego, January 2008, so I packed my bags, and to San Diego we went folks!

I do not like the JMM for one particular reason: the job market. There are so many interviews happening on any given day during the JMM, that they rarify the air with the fear and fried nerves of the candidates, to the point that the environment becomes toxic. I **love** going to conferences, but the JMM has a weirdly tense vibe that is generated by the job interviews. Of course, if you are on the job market, you have to be there to contribute your own bit of dreadfulness to the convention center. There is no way around it.

There is one positive aspect about the JMM that everyone loves: you get together with other mathematician friends that you have not seen in a long time.

Let me first say what the interview process was like with the two small colleges, before we go on to the tenure-track hiring process with UConn.

Small Colleges #1 and #2 set up short interviews with me at the Joint Meetings. The first interview with College #1 went well. The committee was nice and we had a nice, laid-back chat. I was nervous before the interview, but I was able to quickly relax once the interview started, and I thought I made an overall good impression, with nice answers to all of their questions about research and teaching. This interview took place somewhere indoors in the San Diego Convention Center, but not at the official AMS employment center.

The interview with College #2 did happen at the official AMS employment center. I arrived early, and was waiting for my turn to interview with College #2 when I saw the chair of the committee for College #1 walk into the employment center, and sit at a table, with some other institution (let’s call it College #3). While I was curiously spying on him, Chair of Comm. #1 chatted amicably with the committee from College #3 for a while, and then he left. He noticed me on his way out, and I smiled, but Chair of Comm. #1 turned livid and ran out of the employment center in a hurry, barely acknowledging my presence. Weird. Maybe I had not made such a good impression after all? Only much later I learned that Chair of Comm. #1 was actually interviewing for a position at College #3, where he would eventually be hired later that same job season.

At any rate, after the Chair of Comm. #1 left, I was called to the table of College #2. The interview was isomorphic to the interview with College #1, so it went reasonably well. I had a nice impression from the committee and I thought, I hoped, I made a good impression myself.

Soon after I returned from San Diego, I received the good news that both College #1 and #2 were inviting me to on-campus interviews… in the same week of February.

*Alea iacta est.*

College #1 was a small private college in Massachusetts, and the interview was on Monday of what will be forever known as *Helluvaweek 2008*. The day-long interview started bright and early at 8am, with a brief chat with the chair of the search committee, the mathematician who as we now know was in the market to go elsewhere, while trying to convince me that College #1 was a great place to work at. By 8:30am I was being grilled by math faculty. By 9am I was being interrogated by the Provost, and by 10am I was being interviewed by the Dean. Lunch is never a break for the candidate during a job interview: in this case undergrads who were part of the search committee took me out to lunch to the student union, and asked me a few relevant questions. After lunch, the chair of the committee gave me a tour of the campus, and after the tour, I gave a 50 minute **talk on the congruent number problem** (aimed at undergrads). After my talk, a few faculty members took me out to dinner, and the day was over.

The chair of the committee did land the job and moved to College #3, and the funny thing is that College #3 eventually called me to see if I was available as well, because they were hiring two people (they called me, though, after I had already verbally accepted the UConn job). It would have been hilarious if he and I ended up working at College #3.

By the way, I never heard from College #1 after my interview. I thought it went really well, but I did not get the job.

College #1 interviewed me on Monday of *Helluvaweek 2008*, and College #2 interviewed me on Thursday of the same lovely February week.

College #2 was a small liberal arts college in New Hampshire. The Chair (and chair of the search committee) was very laid-back, which is typically a nice quality, but so laid-back that the details of my day were very scarce and reaching him was hard even by phone. The day before the interview was supposed to take place, I had to call several times to get an exact location of when and where to meet in the morning. About the interview itself, essentially, all I was told is that I was to give a short (30 minutes) presentation to an audience of undergrads during their number theory class.

After receiving a few more details about my day when I arrived on campus in the morning, the day went well, with the typical packed schedule of meetings with faculty, Dean, and students. I gave my 30 minute version of my job talk aimed at undergrads (again, on the congruent number problem), and we finished the day with an early dinner, before I started to drive back to Ithaca.

I was happy with the interview, and I thought I made a good impression.

A day later, I received an email from the Chair of College #2, saying that there was a problem:

“*We liked your talk so much that we wanted to know more about the congruent number problem. In a quick google search the second site to show is *[some site about the congruent number problem that no longer exists].* Your talk followed this one very closely, yet you did not give attribution. We would not accept this from a student, so do not see how we can from a faculty member*.”

The absurdity of this message still dumbfounds me to this day. The congruent number problem has been studied for over a thousand years, and, of course, there are **pages** and **pages** written and **published** about this particular problem. There have been also very many **talks** on the subject over the years, just like there are so many talks about **Fermat’s last theorem**. And if you ask a number theorist to put together a 30 minute talk for undergrads about the congruent number problem, you will find that most of us would put together very, very similar talks (what is the problem, some examples, the connection to elliptic curves, **Zagier’s example for n=157**, Tunnel’s criterion, etc). But this was not a typical talk, and much of what was in my talk was not in the website he found, or any other reference, so the accusation was not even true. In my talk I discuss the life and work of Leonardo Pisano (aka Fibonacci), and how the court of Holy Roman Emperor Frederick II challenged Fibonacci to solve the congruent number problem for n=5. None of this was in the crappy website that the Chair found, but the website did have the basics about the congruent number problem (e.g., Zagier’s example), which I did cover.

But nevermind all that. I had 30 minutes to give a presentation to undergrads within the setting of a course in number theory, and it is highly unusual to give a list of references of all materials consulted to give a talk, much less a lecture in a class. Of course, if I had been asked to provide references, I would have compiled a complete bibliography for his students (for instance, Koblitz’s “**Introduction to Elliptic Curves and Modular Forms**” uses the congruent number problem as a motivating example).

I asked and explained the situation to multiple people, at Cornell, at BU, and all agreed that this was a truly strange, incomprehensible behavior by the Chair of College #2. As it turns out, I know the PhD advisor of the Chair of College #2, and he also agreed that some cables must have been crossed in the chair’s head for such a ridiculous response and accusation.

Needless to say, I did not get the job at College #2, nor would I have wanted to touch that job with a ten foot pole.

Let us backtrack to the Joint Meetings in San Diego, where I had my first brief interview with a UConn colleague.

During your mathematical “upbringing” you meet lots of people, and you do lots of things, and it is not clear what connections, if any, might be useful later. Nonetheless, I strongly encourage students and postdocs to cultivate all these relations and connections, because you never know what might end up being of importance later in your career.

A few serendipitous connections made me an ideal candidate for the UConn job. My research lined up quite well with the (very broad) interests of Keith Conrad, who already worked at UConn during my job interviews, and I had met Keith a few times at BU, and also we had invited him at Cornell at least once to give a talk. On the other hand, UConn was looking for a person with a strong educational background who would be hired as associate director of their Quantitative Learning Center (a large tutoring center that offers help in math classes but also in Chem, Physics, and Stats). As it turns out, I had a few very relevant and unusual qualifications for that part of the job: I had worked for the **PROMYS Program for Teachers** for 4 summers while at BU, and I ran the “Calculus Afterhours” tutoring program during my year at Colby. So, naturally, I was interviewed for the job at UConn.

My friend and colleague Tom Roby, who was the Q Center Director, set up an interview with me in the back side of the Convention Center, at a lovely outdoor area.

Tom is a really friendly, congenial person, so the interview immediately became a really interesting conversation, and we were soon having a great time, enjoying the beautiful San Diego weather. Many other mathematicians were also chatting at nearby tables, and I was glad to be outside, and as far away as possible from the dreadful AMS employment center.

Tom asked me typical interview questions, but both of us enjoyed following whatever tangents came up in conversation. I remember that I was trying to concentrate on one of his tangents, but something else was distracting me. Behind Tom, sitting at a ledge, a mathematician I didn’t know was acting strangely. She first seemed sleepy, then disoriented, and all of a sudden, she collapsed. With a loud thump, she fell face first on the concrete floor. I immediately stood up, ran past Tom, and approached the woman on the floor. “Are you ok??” No response. I tried to turn her around and saw that she was starting to convulse. Tom and several others stood around us in shock. Fumbling through my pockets, I pulled my phone out, and started dialing 911. Then I looked up to see the small crowd of mathematicians who were just there, looking at us in disbelief, and I yelled at Tom and everyone else “DON’T JUST STAND THERE! GO GET A SECURITY AGENT FROM INSIDE!!” and Tom, and someone else took off, running indoors to get some help.

The 911 dispatcher asked for some basic information and assured me that help was on the way… but I had to make sure that the woman on the floor was not biting her tongue. I helped her turn around, and I was glad to see that she did not seem to be biting her tongue, though a fair amount of foam was coming out of the mouth, which was surely not a good sign. The dispatcher asked me to put something between her teeth. The only thing I could find was her purse, so I forced the strap of her purse between the teeth. The dispatcher now asked me to stay on the line, and narrate any other developments, until help arrived. The woman on the floor started to wake up, and after a few seconds she realized what had happened. I tried to assure her she was ok, and that help was on the way. She told me that, unfortunately, it was not her first seizure. Her face was scraped and bruised, but she was starting to feel better. I was glad she fell towards the concrete floor, and not towards the other side of the ledge where there was easily a 20 feet drop.

Tom came back with some security officer, and the paramedics were there shortly after. They helped her into an ambulance, and they whisked her away to a hospital.

Tom and I sat back down, still rattled. And then I realized I literally screamed at my interviewer and ordered him to go get help. I apologized, and he laughed it off, just complimenting me on my quick thinking and composure in an emergency situation. “You passed the test,” he joked. The truth is that I am no stranger to passing out unexpectedly, so I only did what other strangers have done for me on multiple occasions.

Though my interview with Tom went well, I did not know if the commotion and screaming might have screwed up my chances. However, I did get a message from the Chair at UConn Math that I was invited to an on-campus interview on Wednesday of, you guessed it, *Helluvaweek 2008*.

PS: While I was waiting for the paramedics to arrive, I was able to have a look at the name tag of the fellow mathematician on the floor. So a few days later, I wrote an email message to her to see how she was doing. She was very grateful for my help, and I was very glad to hear she was bruised but fine. She was happy to hear that I got an on-campus interview despite all the chaos during my JMM interview.

The UConn on-campus interview was exhausting (I had to give two 1-hour talks, one for students, one colloquium for faculty) but fair. Just like after the other two interviews, I left with a good impression, and I thought the day went well… but hey, what do I know given the results of the interviews with Colleges #1 and #2.

In mid February, the Chair at UConn called me with excellent news: a tenure-track offer. I was elated, and I am still so happy that I was hired at UConn and that I get to work at such a great place.

As I mentioned in my previous post (**My Search For Jobs, Round #1**) my first job after grad school was a visiting professorship (a Faculty Fellow, they called me) at Colby College. After the sobering experience of my first round applying for jobs, I felt lucky to have landed the job, and I still feel lucky today that I landed it. Unfortunately, the gig was one of those 1-year positions that are *likely* to be renewed for a second year, but you need to reapply for jobs anyway, just in case. This means that by the time the Fall semester started at Colby, I was already preparing for the job market, and stressing out about the outcome. Moving to and settling in a new town and State, starting a new job at a different institution, trying to publish my thesis and work on new research, requesting a “green card” (permanent residency), and applying for jobs, simultaneously? Yep, it was a lot.

As I started to put together my materials, and I updated the elements of my file, I couldn’t but obsess (more like panic) over the fact that I sent out close to 70 applications for jobs in the previous season, out of which I got exactly one response. What was wrong with my applications? There must have been something that I could fix. Maybe my teaching statement came off the wrong way. Perhaps there was some egregious mistake in my research statement. Probably, my work was not interesting enough. All of these thoughts were perfect fodder for my impostor syndrome.

I requested more feedback from my advisor, from Fernando and other faculty at Colby, as well as other people I knew, and I made the changes that were suggested. However, these amounted to little modifications, nothing drastic that improved my file in a palpable way. One section of my CV did change, however: I worked my butt off to get more papers out before applications were due. When I started to apply for the first time, in the Fall of 2003, I had two papers in preparation listed in my curriculum vitae. By the Fall of 2004, **one paper was accepted**, and three others had been submitted. I crossed my fingers that this would suffice. In the 2003-2004 job season I sent out about 70 job applications. In the 2004-05 season, determined to not leaving a stone unturned, I applied to over 100 jobs (including jobs at UConn!).

*Apply for jobs, teach, do research — apply for jobs, teach, do research — lather, rinse, repeat, for a few months, and try to not stress out so much that you lose your mind.*

In early January 2005, the first piece of encouraging news came my way. A university reached out to me! For a tenure-track job!! I was really excited. The search committee for the Mathematics department at **Seattle University** contacted me for a phone interview. It went well, and soon after they invited me to an on-campus interview.

Now, Seattle was not the ideal location my wife and I dreamed of. Her family lives in Massachusetts, and my family in Spain, so Seattle was in the wrong coast for our goals. Still, the job looked interesting, and Seattle is a cool city (neither one of us had been to Seattle, though), so we decided to go ahead with the interview. *** IF *** I got the job, then we would have to decide how long to stay in Seattle before trying to apply to jobs back on the East coast.

As far as I can remember, the interview at Seattle University went well. I gave a talk to faculty and students about elliptic curves, which I had already given during the interview process at Colby. The most memorable moment (perhaps the most overwhelming part of any interview I’ve been part of) was a meeting with all the members of the SU math faculty in a conference room, where everyone grilled me with questions about my research and my teaching, for what seemed an eternity.

At the end of the interview day, I met one more time with the Chair of the department, who assured me I had done a great job during my interview, and she *seemed* to hint that I was likely to get an offer. Of course, the words were vague and hard to decipher, so during my flight back home I speculated endlessly about the meaning of her phrasing. All I could do was hope for some good news in the next couple of weeks. However, I was the first candidate they brought to campus, so I had to patiently wait for some other unknown variables to potentially ruin my chances.

*Time stands still when you are waiting for a message back from a hiring committee.*

While I impatiently waited for news from Seattle, a wild email message appeared in my inbox. Not from an SU account, but from a Cornell University account. **Ravi Ramakrishna** was wondering if I was still available because, if so, there was an offer for a 3-year postdoc at Cornell coming my way.

Impostor Syndrome (IS): “It is a mistake, he sent a message to the wrong person.”

Me: “Hmmm, no? Look, it says my name in the message. Not a lot of “Alvaro”s in number theory in the USA…”

IS: “Fine, but he* must *be mistaken.”

Me: “Well, he is an expert in Galois representations, maybe he had a look at **my thesis on Galois representations** and found it interesting??”

IS: “Even if that is the case… When you arrive to Cornell and he realizes you know

Me: “I … I do know a bit… obviously not as much as he does… isn’t that the point of a postdoc anyway?”

IS: “Yeah, but you do not belong there.”

Me: “OK, we are done here.”

If you have ever tried to debate your Impostor Syndrome, and you probably have, you know how it goes, and you can relate to the rapid fire of emotions – a combination of elation, incredulity, and insecurity – that went through my head as I was reading and re-reading Ravi’s message. I emailed Ravi back: “YES. HELL YES. I AM AVAILABLE FOR A POSTDOC AT CORNELL WORKING WITH YOU.” Or probably something similar in nature but more formal and measured.

A day or two later, I got a phone call from the Chair of Cornell’s Department of Mathematics, with an offer for a 3-year H. C. Wang Assistant Professorship. And a few days after that, I received the formal offer in the mail, and took a silly picture to immortalize what I knew was one of the most consequential moments of my career.

I had yet to hear back from Seattle University. Even though the possibility of a tenure-track position was quite enticing, I knew that I could not pass the fantastic opportunity of a postdoc at Cornell. So I contacted the Chair at SU, to let them know that I was withdrawing my application for their job, and to thank them for considering me for their position.

By early February, my job search was done. My wife and I drove to Ithaca, because we were so excited we could not wait to be there.

The outcome of the search was just about the best I could think of. I felt the privilege then, and I know my privilege now to have been given the chance to work there. My impostor syndrome reminds me that I was so lucky that year, that the planets aligned so that other candidates, far more qualified than I was, were unavailable by the time the offer came down to me. Needless to say, I am extremely grateful to Ravi and the Cornell math department for the incredible boost to my career.

Be that as it may, I took full advantage of the opportunity given to me, to learn and do as much research as I could, to improve my teaching, and to be as prepared as possible for the next round in the job market.

If you ask any math PhD about their first round of applying for academic jobs, they will almost surely answer with an emotional roller-coaster ride of a story. As I mentioned in a previous post (**How to Apply for Academic Jobs in Math**), the unfortunate *reality* of finding a job is that the job hunt is stressful, a very long, drawn-out process, with uncertain results, and very hard to go through, for you and your loved ones around you.

Perhaps, a bit peculiar to my own case is that in addition to the stress of the job search and finishing my thesis, I added a wedding on top of everything else: I got married on a Saturday, and defended my thesis two days later, on a Monday. As the reader may have guessed, I was really close to a nervous breakdown by the end of it all.

Why would anyone in their right mind combine a wedding and a thesis defense, you ask? Well, my family lives in Spain, and they very much wanted to be part of both events. However, traveling twice within a year was out of the question (travel cost, limited vacation days), so we decided that it would be best to celebrate the wedding and defend the thesis within days of each other, in order to maximize family attendance to both events.

As ill-conceived as that plan might sound, in the Fall of my last year in grad school when the plans were crafted, I was young, single, and naive, and I was very excited about the prospect of obtaining my PhD, getting married, getting my first job, and having my entire family and closest friends visit for a couple of weeks in the Spring to celebrate everything in one epic party. What could go wrong?

Anyone who has planned a wedding or applied for jobs or prepared a thesis defense can tell you all the things that would go wrong. If only those people had put together an intervention to tell me what a stupid idea was to try to coordinate these life-changing events at the same time…

All I will say here about organizing a (fairly large) wedding in Boston, in the span of less than seven months, is that it is a challenge I do not wish on my worst enemies. To everyone’s surprise, including my own, I cared way too much about every single wedding detail, so much so that I became the “groomzilla” no one talks about in wedding movies. But this blog is no place to rehash my angry fights with the florist, so let us move on.

Let us instead back track to the Fall, when I started working on the graduating and job application process.

When I first applied for jobs (early 2000’s), one would go department website by department website, and find out if they were hiring. If so, the applicant would send a manila envelope by snail mail, with their printed-out materials, including a customized cover letter. Thankfully, the Boston University math department’s staff (thanks, Angela!) consolidated my materials into envelopes, so I just had to supply a set of labels with the addresses of the departments that would receive my package.

I just checked my records, and I had applied to at least 60 jobs from October to February. I never heard from any of them except for the occasional formal rejection letter, and a very few personalized rejection messages. I was crushed.

I was perfectly aware of my own potential, and I knew well that many prestigious postdocs were a reach, at best, for someone with my credentials. But I had applied to many, many places (including UConn!), and I couldn’t even conceive that I would not hear positively from a single institution. I was devastated. My advisor encouraged me to persevere and continue looking for jobs, and keep applying. So I did. My advisor also told me that, if I did not get a job by May, then I could extend my graduate studies by a year (even if I defended my thesis, we would delay the submission of the dissertation by a year). It was comforting to know that I had such a safety net to fall back on, but it would be a bitter pill to swallow, since my family was coming over to see me defend my thesis in April.

The weeks leading to the wedding and the defense were extraordinarily hectic, and truly a series of unfortunate events. My family and friends, most of which had never been to the USA, arrived a week before the wedding. Thankfully, I prepared and practiced my thesis defense before their arrival, because once they landed, I had no time to myself. Moreover, everything that could go wrong, went wrong. One of my friends was detained for a day by immigration officers at the airport as soon as she landed because her name matched a terrorist’s name (a very common name). Another friend lost his passport and wallet. Unusual torrential rains alternated with the usual cold wet April Boston weather. Half of the Spanish guests were to stay at a house by the ocean, and the day I drove them all in a van to the house, I forgot to bring the key to the house. No big deal, except that our van got stuck in the mud (did I mention torrential rains?) and we all got drenched in dirt from head to toe until we managed to get the van out of the mud pit. The same cursed van, late on the eve of the wedding, got a flat tire. The day of the wedding, half of the wedding party (including my parents) got lost on their way back from a hair appointment and barely made it on time to the ceremony. Argghhh!

If I was a superstitious man, I would have thought that the signs warning me against this wedding were unmistakable. However, once we all miraculously made it to the church, just in time, the tide turned in our favor, and there were no more unfortunate events. The wedding was in fact wonderful and we all had an insane amount of fun during the banquet.

Two days later, I defended my thesis. A large contingent of family members and friends attended the actual thesis defense. As far as I know, my defense was the first one at BU to feature an introduction in Spanish for the sake of the audience, with the permission of my thesis committee. The presentation went according to plan, and I satisfactory answered all the questions that the committee members had for me. Afterwards, we celebrated at a nearby bar, with pool tables, drinks, friends, family, grad school buddies, committee members. Ah, the memories.

I defended successfully in early April, but I had no job prospects. None.

By late April, I had given up, and I had resigned myself to staying an extra year in grad school and start applying all over again the following Fall. My wife and I went on a mini honeymoon for a few days (the real honeymoon would be later in June) and when we came back, I had a message from someone at **Colby College** (Waterville, ME) waiting for me. Apparently, they tried to contact me the very first day I was away on our trip, and they were about to move on to the next candidate when I returned from our mini-vacation and got back to them. The job was a one-year visiting position (potentially renewable for two years), and I was invited for an on-campus job interview.

With the help of my advisor, I prepared for my job interview, where I had to give a talk to the Colby undergrads (of course, about elliptic curves). Also, during the interview, they asked me about my teaching and tutoring experience, because part of the job would be running their “Calculus Afterhours” program (Calculus help sessions). Thankfully, my experience as a mentor in the **PROMYS for Teachers** program was exactly what they wanted to hear.

Shortly after my return back home, I got a phone call from Tom Berger, the Colby math chair at the time, with the good news. They wanted to offer me the job.

Was this the right job for me? Certainly, it was not the job I had in mind when I started my search. Thus far I had only been associated to fairly large universities: Universidad Autonoma de Madrid (about 30,000 students), Imperial College London (about 17,000), and Boston University (about 32,000). Colby, on the other hand, enrolls about 1,800 students. Since I did not have any experience with small liberal arts colleges, I did not know what to expect. However, my advisor thought it was a good professional move, for two reasons. One, Colby is **one of the best** liberal arts colleges in the country, so the teaching experience at Colby would be a great addition to my resume. And two, **Fernando Gouvêa** works at Colby, so I would have a number theorist (and historian!) in the office next to mine.

I took the job, and even though this was my one and only one job offer, I am so glad it came through when it did. I had a great time at Colby, socially and professionally. It was in fact an eye-opening experience about the beauty of small liberal arts colleges. And even though I was only there for a year, I believe my time at Colby was highly influential in the way I teach, in the way I interact with students, and in the rest of my career, in no small part thanks to Fernando, and Tom.

The job market works in mysterious ways.

Here is a list of contents with links to the appropriate sections. Feel free to skip around:

**Introduction and Disclaimers.****Know your Audience**.**Research Potential.****Networking.****About the Research Statement.****Educational Experience.****About the Teaching Statement.****Colleague Potential.****The Job Market.****Mathjobs.org.****The Joint Math Meetings.****Preparation for Interviews.****Conclusion**

This post will always be “in-progress” as I intend to come back to it and add more details and commentary as I think of other items that need to be mentioned here. I would welcome any comments, feedback, and suggestions! This post, however, represents my own point of view. If you are a student, I hope you find these tips useful, but please always consult with your PhD advisor (and other folks in your department) about the best way to put together job application materials.

There are many other online resources on the subject that students may want to consult, for example (and in no particular order) see Duke’s “**Applying for Jobs,**” Frayer’s “**A Search for an Academic Job…**,” Forde’s “**The Academic Job Search**,” Sawyer’s “**A Timeline for a Job Search in Math**,” Narayan’s “**The Academic Job Search in Math**,” Virginia’s “**Job Hunt**,” Katz’ “**Getting a (Teaching-Research) job**,”, or Kaabar’s “**Preparing for Job Applications**,” Folsom and Kontorovich’s “**Advice for the Campus Interview**,” among others.

This post concentrates on academic jobs, but it is important to remember that there are also a lot of great non-academic jobs out there for mathematicians. As **someone mentioned** on Twitter, and rightly so, “we do a disservice when we perpetuate the academic-only mindset.” For more details on non-academic jobs, see for instance Lamb’s “**What Are You Going to Do With That?**” or the **Big Math Network** and its job guide.

In addition, this text mostly concentrates on applying to jobs in the US… but there are a lot of **great jobs in other countries** too! If you would like to apply to positions outside of the US, discuss this with your advisor, in order to find the best way to locate jobs that match your interests. Since many non-US jobs are not advertised in mathjobs.org, we will list **below** some additional sites where to look.

I shared **excerpts of this post** on Twitter, and many other mathematicians and students commented and provided some of their own views. I will be editing this document to incorporate some of those thoughts here as well. There were also interesting offshoot conversations such as **this one** or **this one.**

Before we begin, one **warning** about the unfortunate *reality* of finding a job. It is hard. The job hunt is stressful, a very long, drawn-out process, with uncertain results, and very hard to go through, for you and your loved ones around you (for my own personal experiences, see the **series of posts that begin with this one**). However, there is a lot you can do as an applicant to tip the scale in your favor and, as suggested above, the earlier you begin tipping the scale, the better results you will achieve in your search. So let’s get planning ahead!

If you are going to take away one piece of advice from this post, let it be this one: **start thinking about your resume as early as possible during your grad school years**. How are you planning to present yourself to potential employers? What are going to be the highlights of your job applications? The sooner you figure out what your best skills are, the sooner you can start polishing those to make them shine. But you also need to spend time improving all the other skills that hiring committees consider essential. Your file should not be supported on one single item (e.g., your amazing research) but, instead, you should be able to demonstrate that you have been trained to be a well-rounded mathematician: a colleague, an educator, and a researcher.

However, as

Before we go into specifics on how to apply for jobs, let us think for a moment about the “consumers” of PhDs. First of all, those who will look at your file are very much like the professors in your own department. So look around you, meet faculty, and talk to them about jobs! What are they looking for during a hiring season? The more opinions you get, the closer you will be to having a good picture of what the ideal candidate may be.

One thing is clear: **it is no longer enough to graduate with a “strong thesis,”** whatever ‘strong’ means. You need to aim to be the whole package. In order to be marketable at a wide range of institutions (and here I mean institutions in the USA for the most part), the ideal candidate needs to have demonstrated excellence in research, ample educational experience, and have great colleague potential (not necessarily in that order). Note that **most** (about 58% of) tenure-eligible faculty jobs in the US are at non-PhD-granting institutions where teaching is a great part – if not all – of the institution’s focus (see the **AMS 2017 report on the profession**, Figure D.2 on p. 1722; see also the **AMS 2017-18 report on academic recruitment**).

In the rest of this post, let us try to break these requirements down and give tips on how to improve your file with each of these goals in mind.

Yes, it is true, you should try to produce the “strongest thesis” possible during your PhD. But what does that even mean? “Strongest” here, in my opinion, actually means “most interesting” to a wide audience of mathematicians in your area (or, even better, including others outside of your immediate area!). In most cases, the PhD student is at the mercy of their PhD advisor and the problem that was suggested to them. But you, as a student, can also be in control by trying to figure out how your problem fits into the larger picture of your research area. How? Start by asking your advisor and other faculty in your department. Then go to conferences and attend lectures related to your problem. Read articles in the arXiv related to your thesis. And make notes of how your problem fits into the larger puzzle. Once you figure it out, then try to “spin” how you talk about your research, so it is clear to the audience that you know how your own work belongs to and enriches the larger framework. This will increase the interest in your research problem, and it will show that you have a breadth of knowledge that some PhD candidates unfortunately lack (or at least they do not know how to convey it). Even better, start talking early to other students and mathematicians that are working on problems related to yours. This might have the very desirable side-effect of finding a potential collaborator, so that you can start a project in parallel to your PhD work (warning: consult your advisor before starting other projects that may consume your time). This brings me to a new important point: networking.

Networking as a grad student (as a postdoc too!) is really, really important. Start in your department: the first people you will communicate with and meet are the staff members in your department offices. Get to know them! Be friendly, ask for and follow their advice, be thankful for their help. The staff will be critically needed at several stages during your PhD, so be mindful of that.

Once you start your program, get to know your fellow grad students, including those in areas far from yours. Of course, it is healthy to be social and friendly to your peers, but it also helps to be able to talk to people from other areas about your own research and theirs (this is good practice for job interviews where you will talk to people from all areas!). Building a community around you will help you all succeed.

Start attending your local seminars, in your math area, and related areas. This might seem pointless at first: you will get lost early in the talk. That’s ok and expected to some extent (unfortunately the quality and level of math talks is somewhat unpredictable). Some talks are more specialized than others, but you should be able to get some sense of what the speaker is trying to convey (when you do not, ask! Ask the speaker or your advisor about what was going on in that talk). The most important point here though is to meet the speaker after the talk, or even better, during tea, lunch, or dinner (hopefully subsidized by your advisor or research group!). It’s important for you to attach faces to names and, viceversa, it will be useful if other mathematicians recognize your face and/or name at a later time (e.g., when your advisor suggests your name for their local seminar). You never know who you will cross paths with in the future! **One of my first papers** was a collaboration with **a mathematician** that spoke at the number theory seminar at **Boston University** while I was a grad student, and was looking for help with a paper he was working on. So always be on the lookout for opportunities that match your interests!

Next, start going to conferences in your field. During the conference breaks and meals, mingle and meet as many grad students in other departments as possible. Some of these people will be your future colleagues, collaborators, confidants, people who you might email questions to, perhaps write papers with. If you are lucky, they might be organizers of local grad student seminars that you might be able to be invited to and speak at.

Of course, during a conference or seminar, also make an effort to meet speakers. If your advisor is around, ask them to introduce you to people. Try to ask questions about the math you see in talks – most people are happy to answer students’ questions!

When you have results towards your thesis, start by giving a talk to your fellow grad students about your area and a bit about your findings. Ask your advisor for a spot at your local seminar, so you can practice giving a research talk. Let people know you welcome feedback on your talk. There is only one way to get better at giving talks: give many talks and listen to people’s advice. Once you are ready, ask your advisor and other faculty for help getting you to speak at other institutions’ seminars, and conferences that have 15-20 minute spots for grad students (e.g., **regional AMS meetings**). Keep an eye open for such events and propose talks to the organizers.

When networking, keep in mind that it would be wonderful if by the time you graduate, you can get a letter of recommendation about your research from a faculty member who works outside of your own institution. For example, your research is probably related to someone else’s work, let’s call them Professor X. You may have communicated with Prof. X by email, asking them questions, or maybe you have sent them a copy of your thesis. Perhaps Prof. X invited you out to speak at their seminar, or at a conference they organized, and they had a chance to attend your talk. The more interactions you have had with Prof. X, the easier and more natural it will be to ask for a letter of recommendation, and it is more likely that Prof. X is expecting to be asked for a letter. So, your job (with the help of other professors in your department) is to make all these interactions happen and be meaningful, so that Prof. X accepts writing a letter and they can write a strong letter for you.

As part of your application, you will need to include a research statement that summarizes your own work and describes briefly your future projects. This document is typically 5-page essay, plus a page of references (Google “**research statement in math**” to see real-life samples). The document and level of sophistication of the math within depends on the schools you are applying to. In other words, who is the audience of your statement? *Ideally* (but this is a lot of work), a candidate would write two statements: one for strictly research postdoctoral positions, and one for visiting positions at colleges and universities that may be teaching focused, or perhaps a department that does not have a mathematician working strictly in your topic. The former kind of statement would be more technical than the former, and it would be written mostly with specialists in mind. The latter kind of statement would be much more general and gentle, so that anyone in an area relatively close to yours can read it and get a good idea of what you are working on. Either way, your statement should begin with an eloquent introduction that motivates your research and relates your work to at least one of the major current research trends, and more specifically to other important published works. The introduction should also showcase your breadth of knowledge we discussed above in the section for **Research Potential**.

At least in my opinion, your statement should be thought of as an advertisement of your work and the techniques you are fluent with, and there is no need to go deep into technicalities that only a few specialists may be interested in. Those same specialists can read your papers in your website or the **arXiv**. So instead, in your statement, you should make an effort to showcase your theorems, their consequences, and how they relate to other big results. It is also a good idea to end your essay with a summary of other work in progress or ideas for future work, so that the reader knows that you have the research potential we hope for, beyond your dissertation.

A great research statement also establishes that you are ready to be an independent researcher. In addition, it is a very positive sign if you already have an incipient network of collaborators, and this comes through in your statement. See the section above about **Networking**.

Before you get started writing your own statement, though, ask your advisor and friends for samples of previous (successful) research statements, so you can have a look and get a better idea of what to do.

Here is a sample structure of a research statement:

**Brief intro, and summary of the contents of the document.****Introduction to your area of research.****Statement of results, and corollaries. Comparisons with similar works by others.****Future directions.****References/Bibliography.**

Many of us in the profession care deeply about our teaching and our students, and we strive to constantly improve our undergraduate and graduate programs. Thus, we expect the same from those we hire, even visiting faculty and postdocs, including research postdocs. No matter how great your research is, if you are unprepared to teach your own courses (and do a good job at it!), then you are not a good fit in a department where we value our under/graduate program and we care about our under/graduate students.

You should consider your graduate program as an apprenticeship to be a professor, and as such, you should use your years in grad school to fine tune your teaching skills. Hopefully, your department runs a graduate teaching program, to help you improve your classroom technique, and you should take advantage of all such opportunities. In particular, it is important to diversify your teaching experience. By this I mean that while it’s certainly easier to be a TA for Calculus 1 every semester, it is much more important for your CV that you have experience teaching a diverse range of courses (for example, it’s desirable to have experience teaching Calculus up to Multivariable, and if possible Linear Algebra or Differential Equations). It is also important to have as much experience as possible as solo instructor (so not just a TA), so seek out such assignments.

Keep in mind that you will need at least one teaching letter of recommendation in your file, so start thinking early about who that writer might be. The wider a range of courses you’ve taught and the more you’ve stepped out of your comfort zone to teach these courses, the easier of a job your letter writer will have. *What if your teaching evaluations are not that great?* As long as you do care about your teaching, then there are lots of resources out there to help you improve your teaching technique. Your department probably has a graduate program director (or a director of graduate teaching), who can point you in the direction of useful help services. Most institutions have a “**Center for Teaching Excellence**” and there you can find workshops to attend, and staff that are dedicated to help you improve your classroom technique. You should invite either someone at this center or someone in your department (e.g., your advisor, fellow grads) to visit your classes, so that they can give you pointers on what to improve. Many departments have faculty whose job involves doing such classroom visits.

You should also think hard about the pedagogy of your research talks. Your advisor can also help, by watching you give practice talks and give you pointers. You can also ask your fellow grad students for help, and trade being a test audience for each other as you practice giving 15-20 minute talks. In your talks, always include a nice introduction, and plenty of concrete examples that clarify the concepts you are describing.

Note that *educational experience* is a broad umbrella and it includes *all* educational experience and initiatives! You can contribute to the educational mission of your institution and department in many ways, and as before, you should strive to participate in as many initiatives as possible, particularly those that match your interests. For instance, other contributions beyond the classroom are: outreach activities, online resources, summer programs, instructional conferences, learning seminars, directed reading programs, tutoring, survey-style writing, etc. All of these can be of great value: they show you care about the community at large, and they can be an interesting highlight in your applications, and a valuable topic to bring up and chat about during interviews.

As part of your application, you will need to write a teaching statement or teaching philosophy statement. You can** find samples online**, and other resources on how to write it (see for example Oxley’s “**Writing a Teaching Statement**“).

Use your teaching statement as the opportunity to summarize all the educational activities (see prior section) that you have participated in during your grad program to grow as an educator. While we do want to hear about your own philosophy, and your own approach to teaching students, many of us are more interested in a teaching portfolio, where you showcase all the different educational projects you have engaged in, and more importantly all those others that you have *initiated or helped put together. *

First, describe your teaching experience, what courses have you taught, what was your role, your approach, your methods, etc. Nowadays it’s crucial that you are tech savvy, and that you have experimented with different types of educational technologies: what devices do you use, what pieces of software, do you use online discussion boards? How do you structure assessments in your classes?

Second, discuss your results. How did your methods work out in your classes? What did not work (introspection is great) and what worked really well? You do not need to share the entirety of your teaching evaluations, but if they are good, then include a summary box with your scores. Also, it’s nice to see a few quotes about your teaching from students. It’s also a good place to include some self-reflection on what you do well and you need to improve on (together with a plan on how you plan to grow).

Then, discuss other educational components that are somewhat out of the box (see the list of contributions beyond the classroom I provided above).

Finally, and this is very important, your statement should touch on **diversity and inclusion** in mathematics (not just in a note at the end of the statement!). What are your thoughts on the subject? How do you think you could contribute in efforts to improve diversity? How have you already contributed (e.g., outreach programs)? Do you have any relevant experience in making content accessible to broader audiences? How can we improve retention of underrepresented groups in math? *How can you help?* Some institutions now require a separate diversity statement, but I would encourage you to approach the topic in your teaching statement as well. As with everything else, students should start thinking about issues of diversity and inclusion early in their degree. As **someone pointed out** on Twitter, “we can all spot the difference between a diversity statement merely full of all the right buzzwords and platitudes, and a diversity statement borne out of real efforts and experiences.”

Below you can find a summary and a sample structure for a teaching statement. I suggest using sections with clear headers in boldface so that the reader can easily navigate to a section that might catch their attention:

**Brief summary of contents of the document.****Brief intro and brief teaching philosophy statement.****Teaching experience.****Methodology.****Results, praise, and evaluations.****Other educational contributions.****Commitment to diversity and broader impacts.**

If you are invited to an interview on the phone, at the **JMM**, or on campus, then you are on paper qualified (probably over-qualified…) for the position. At this point, the rest of the interview process will try to establish that you have great colleague potential. Mathematicians are people, and we like to hire nice people. We will be working with you anywhere from a few years to the rest of our careers, so we want to make sure you are going to be a nice colleague to work with. How can you possibly prepare for this?

It is quite simple. You can practice this facet of academia just like any other skill. As a grad student, you are part of an academic department, so make sure to be *an integral part* of the department life. Attend social events, tea breaks before/after talks, colloquia, grad student events, department parties, etc. Even better, help organize these events! Also, propose similar events that you think would be welcome by members of the department (e.g., grad seminars, one day conferences, invited speakers, panel discussions). Particularly, we are interested in initiatives aimed at improving the atmosphere in the department and/or the departmental life of underrepresented groups (e.g., creating a AWM chapter, joining NAM).

Being a good colleague does not start and end in your department. There is an entire mathematical community that you can be a part of, either online, or through events such as conferences. We want you to be visible, and as engaged as possible in the community, e.g., you can help with the organization of conferences and summer programs, help writing grants, or provide online resources for other fellow grad students, for example.

The job search process starts early in the summer before your last year in grad school. The first step is to get the green light from your advisor, who should confirm you are likely to graduate in the upcoming year. Once that’s settled, the candidate should spend the summer getting materials ready.

**A professional website with: your papers and research interests, materials for the courses you are teaching, a public version of your CV, other materials and links to initiatives you are part of.****A curriculum vitae.****A research statement and a publication list.****A teaching statement and/or a teaching portfolio.****A diversity statement.****A generic cover letter that you may personalize later for each institution you apply to.****If possible, an early draft of your thesis.****Find and ask letter writers for a letter of recommendation.**

* An important piece of advice*: circulate your first drafts of application materials early among faculty, fellow grad students, friends, family, so they can give you their best advice and feedback. It takes time for people to read these papers and give you valuable feedback, and then you need time to edit and revise these documents to get them ready for prime time! So get started early summer!

In addition, in August or so (not much later), it is time to decide who will be your letter writers and ask them if they would be willing to write a letter for you. You will need a minimum of three research letters and one teaching letter, but keep in mind that some institutions may require a total of five letters. Ask your letter writers what materials they would like to see when preparing their letters. At the very least, they will want to see your research and/or teaching statement, and it would be great if they can offer feedback when they look at your materials. They will probably need at least a month to write your letter, so keep the timing in mind. Last thing you want is to ask for a letter at the last minute.

Choose your letter writers wisely. Who knows your work best? An ideal letter writer is a person who

Once the application materials are ready, it is time to start looking at **mathjobs.org** to see what jobs are available and what deadlines are coming up. The first deadlines are as early as September, but most deadlines are after mid October. However, new job ads will keep appearing until as late as April or May, so **keep looking at mathjobs until you have a job lined up!** Do not give up. Applying for jobs is a marathon, not a sprint, so be mentally ready to be looking for jobs for months. Talk to your advisor early on about what jobs are available and what jobs they think you are a good fit for. Also, keep a spreadsheet with jobs and deadlines, which you should update frequently, to avoid missing any deadlines.

By the way, while most jobs are posted on mathjobs, some institutions do not post their openings to that site. For instance, some jobs appear only in the **jobs section for the Chronicle of Higher Ed**, so you might want to look there also from time to time. In addition, if you are interested in a particular school, check out their website for job ads that may not have appeared elsewhere (or send a message to someone in their department).

Jobs that are not in the USA are rarely posted on mathjobs. Ask your advisor for help in locating positions outside of the US, as some of them are advertised in area-specific ways. **Someone on Twitter** suggested the following sites instead for non-US math academic jobs: **European Mathematical Society’s job ads**, **europeanwomeninmath.org**, **Times Higher Education**, **math-jobs.com**, **Australian Math Society**, etc.

When applying for (temporary/visiting) jobs, if at all possible, try *not* to restrict yourself either geographically or on the type of institution you apply to, so that you can cast the widest net possible. However, it makes no sense to apply for jobs that you would not accept if they offered them to you, so be mindful of what you apply for, so that your application does not become noise. Some locations may not be ideal for you, but you should think of a postdoc or visiting professorship as an internship that will help you get a tenure track position closer to the geographical area of your choice so, in the grand scheme of things, three years in an institution that is far away but advancing your career (either because of the teaching experience or the research collaborators or both) is worth the time investment.

This is almost a tautology but your best chances of landing a job are at positions that you are a very good fit for, so spend extra time working on those applications that you are particularly excited about. First, spend some time learning about their department, who works there, and what kind of research they work on. Then,** write cover letters that are specific to these institutions**, where you explain why you think you are a good fit for the position. (Some of my colleagues are indifferent about institution-specific cover letters, but some of us pay a lot of attention to the cover letter itself.) Let your advisor know what jobs you think would be best and, if they agree, see if they can write a message to someone they know in the department, with a personal recommendation in the form of a brief email message. The reality is that a personal connection to the job goes a long way.

I always encourage students to send a brief email message to people in their research area at institutions with job openings that you consider yourself a good fit for. Of course, first, learn about their research and what they do, and then send them a brief email message, letting them know who you are and why you would be excited to work at that college/university, or why you would be excited to work with them specifically. One brief email message can’t hurt, and being proactive shows initiative and drive to succeed.

During your last year of grad school, plan to attend the **Joint Math Meetings** in January. Since this is the largest annual mathematics gathering, many institutions send representatives to (briefly) interview candidates during the JMM. Thus, indicate in your cover letters that you will be at the JMM and that you’d be happy to meet with their people during the meeting. If possible, try to give a talk at one of the JMM sessions, for greater visibility. By the way, **apply for student funding to attend the JMM!**

And for the love of God, do not wear a suit at the JMM (here you can replace the word ‘suit’ by any other type of formal clothing that makes you feel uneasy). **IF** you own a suit, you wear it often, and are comfortable wearing a suit, then by all means, wear a suit to the JMM. Otherwise, if you are not used to wearing suits, don’t. The last thing you want to is to feel uncomfortable in your own clothes (everyone can tell and it makes everyone uncomfortable). You can definitely dress up for an interview, but some business-casual clothing suffices! Just wear what makes you comfortable and confident and professional. (I bring up the topic of dress code because, invariably, students ask about it before heading to the JMM.)

When you finally receive the happy news that you are invited for an interview (by phone, video-conference, or in person), it is time to prepare, prepare, prepare. First, review your own materials. What are the main points that you want to highlight during the interview? Then, study their department website, their faculty composition (who works there, and what they do, what they known for), the structure of their under/grad program, etc. Then go back to your own files: what makes you a good fit for this particular department? Why do you think they chose* you*?

Prepare to answer some typical interview questions (please prepare concise answers ahead of time and be ready to expand if they ask). What attracts you to their department? Can you summarize your research and/or teaching experience? Can you describe a positive/negative teaching experience and how you handled it? Do you have experience teaching large classes? Do you have online teaching experience? What are some research problems you plan to tackle soon? How do you plan to keep up your research productivity in their department (which may or may not have people in your area)? Have you been involved in research projects with undergraduates? Would undergrads be able to participate in your research? Have you been involved in any efforts to improve retention of underrepresented groups in mathematics? Etc.

You should also be ready to ask questions! An interview goes both ways. Some typical questions include: what is their research expectation of a person in the position they are hiring for? What is a typical teaching load in their department? What courses do postdocs/visitors/tenure-track faculty usually teach as their load? Are there funds available for research/travel/student funding? Are there seminars in your area (if not apparent in the website)? Are there service expectations (such as advising, committees, etc.)? What jobs did previous postdocs get after they left their institution? However, avoid asking questions with answers that one can easily find out by studying their website.

During on campus interviews, you might be asked to give a talk, either a research talk or a colloquium-style talk, or a presentation aimed at undergrads. Discuss a suitable topic with your advisor, and practice the talk several times in front of an audience several times before you travel to the interview. The best job talks are those that are not overly technical and engaging, but do give a clear sense of the depth and breadth of knowledge of the candidate.

In summary, be prepared. Be professional. Be enthusiastic. Be confident about your skills!

The job market can be hard to navigate, but there are many people around you who have experience in finding jobs, and who can help you in your own search. Unfortunately, there is a lot that is out of the candidate’s control (what institutions will have openings, how many jobs of each category are available, how many candidates get interviews and when), but there is a lot one can do to be better prepared and better positioned when the job season finally arrives. Unfortunately, **there is some degree of randomness/unfairness** in any job search, so a failed search can happen, and it should not be attributed to lack of preparation. Students sometimes stay an extra year in grad school after one failed search, and the next year they find a great job, so that may be an option. Even those that did everything right might be unlucky in their search. The trick is to tip the odds in your favor.

The reason to advocate for a well-rounded resume is so you are marketable at a wide range of institutions and types of positions, research, research/teaching, teaching jobs, PhD-granting and non-PhD-granting institutions, colleges and universities, etc. As **someone mentioned**, “be prepared to end up at a non-PhD granting research institution.” My first job was a visiting position at a small liberal arts college, something I did not anticipate when the search started, and I had a fantastic experience there (**Colby College**).

The end result of the search may not be what you originally hoped for. But, in my experience, the great majority of the candidates that I thought were well poised to succeed, did succeed in their job searches. So good luck and always keep asking for advice from those who have succeeded before you!

Claudia Suárez woke up and immediately knew she had solved it.

It had been one of those nights when her subconscious mind kept working on mathematics all night long, while her conscious self tried to rest in vain. She had been tossing, turning, tossing and turning, while her neurons, taking no prisoners, kept inexorably marching towards a proof of the conjecture she was obsessed with. A few times, she woke up briefly, in a daze, and fell back into a light slumber only to wake up again after what seemed just a few minutes of poor quality sleep. During the rare occasions when she actually reached a deeper part of a sleep cycle, her dreams confounded plausible reality with abstract concepts from her research in disconcerting ways that triggered spikes in her anxiety levels. In what was a recurring dream, a surgeon had to remove a cohomological theory out of her brain to save her life and, with the first incision, she would abruptly wake up still feeling the pain of sharp needles digging into her scalp. Groggy and confused after such a disturbing nightmare, she had a few sips of water trying to clear her head, and then tried to sleep again, but the tossing and turning resumed, while the thoughts raced through her head, running over all the possible ways in which she could put together the last pieces of her mathematical puzzle.

Now that she was fully awake though, she knew she got it. Claudia knew how to do it. The solution was natural, elegant, and she had a strong hunch that it would work. The idea was in fact so natural that she kicked herself for not coming up with it any earlier. There was barely any light outside, but she could not wait any longer to map out the entire argument on her notepad, and then type it up in all its glory, to make sure all the details checked out.

Richard was still fast asleep in bed besides her — it was just a few minutes before 6am — so she got up as quietly as she could manage, and tip-toed her way out of the bedroom and into the spare room that served as an office in their apartment. When she turned a light on, she was surprised to find a full cup of coffee on the desk, next to a hand-written note from Rick — “You can do it!” — decorated with a goofy-looking smiley face. The coffee was cold, just the way she liked it in a summer early morning, but clearly brewed not long ago… Did he brew some coffee for her before going to bed? She could not recall, but it was a sweet surprise nonetheless, so she made a mental note to give him a big hug as soon as he woke up.

Claudia hesitated before turning her laptop on. If she was right, if this new argument worked, the paper she had worked so hard on for so long would be finally complete. This article would be a huge deal in her field, and she knew it well, as it would settle one the longest standing conjectures in her area of research. Of course, it would also be game-changing for her career. She had done quite well for herself until now (she was a postdoc at a prestigious research institution), but this paper would catapult her name to the annals of mathematics, and not just the journal with that name.

Prior to this point, she had spent many months, years in fact, learning about the conjecture, reading books and papers on the subject, going to conferences related to the conjecture’s topic, computing examples, trying different approaches, and learning new methods. A few months ago, she had a breakthrough and found a way to prove a special case of the conjecture. Even a proof of a special case would have made quite a splash in the field, but soon she realized that, perhaps, the same method could be pushed to prove the entire conjecture. So she kept her breakthrough a secret, and months of hard work ensued, which seemed to get her closer, inch by inch, to a full proof. And during the last week or so, she was so close to a complete argument that she could taste it. The last few details, however, had eluded her completely.

Until now.

“One step at a time,” she thought.

The laptop screen came to life, flooding the dark room with a dim artificial light. She found the file with the latest version of the manuscript, and started to LaTeX the rest of the proof. The argument was clear as day in her head, but there was a lot of work to do. A couple of sections of the paper had to be completely reworked because, she now realized, a brand new point of view had to be employed. Perhaps the most important contribution in the entire paper would be the fact that the objects the conjecture dealt with could be realized in a totally different way. It was this connection between seemingly distinct mathematical objects that had been brewing in her mind for the last few weeks. Now it was a matter of verifying the delicate details of the relevant isomorphisms, then establishing a dictionary between objects, and translating the problem from one category to another. Finally, some of her previous work could be used to deal with the objects on the other side of the bridge she was building.

Claudia typed frantically, without bothering to look at a clock. After so many days staring at a blank screen without progress, it was exhilarating to move forward, and write down the theorems, corollaries, and proofs that she had hoped for. Her heart raced with each proof she concluded, as she approached the end of the paper.

At some point, Rick had walked into the room, without her noticing. Claudia looked at him with a beaming smile.

“I think I got it, Richard,” she said.

He replied with a warm smile, somewhat tainted by a hint of melancholy, and gestured *go on* with one hand towards the screen, encouraging her to keep doing whatever she was doing. Rick seemed to have aged years in the last few months, and Claudia felt a pang of guilt in her gut. Unfortunately, their relationship had suffered badly as of late, as she entrenched herself deep into research. Claudia knew her moods had been foul at times, too many times to count, and their interactions were strained to say the least. But the conclusion of this article would change everything, and she promised to herself to take some time off, go on vacation with him, and patch things up. This morning, though, she was so happy and excited about her work that she could barely contain herself, so she smiled back and nodded at Rick, and turned around to keep typing. He stayed in the room, sipping on his own cup of coffee, patiently watching her work.

Claudia lost track of time once more. Maybe an hour or two passed by, or was it four or five hours – she could not tell. One moment she was starving, and an indeterminate amount of time later she had an empty plate of crumbs next to her on the desk, and her hunger was satiated. Did Rick bring a sandwich? She was so engrossed in her work that she could not recall whether Rick had brought food or when she had devoured the items on the plate. It did not matter. She kept working, proving lemmas, reorganizing sections of the paper, computing nice examples to go along the results, checking details, drawing illustrative diagrams, including items in the bibliography, compiling and recompiling her LaTeX document.

Until she was done, until the paper was perfect. All the ideas fit together in an extremely harmonious way, and formed a beautiful theory and proof of the conjecture. Tears of joy started rolling down her cheeks. Rick was still there, crying in silence, enjoying her joy. Claudia got up and Rick embraced her, and both cried together, until she started bouncing up and down.

“I DID IT! I FUCKING DID IT!” she screamed, and they both laughed, shaking off the tears from their faces.

“I need to post it on the **arXiv**,” she spoke to herself, and sat down in front of her laptop one more time.

“Claudia, no, hold on,” Rick said, but his voice was drowned in her own thoughts.

Her mind was racing, anticipating the reaction of other colleagues, when they saw the arXiv posting of her paper. How proud her parents would be, how happy friends would be for her. She imagined herself giving talks on her proof at the most prestigious seminars and conferences, and her heart rate accelerated — how nerve-wracking that would be. She even started thinking about the most pedagogical ways she could explain the proof to others.

“Claudia, please wait,” Rick pleaded, but it was too late.

Claudia logged into her arXiv account, and her breathing stopped. The screen displayed the list of papers that she had previously posted online and, at the top of the list, there was a link to a paper with the exact same title as the new paper she was about to upload.

“Claudia…,” Rick’s voice was coming from a place very far away.

Dumbfounded, she clicked on the link and the arXiv displayed a page, with the title of the paper, and an abstract, which read exactly like the abstract of the paper she was about to upload.

*In this article, we settle a long standing conjecture…*

Thinking that it must be some sort of bizarre bug in the arXiv, she clicked on the PDF link. The paper was downloaded in an instant, and appeared on her browser. There was no doubt it was the exact same paper, perhaps with very minor differences, but it was the same paper. Her paper. She was the sole author on this one too, and the paper claimed a full proof of the conjecture.

“Claudia, please stop and look at me,” said Rick.

Choking up, she turned around.

“I… don’t understand… what…”

Rick held her hand, and continued.

“I am sorry, I was trying to tell you. You already proved it… once.”

As hard as she tried to process what was happening, nothing made sense, and Claudia started to hyperventilate. Her brain started to fire fast electric waves in every which direction. Increasingly upset and agitated, Claudia started shaking uncontrollably until her entire world went dark.

She woke up on her bed, with Rick by her side.

“What happened?” she said. Her mouth was dry and there was a hint of the metallic taste of blood in her saliva. Her body was weak and heavy.

Rick began explaining, the way he had done many times before. Over time, he had perfected his speech, and he knew when to stop for her questions, and when to pause for a few seconds to allow her to absorb the difficult news.

“The day you proved the conjecture for the first time, you had a seizure. You were rushed to the hospital, where they found a large tumor in your brain, and severe internal hemorrhage. They had to operate right away to release the pressure building up in your cortex, to save your life, and remove as much of the tumor as they could.”

“When? When was that?” Claudia asked, afraid of the answer, while gently touching her scalp and feeling a large scar with her fingertips. The area was tender, it felt like needles were poking the skin.

“That happened ten years ago,” Rick said, and waited for her to process what that meant.

After a long pause, Rick explained that as a side effect of the surgery, she had lost the capability of retaining new memories. The damage was so severe, that her brain essentially reset every night, and her mind went back to the morning of the day she had the seizure, the day she proved the conjecture.

“But… my paper… it was not finished in my laptop when I woke up… how… Wasn’t it finished already?”

“I decided long ago to deep freeze your computer, in the state it was the morning of the seizure. The computer and its contents are restored to the same saved configuration every night.”

“Wh… Why?”

“Because that was the happiest day of your life… the first half of the day, obviously. The morning, the afternoon, the hours you spent finishing the paper and completing a proof of the conjecture. I’ve never seen you any happier than that, not even close.” He paused. She felt a bit of guilt, and he knew it because they had had this conversation many times. “It is okay, Claudia, I know how important your work is to you, and I respect that.”

They looked into each other’s eyes for a few seconds and then, he continued.

“Many nights, you dream of the proof, and wake up before me, jump out of bed, ready to prove it, blissfully unaware of the surgery, the coma, and the last few years. The proof itself is what your brain was trying to process when the trauma happened, so it is still probably there but fragmented. Somehow, though, you can build the proof together again, and it makes you immensely happy. I… I can’t bear to take that away from you.”

They embraced for a long time. When the hug ended, there was a silence, while she tried to assimilate the strangeness of the whole situation. Claudia was afraid to ask her next question, but he knew what was on her mind, so he reached for his phone, opened the YouTube app.

“Watch this playlist,” and he handed off the phone to her.

The first video was a message from her parents, assuring her they were doing well, sending their love, thanking Rick for taking such good care of their daughter and, at the end of the message, they went through a quick recap of family updates that spanned the last few years.

“We update the videos every few months,” said Richard, “and yeah, your parents are `youtubers’ now,” which confused Claudia, who didn’t quite recall what a youtuber was, if she ever knew what that meant.

When the second video started, a woman introduced herself as the neurologist that was treating her. The doctor explained her medical situation, the reasons for the emergency procedure, the outcome, and what she could expect in the future. In short, the brain was resilient and there was hope that it would recover to the point that she could retain memories, but it could take many years for this to happen.

In the third video of the playlist, Sarah Beischel, one of her colleagues and a good friend, appeared on the screen. She looked older than Claudia remembered her but, then again, Claudia had not looked at herself in a mirror yet, nor she was looking forward to it. After saying how sorry she was about her “condition,” Sarah narrated what happened after the paper was published to the arXiv. As Claudia had predicted, the article caused an immediate uproar. Without knowing the reason for Claudia’s silence after the paper was uploaded, her colleagues imagined her inbox was flooded with request to speak at seminars. After the community found out she was in a coma, funds were collected using a GoFundMe to help Richard with the medical expenses, quickly reaching and far surpassing the initial goal. In the meantime, Sarah submitted the paper to the Annals of Mathematics on behalf of Claudia, and it was quickly refereed by a team of experts. The article was accepted in record time, with a few very minor corrections, which Sarah took care of. Thus, the conjecture was indeed settled, and it was now called Suárez’s Theorem. The collection of tools that Claudia had invented in the paper were now commonly referred to as Suárez’s Cohomology. Sarah and a few other colleagues organized a five-day conference in Claudia’s honor, to go over the proof of the conjecture and do a deep dive into Suárez’ cohomological theory and its consequences. In the video, Sarah displayed the poster for the conference, which included a rather impressive list of speakers, with all the biggest names in the field. Claudia shuddered at the thought of all those mathematicians knowing who she was, and discussing her work. Sarah, excitedly, continued her monologue, letting Claudia know that her article and her theory had had all sorts of follow-up papers with very interesting applications, so some of the talks in the conference were announcements of other results that one could deduce from her work.

“The best part though?” Sarah asked rhetorically and continued “the best part is that you were part of the conference.”

Claudia woke up from her coma a few days before the conference started, and on the last day of the event, they were able to connect briefly via video-conference with Claudia at the hospital. On that day, Richard explained to her what had happened, and Claudia got ready to address the audience of the conference, for a few minutes.

“It was the most emotional conference I’ve ever been to,” said Sarah, tearing up a bit. “After admiring your work and its consequences for five days, we were all bawling our eyes out when you came up on the screen. “

Sarah’s video ended, and a new video began. It was an official video from the International Mathematical Union. As soon as Claudia saw the ICM acronym on the screen (short for International Congress of Mathematicians), a chill traveled down her spine. Her eyes were wide open and turned to look at Richard, with a “*is this what I think??*” expression. Richard paused the video, went to her desk, and picked up a small blue velvet box from one of the drawers. Claudia was trembling when Rick gave her the box, and he worried she was about to have another seizure. She was overwhelmed and shaking but, thankfully, it was not a full-blown event. The box contained a medal, the Fields Medal. Rick gave her another long hug to calm her down.

“You are an amazing woman.”

The video resumed, and she watched the ceremony, holding the large coin in her hands. Claudia was on a wheelchair and Rick helped her onto the stage, where she received the medal. She looked disoriented on the screen, but her smile was radiant, knowing full well the honor that was being bestowed upon her.

The last video began playing, and three young women appeared on the screen. She did not know any of them, and she was immediately concerned that her memory was failing her. As soon as the video started, one of the women assured Claudia that she did not know them, but her work had had a huge impact on their lives. They were three Ph.D. candidates, from three top institutions (two in the USA, one in Europe), whose dissertations were intimately related to Claudia’s work. They just wanted to thank her for being such a brilliant role model for many women in mathematics.

When the video playlist ended, Claudia was overwhelmed with a smorgasbord of emotions.

“Do we go through this every day? Is every day like this?” she asked sheepishly.

“Some days are better than others,” Rick replied honestly. “The days when you prove the theorem once again are the best, because you are the happiest you can be. Some days your head hurts so much that you cannot get out of bed, though. Traveling is hard on you, because you are extremely confused when you wake up elsewhere, but we have gone places, not too far though, such as the trip to the ICM in Montreal.”

Claudia had many more questions but her head was pounding. They lay down in bed to rest, and she fell asleep in his arms.

When she regained consciousness, it was dark outside. Her mind was foggy, but she was aware that they had had an intense day. She actually remembered bits and pieces of the videos they watched, she knew about a condition, but she could not recall what exactly had happened in the morning.

Rick had prepared a simple dinner, mostly leftovers, and they ate together while Claudia asked him a few more questions about their life, such as it was.

After dinner, Claudia was exhausted. She hugged Rick for a long time, gave him a kiss, thanked him “well, for everything,” though she was unsure what the scope was, and went to bed. A headache kept her awake, and she tossed and turned, until she fell into a light slumber. She woke up, sweating, and had a sip of water, and then tried to sleep again. This time, she slowly faded into a deeper form of unconsciousness, while math concepts started to creep into her thoughts.

In the spare room, her laptop turned to life on its own, and started running a process to revert any changes to files that happened during the day. At the same time, in Claudia’s dreams, a surgeon was trying to save her life by removing a cohomological theory from her brain. With the first incision, she woke up, now sweating profusely, still feeling the pain of sharp needles digging into her scalp. Groggy and confused after such a disturbing nightmare, she had a few sips of water trying to clear her head, and then tried to sleep again, but the tossing and turning resumed, while the thoughts raced through her head, running over all the possible ways in which she could put together the last pieces of her mathematical puzzle, to prove the conjecture.

Claudia Suárez woke up and immediately knew she had solved it.

Here is a summary of the sections below, for easy navigation to skip to the sections you might be most interested in:

**TL;DR: My Personal Summary**: this is a blog after all, so here is my personal take on CTNT 2020.**A Bit of CTNT History**: some remarks on how CTNT first came to be.**Here Comes Covid-19 to Ruin Everything**: how the pandemic ruined our plans for an in-person event.**What IF? What If CTNT Went Online?**The organizers weigh the pros and cons of putting CTNT together online.**We Stand on the Shoulders of the Community**: acknowledgements of those who put together online events before us and shared their experiences and suggestions with the community.**Online Organizing Begins: About Participants and Speakers**: the first steps, contacting participants and speakers.**Technology for Lectures**: our decisions on how to setup the lecture room.**Additional Technology for the Summer School**: the setup during the summer school portion of CTNT.**Additional Technology for the Conference**: the setup for the conference beyond the lecture room.**Schedule**: some remarks about the summer school and conference schedule**So, How Did It Go?? Part 1: Deliverables**. A list of objective products created by CTNT collaborators.**How Did It Go, Part 2: The Summer School**. A discussion on what worked and what didn’t during the summer school.**How Did It Go, Part 3: The Conference**. What worked and what didn’t during the conference.

Let me start by thanking my co-organizers and all the CTNT lecturers for the huge amount of time and effort that they have generously donated to make CTNT 2020 happen. THANK YOU.

When we started planning CTNT online, my worst fear was the following thought: “*What if CTNT online is such a success, that we never ever again get funding for an in-person event?*” Luckily, my impression is that CTNT 2020 has been a very successful *online* event, but some of the crucial social aspects of the face-to-face summer school and conference are, in my opinion, irreplaceable, and no matter how hard we try, we will not be able to reproduce the connections and camaraderie generated by spending a week together during a summer school and conference.

The mathematical content produced by the summer school and conference is excellent. However, we the organizers failed to manufacture effective social opportunities, as much as we tried. Rather, attendants chose not to engage in the opportunities we did provide, which I assume is due to the fact that these opportunities were not appealing enough to engage in such social engineering.

On a personal level, it suffices to say that I have lost 3 pounds in two weeks. It has been extremely stressful and hectic. I was so busy with technical aspects of the summer school and conference that I was barely able to listen to any of the talks (thankfully, we have videos!). Needless to say, all of this effort would not have been possible if my kids were any younger and my wife was not available to take on all childcare for the week (all the tasks that we usually share). So, THANK YOU, Marisa.

I do believe that all our efforts have paid off, I am very happy we were able to make it happen, and would do it again if need be. Let us all HOPE that there is no need to make it happen online again, but if it must we have this experience to fall back on.

However, some benefits of the online format are undeniable. Our content has reached many more students and faculty than ever before (we had people joining us for the conference from North America, South America, Hawaii, Asia, Africa, Europe, Australia, New Zealand…). Thus, we will consider adjusting future in-person CTNT conferences to have a larger online presence as well.

CTNT, the **Connecticut Summer School and Conference in Number Theory (more info at ctnt.math.uconn.edu)**, is a summer school in number theory for advanced undergraduate and beginning graduate students, followed by a research conference. The program’s goal is twofold: (a) expose undergraduate and graduate students to important ideas in number theory, and (b) help students join a network of student peers and faculty interested in the same area of mathematics. The research conference following the school is open to participants of all levels, including senior and junior faculty, and interested students. In particular, summer school students’ participation in the research conference allows them hear about recent developments in the areas of arithmetic geometry and number theory.

The CTNT program started to take shape in 2014, when **Keith Conrad** and I organized a one-day conference called “**Elliptic Curves @ UConn** **2014,**” which was meant as a test to see what kind of interest there was for an instructional event in number theory in the Northeast. About 60 students joined us for the day, which made it clear to us that there was enough interest to try to put together a larger, more ambitious event, similar in spirit to the very successful **Arizona Winter School**, but aimed at an audience of students beginning in number theory (the** AWS is aimed at advanced graduate students**, instead).

Encouraged by the success of the 2014 event, we (Keith Conrad, **Amanda Folsom**, **Liang Xiao**, and I) applied for NSF+NSA funding to put together an week long event, which became the **very first edition of CTNT in 2016**. And then two years later, Jennifer Balakrishnan, Keith Conrad, Liang Xiao, and I, put together **CTNT 2018** with NSA+NSF+Number Theory Foundation funding, together with UConn funding. It is a huge amount of work to put these events together (in person), so our plan was to put together one of these events every other year, in even years but…

In the Fall of 2019, **Jennifer Balakrishnan**, Keith Conrad, **Christelle Vincent**, and I joined forces to apply for funding to put together CTNT at UConn for June 2020. We received funding from the NSA, the NSF, and the Journal of Number Theory, and we started the preparations for CTNT 2020: fix dates, invite summer school lecturers, guest speakers, and conference plenary speakers, reserve dorm space, create and mail a poster, announce the event, set up and collect registration entries for the summer school and conference, etc.

By early March, however, Covid-19 was a very serious concern (see the **Pandemic Logbook** entry of this blog), and on March 27th we decided to cancel the in-person CTNT 2020 program that we had already worked hard to put together. (A month later UConn announced that all overnight summer programs on campus in 2020 would be canceled, so if we had not already decided to cancel our in-person event then the decision would have been made for us in late April.)

Once it was clear we would have to cancel the CTNT in-person event, the organizers started discussing whether an online event was possible. Our conversations were fueled by seminars and conferences, such as **AGONIZE**, and the meteoric popularity of mathseminars.org, now **researchseminars.org**, which exemplified the **need and desire for online math content** during social isolation.

It was clear that students would appreciate the opportunity of an online program, but could we actually pull it off? Many hurdles became immediately apparent:

- We had to “reinvent” CTNT in late March, for an event to be held in early June.
- A majority of organizers have young children at home, some
*very*young, and no childcare. For those who have kids of school age, we are**basically homeschooling our kids**. Are we going to have time to organize a CTNT that may be potentially larger than a face-to-face format? - We might have to find a new line-up of summer school instructors and conference speakers from scratch, as we cannot assume that everyone who already agreed to speak will be able to contribute a mini-course or conference lecture online (e.g., many of the speakers are probably in a similar situation at home without child care).
- The technological hurdles seem daunting, particularly for a summer school.
- How many students can we accept into an online summer school and still be able to interact with all of them? How big can the conference be?
- What happens to our funding for 2020?
- Are people going to be able to concentrate on math and/or organizing knowing that thousands of people are dying as a consequence of the pandemic?

However, there are some strong arguments in favor of moving forward with CTNT:

- Many REU programs and undergraduate programs have been cancelled (e.g., PCMI 2020, though other programs would later come back online, like ours). Undergrads are under the impression that a summer research experience (or similar summer math programs, such as CTNT) is crucial to their success applying to grad school, so those who were hoping that CTNT would be something they could mention in their grad school applications are worried about this opportunity disappearing.
- One goal of CTNT is to facilitate (undergrad and grad) students to network with their peers that are also interested in number theory. In a time of social isolation, it seemed even more important to put together a program that allowed students to meet others.
- Another goal of CTNT, in particular a goal of the conference, is to promote research by graduate students and postdocs, by featuring their talks next to lectures by junior and senior faculty. Since we all anticipate that the job market will be particularly tough, it seemed rather important to promote young mathematicians, perhaps more than ever before.
- Overall, it seemed imperative to add mathematical resources and opportunities under these circumstances. We did not want to “give up” on 2020.
- A relatively minor point: we love our event, and we love interacting with the students, and then with our colleagues that join us for the conference, to tell us about their latest research.
- Math content might actually give people an opportunity to stop thinking for a while about the pandemic.

After weighing the pros and cons, we decided to move forward with an online program, during the same dates as the original face-to-face event.

Thankfully, we were not going totally blind into organizing an entirely online mathematical event. Others had already done it before us, and then written extremely useful documentation about it. In particular, Daniel Litt has written** about AGONIZE** and **about WAGON**. In addition, Andrew Sutherland and Bianca Viray put together a very useful **panel discussion on online conference organizing**. We are indebted to all of them for the very useful ideas we got from from the panelists and the write-up about their event afterwards. Note, however, that we did not have a point of reference for an online summer school.

The first step in organizing CTNT was to gauge interest in an online summer school. We had 80 applicants that we were going to admit, so we sent a survey to see how many of them would be interested in an online program instead. Only 2 people declined our offer, and 78 students were admitted into the online CTNT. So, clearly, there was an overwhelming positive response.

Next, we contacted the original line-up of summer school instructors: one instructor informed us with regret that it would be impossible to put together a course during isolation and without childcare for a young kid (completely understandable!). The other 5 instructors tentatively agreed to teaching mini-courses, but they had to do a lot of research about delivery methods.

Then, we contacted the plenary speakers for the conference. Almost everyone was either interested to hear more about what we had in mind, or they tentatively accepted the invitation, until they had a better chance of experimenting with online teaching/lecturing technology. A few speakers declined our offer, understandably, due to the overwhelming circumstances, lack of childcare, due to their areas being particularly hard hit by Covid-19 deaths, or all of the above.

We created a new registration survey for the online conference, and we requested contributed talks.We received many more (great!) talk requests than we could possibly accommodate in a three-day conference. We selected 11 contributed talks and all others who submitted talk proposals were given the opportunity to submit a pre-recorded 20 minute talk that that would be featured on the conference website. Four people took us up on the opportunity to submit videos, and those are **listed at the conference website** and are part of the** YouTube playlist of conference videos**.

Once we had converged on a new line up of mini-courses and speakers, we altered the poster to create a new poster for the online, one-of-a-kind (hopefully) CTNT 2020 Online!

We asked all participants for the conference to register using a brief survey, so that we could gather some data about who would be coming to the event, so that we could communicate with all participants quickly by email if needed, but also as a measure to protect our event from trolls.

One of the most difficult challenges in organizing CTNT 2020 online, and one that we debated the most, was the technological framework for the summer school and conference. The first and most crucial decision was picking a platform for the lectures. We decided on **WebEx** for several reasons:

- UConn’s IT department recommended using WebEx. When I asked why UConn didn’t have a Zoom license, their response was: “
*The cost is more than Webex and the features aren’t that much better and when it comes to security, Webex is much better than Zoom. I think people are comfortable with it because they did a good job marketing to everyone during this time and have gotten into their personal lives. Therefore they think it will work well in their professional life as well. But that is not always the case*.” - Indeed, when we looked into this, we found that there are people that have serious concerns about the security of Zoom (see, for example,
**this Forbes article**, and this Guardian article where security experts opine that “**Zoom is malware**“). - We had a Zoom license available to us through BU, but that meant our colleagues at BU would be burdened with the tech infrastructure, and I preferred to take on that part of the work myself. Alternatively, we could have purchased a Zoom webinar license for UConn, but that would mean taking funds away from CTNT 2021 that would otherwise be used for student support. And anyway, the security issues of Zoom were enough to look for alternatives.
- We do not like the way Zoom handles breakout rooms (either randomized or a huge hassle for the host), so we were going to need a different technology for breakout rooms anyway.
- 3 of the 5 mini-course instructors had used WebEx this past spring for teaching at UConn, making them quite familiar with many features of WebEx and confident they could help the other mini-course instructors and conference lecturers get comfortable with it.

Once we had settled on WebEx, we had the option of running regular online meetings where every participant except the host and presenter are on equal footing, or run a Webinar-style meeting, where the host has tighter control over who can participate at any given time. Since we did not know how large the meeting would be, we decided early on to do a Webinar-style where the host would be able to manage audio and video to maximize bandwidth and lecture quality. Thus, the setup for lectures was as follows:

- The conference organizers were “panelists” as well as the lecturer. The panelists always have access to use the microphone and to share a screen (e.g., lecture slides), and can send chat messages to any attendees.
- The rest of the audience were “attendees,” who had access to listen to and watch the video and shared content by the presenter and panelists, chat privately with panelists, and ask questions for the presenter in the Q&A box.
- If an attendee “raised their hand” (virtually), this signaled the host to give mic permissions to the attendee to ask a question out loud (when at an appropriate time during the talk to interrupt the lecture). The majority of questions (as many or more than in an in-person conference, I would say) came in text form in the Q&A to be read out by the host.
- All lectures were recorded. The Q&A sessions after each talk were not recorded, so that people could ask freely without the anxiety of being on the record with a “dumb question” (no such thing as a dumb question!). We often encouraged people to ask questions, during talks and after talks.
- All chatter and remarks were encouraged, but not on WebEx, where it would be distracting to those paying attention to the talks.

Of course, we wanted the Summer School part of the event to be as interactive as possible. We wanted to maximize interactions with the instructors, and interactions among the students. How does one do that? We eventually decided on a combination of platforms to achieve our goals:

**WebEx**: already described above, this is where we held all our summer school mini-courses.**Piazza**, the educational-oriented Q&A platform. The CTNT page on Piazza served several roles: first, we stopped sending emails to all 70+ students, and all announcements before the CTNT summer school were posted on Piazza. We also posted all sorts of resources (such as links to meetings, or rough lecture notes) that were not ready to be posted on the public site. Once the summer school started, this platform was where students could ask questions to the instructors, and the instructors (and others) could post and collaborate on answers.**Blackboard Collaborate Ultra**(BCU) rooms: BCU is a real-time video conferencing tool, which is browser based, so users do not have to create an account or install any additional apps. The great advantage of this software is that we could create rooms that the students could join by themselves, and share content with each other without the need of a host. We created a number of these rooms, for example one for each mini-course, so that the students could discuss the lectures with the instructors, and also meet in the rooms on their own to work together on exercises.**Zulip**streams. At the first two CTNT summer schools, students organically organized themselves and created text-based channels of communication (on WhatsApp or GroupMe), so we figured that we should get ahead of this, and create one unified text-based method of communication. We decided on Zulip (recommended during**this panel**) because it is open-source, handles LaTeX well, etc. Zulip was used for more informal chatting and texting, and also for announcements and to ask questions to speakers and organizers.

In addition to a lecture room, we wanted to provide spaces where participants could interact socially, and also spaces where students, postdocs, and faculty could interact mathematically. We decided on two additional platforms for the conference. (Conference participants did not have access to Piazza.)

**Blackboard Collaborate Ultra**(BCU) rooms: we created three rooms that were open 8AM-11PM, and anyone could join at any time. The themes were: (1) Meet the speaker, discuss talks, (2) General/Coffee Break, (3) Ask questions, talk math. After each talk, the host asked the speaker where and when the participants could find them in one of these rooms, and in general the speaker joined the BCU room for talks during the break that followed their talk. The organizers also encouraged participants to gather in the General BCU room before the conference started, or during lunch breaks.**Zulip**streams (text-based communication). We invited all registered conference participants to join the CTNT Zulip streams. We made all the previous streams that we used during the summer school private (for privacy and also to avoid clutter), and created new streams to discuss talks, and chat in general.

The CTNT Summer School took place Monday-Thursday, and the schedule was as follows.

We had students connecting from the US Pacific time zone to the Eastern time zone, but we also had a lecturer connecting from China (a 12 hour time difference with US-Eastern). We decided in the end to start lectures at 10am (7am Pacific), which meant that some of the students would miss the morning lectures, and hopefully they could catch the videos later in the day. (Yes, the morning lectures were available in video by the early evening.) Instructors were available either during lunch or after dinner, in their BCU rooms and/or Zulip, to answer students’ questions.

For the conference, we settled on the following schedule.

Before our conference, we debated whether we should move the conference to be within M-F, or keep it to the weekend, and in the end we decided to do a full day on Friday (usually we do a half-day on Friday during an in-person CTNT) and two full days during the weekend, hoping that some people could find time on a weekday or a weekend to participate in the conference.

We think it went very well, overall! Let’s talk objectively first, and then we can talk about subjective experiences.

First, we describe the outcomes of everyone’s efforts, that is, the CTNT by-products:

- Five mini-courses in number theory, with a total of 4+4+5+5+7 = 25 lectures, which are
**available in YouTube playlists**. - Resources for the mini-courses (references, course notes, slides, exercises), which are
**available on our CTNT website**. - Twenty five conference videos (21 live lectures and 4 pre-recorded talks), which are available in a
**YouTube playlist**. - Materials that accompany the talks (abstracts, slides), which can be found on the
**conference website**.

The quality of the conference talks was excellent. Some of the slides were works of art:

In terms of numbers, there were:

- 78 students (undergrad to grad) registered for the summer school. At any given time, there were about 50-60 students attending a lecture.
- 4 organizers, 2 of which were summer school lecturers.
- 3 additional summer school lecturers.
- 25 conference speakers (11 plenary, 10 contributed talks, 4 recorded talks)
- 253 additional people registered for the online conference

for a total of 363 people. See **here** and **here** for the group photos of the summer school and conference, respectively.

As mentioned above, the goals of the CTNT Summer School component were (1) to introduce students to topics in number theory of current interest and/or basic tools, (2) create a network of peers, and (3) introduce students to postdocs and faculty, and their research. How did we do? First, some anecdotal evidence:

Next, a quick poll in between lectures, during the last day of the Summer School (Thursday). The question read “What’s your overall experience in the Summer School so far?”:

and some more feedback from a post-survey:

I think the content we offered was great, and the surveys indicate the students agree. However, we were able to recreate the social interactions and networking among students that we foster during an in-person summer school. To be honest, I do not think one can come close to recreating such opportunities in an online event. When the event is at UConn, the students stay in a dorm, they share double rooms, have meals together, play sports together, go to restaurants, etc.

That said, we failed when we created too many different online platforms where the students could interact. In our thought process, we believed that different students would prefer to communicate with us and others in different ways, but in offering more opportunities, we also diluted the audience. Here are some comments by students:

One thing that seemed to work well, and that we wish we had done more of (in retrospect) was a “number theory scavenger hunt” which we had during an introductory meeting that took place one week before the summer school started. We created random teams of students, and each team had to find the answers to three number theory/algebra questions, together. The students had to connect as teams in Zulip, then work together to find answers, and finally submit their responses to the organizers. This simple game created more opportunities to meet some of their peers than many other more structured class-related activities during the summer school.

Let me begin with a couple of unsolicited email messages from mathematicians that I do not know on a personal level:

and some graphs:

and then some written comments:

Most critical comments are about WebEx (see **my comments here** about WebEx vs Zoom) and about our poor attempts to encourage socialization. Admittedly, the Blackboard rooms we created, and where we encouraged people to gather, did not work well. There were only a few people at a time (between 5 and 10), at best, but they were empty most of the time. A few of the organizers disliked the idea of forcing/sending attendees into random rooms, so we preferred to create rooms where people could voluntarily gather… but it seems most people will not naturally go into such rooms, unless virtually prompted to do so.

One aspect that seemed to work remarkably well was the Zulip stream for each talk, and the general stream for announcements and other comments. There are pages and pages of comments on CTNT 2020 talks in Zulip, including many encouraging comments for speakers, new ideas to explore, references to follow-up on, etc.

Alright, I think this is quite enough for the moment! Please feel free to contact me if you have any other comments or questions. Thanks!