Featured

How to do Research in Pure Math

Disclaimer: In this entry I describe my own experience doing research in pure math. Perhaps, others have different experiences, and I would love to hear about those, but I believe this narrative may be representative of a majority of researchers.

The Best Case Scenario

Let us begin describing the best-case scenario, that is, a day when we have a few consecutive hours to dedicate to our research. Of course, this is not the typical scenario, and most days we have to fight to squeeze a few minutes of research here or there, particularly during the Fall or Spring semester.

The first (possibly, the zero-th) and most crucial step is to get a nice, hot cup of coffee ready. The quality of the coffee is directly proportional to our research productivity, so this is not a step to be taken lightly. Whether we are in the office, at home, or at our favorite coffee shop, there should be a steaming cup of coffee next to our laptop before we begin. Ideally, the hot liquid fuel is inside a ceramic cup, but this is not crucial, as paper cups have evolved to the point that they do not absolutely ruin the flavor of the coffee, so coffee in a paper cup is better than no coffee at all.

We take a sip of our coffee, slightly burn our tongue and palate in the process, and open our laptop or wake our desktop computer up. It is time to do research.

Step 1: Prepare a nice hot cup of coffee.

Actually, not quite yet. If you are anything like me, we cannot resist having a look at our email inbox, in a vain attempt to clear it before research begins. This is our first mistake of the day, and one that can be quite costly and time consuming. For some of us, however, it is a mistake that is somewhat unavoidable if we hope to concentrate on our research later on. There are probably a few emails from undergrad students that we can answer quickly, which is, typically, a pleasant experience. There might be one or two messages that require some longer explanations in order to appease some of the more demanding or annoying students, which is typically, an irksome experience that may already turn our mood towards the darker side, but nothing that another sip of steaming coffee cannot fix. Unfortunately, we have also received a couple of messages from colleagues in our department, or directly from the Head, that cannot be postponed and a timely response is of the essence. Probably, we are in a committee that needs our input or guidance. Most likely, a colleague has proposed some bizarre idea at 3:57am that we need to put a stop to absolutely right now before it gains momentum, and before another equally out-of-touch colleague supports the outlandish idea and the argument snowballs into a nightmare.

Let us assume that it is one of the wonderful days when all those clerical, teaching, and committee messages did not consume our patience or our hours alloted to research, and we can actually put all those thoughts aside for the time being to think about research. It is likely that we have a few messages pinned to our inbox, or starred in Gmail, or itemized in a to-do list, with some crucial tasks that are nagging us for our timely attention. There is at least one article referee report that we are supposed to write, that is overdue, and an editor has asked for an update. There are a couple of research collaborators that are checking in. There is a grant proposal that is due in a few weeks and we really, truly, need to get started. There are a couple of email messages that we had not touched before, from our graduate students (our doctoral advisees), and they need our help to move forward with their dissertation research, with their career, and with their life, so we cave and we spend a few minutes answering their messages with new ideas, hints, references, and encouraging words to let them know that they are on the right track.

Once the graduate students are taken care of, we must muster all our strength and put all those other very important items aside (grants, reports, checking-ins), for now, and try to think about our own research project. We will get back to all those crucial items later, but the very first hours of research, when we still have our full energy and complete focus, need to be spent on pure research. Here we go, it is research time, now for real.

Step 3: Procrastinate.

Actually, not quite yet. At this point in the process, and often against our free will, we might inadvertently fall into some sort of distraction, and the first bout of procrastination begins. There are a million ways to get easily distracted, all around us. If we are still on an internet browser, there may be a tab open right next to our email tab, with Facebook, or Twitter, or WhatsApp, open and calling for our attention, or our phone might have some new Instragram notifications that are waiting to be inspected. Perhaps we are just itching to read some depressing news, and this will inevitably become a time sinkhole as soon as we encounter the article that catches our eye today, or we follow the headline that infuriates us more than the others.

We need to be strong and set limits to our procrastination, take another sip of the now regrettably lukewarm coffee, and get back to our business of research. Some classical or instrumental music may help at this juncture, to be able to regain our concentration. Nothing too exciting — a piece that we have already listened multiple times is best so that our cerebral cortex does not have to assimilate brand new stimuli while we attempt to concentrate (one of my go-to choices is Glenn Gould’s Goldberg variations, for example).

In order to put our head in the right mood, we open the one email message we have not yet touched: the math daily Subj-class mailing message that arXiv.org automatically generates every weekday morning. As a breath of fresh air, the message contains a list of just-posted papers (fresh out of the oven – see Mathematical Cuisine) in our areas of interest, as well as others that have been recently revised. One or two papers will catch our eye, and we quickly glance at the abstracts and the introduction, in order to get an idea of what theorems they prove and what techniques they might be employing to get such results. If the preprints are truly relevant, we make a mental note to check the papers in more detail at a later time that, let’s be honest, may never come.

Step 4: Check the arXiv.

And now, finally, we have completed the initial ritual, and we are ready for our very own research. We open our LaTeX editor of choice, and we encounter a number of .tex files that are open in separate tabs. Each file is a piece of an in-progress project that we are working on as a solo author or with collaborators. So the first decision is to pick a project to look at. In my own experience, I tend to fixate on one project at a time (unless time constraints force me to bounce among projects), so before we opened the editor, we knew what file we are going to concentrate on and we try to ignore the rest. Scrolling through our TeX code, we quickly review the status of the project. But before we describe what happens next, let us backtrack a few years.

Step 5: Open your LaTeX editor.

How to Find Yourself Working on a Research Project

The time as a doctoral student in mathematics and, to some extent, the time as a postdoc, serves several purposes. One, we earn a diploma and a formal recognition that allows us to be employable at institutions of higher education (and certain bragging rights). Two, we dive deeper and deeper into one specialized area of mathematics, until we are able to solve a particular problem (the so-called thesis problem), and write an expository solution that it is publishable (the so-called thesis or dissertation). Three, we learn and practice our basic teaching methods, at the expense of the undergrad students that serve as guinea pigs (see Teaching Debut). We also get to improve our oral exposition by giving talks to fellow graduate students and faculty members who, hopefully, will give us the constructive feedback that is necessary to hone our presentation skills. Four, we create a network of colleagues and mathematical acquaintances around us, and we advertise our brand new existence to the community. And five, we develop some intuition about where to find other research-level problems that we can work on by ourselves or with collaborators. All these skills, one through five, are paramount for a mathematician, but number four and five are, in fact, crucial for our survival in the research part of the profession. Either we have amazing colleagues that invite us to participate in their ongoing projects, or they propose ideas that we can work on as a team, or we have developed our own fine sense of mathematical smell, and we can track down the scent of a problem that is within our reach. Unfortunately, many PhD students do not succeed in steps four and/or five, and eventually abandon mathematical research.

There are two categories of research projects in our to-do list: there are those where “the math” has been completed, the proofs are scribbled in pads of paper, or safely stored in our neurons, and they just need to be written up. And then there are those projects where “the math” is incomplete, and some of the proofs are still missing both on paper and in our heads. The former ones are much easier to tackle, and can be immensely satisfying as we fit all the pieces of the puzzle into one coherent narrative. The latter type of projects are an uphill battle, as we fight the uncertainty of whether we may ever reach the hill top.

For the sake of argument, let us assume we are fighting the uphill battle of an open ended project. We have a statement of a theorem in mind, that we are trying to prove (or salvage), and we are trying to work out a strategy that may get us there. Or, perhaps, we are in the very early stages of our research and we are trying to figure out what is the theorem that is lurking in the background of some phenomenology we have unearthed. The battle of the wits begins with some examples that, we hope, are a proof-of-concept of what we are trying to study. Such examples should be general or generic enough that they may reveal all the intricacies of the question we are trying to study. Finding such illustrative examples is an art, and one of the most important steps in our job. With some luck, our doctoral training served us well and we are able to produce these examples with some ease.

After much hard work with examples and some educated guesses, we have arrived at a candidate for a statement/theorem that we are sufficiently confident may be true or, rather, true modulo some minor adjustments that will become clear during the proof. Further, we hope we can prove it with the techniques that are in our repertoire and without having to assume too many conjectures or restrictive hypotheses. It is time then to start putting together the building blocks that will form the proof. Now, the key is to tease apart preliminary lemmas that we can prove as stepping stones towards the bigger results.

We are back in front of our laptop, lukewarm coffee by our side, our TeX editor is fired up and ready, the file for the project we need to work on is displayed on the screen, we have typed up a statement of a lemma that we need to prove next, and… we are stumped. Utterly and miserably stumped. We were stumped a few days ago, and we are stumped today. We sit back, grab the coffee cup, take another sip, and think. This step can take minutes, hours, days, or months, so patience is of the essence. We have considered the usual suspects of approaches and nothing has worked so far. Perhaps we made a mistake somewhere, and the statement we believe true is actually false, so we go back to the drawing board (literally, a blackboard, if at all possible) and work out a few more examples. With the aid of a computer and a computer mathematical system (Sage/CoCalc, Magma, or similar), we can compute dozens or hundreds of examples to put our mind at ease: everything seems to corroborate our intuition, the statement seems fine, so it is just us — we are not ready to prove this lemma yet. Thus, we contemplate it for a bit longer.

Computing examples with Magma.

Unfortunately, at this point life beckons and it catches us at a moment of weakness when we are most susceptible to getting distracted. Our wife is texting us, or our kids’ teachers have emailed us, or a friend sent a hilarious group text and we are committed to find the perfect GIF to send back as a reply,… Any or all of the above distractions lead us away from our noble goals of the day, and the lemma’s proof will remain elusive one more day. Our research time is up and we have to move to the next thing in our lives: teaching, a committee meeting, picking kids up from school, preparing dinner, etc. In other professions, I imagine that a worker can put their job on park for the day, and rest assured that they will continue with their tasks at 9am the next morning. But the mathematician brings home the mental burden of the proof that is eluding us at the moment. We are driving back from work, and the problem is in our head. We are cooking pasta and we continue to ruminate possibilities while absentmindedly stirring the pot. The lights in our bedroom turn off, we lay in bed, and the evasive proof continues to haunt us. By morning, we might have a few good ideas, or most likely, just ideas that are worth pursuing. However, today’s schedule will not allow any time for research, so the best we can do is to quickly scribble some notes and impatiently wait for the next day when there is a chunk of time that we can dedicate to research. Sadly, the brain cannot put the problem aside, we continue to think about it, and we are often unable to fully concentrate on other tasks or people around us. To our families: we are sorry.

After a couple of days, we are able to dedicate a handful of hours to our research, and we finally have time to try all the ideas that the relentless neurons have accumulated in the meantime. Regrettably, none work. We are demoralized, and we procrastinate some more. In an attempt to be productive in other ways, we finish a referee report. When we go back to our research question, we are still fresh out of ideas, and the bad mood sets in. Weeks may go by before we make some progress. The good news is that there is help out there. We have mentors, colleagues, and friends that may be able to help. We send an SOS and a few of them respond with useful ideas, and this buys us some renewed energy and we continue to attack our problem. If our immediate network is of no help, then it is time to use the big guns: Math Overflow. Posting a question on Math Overflow surely is intimidating, but it is worth every penny, as long as we have thought about our problem long and hard before posting, which we have. Math Overflow is not the miracle cure for the research blues, but sometimes it helps a bit, and sometimes it helps a lot. Either with the help of our network or our extended overflow network, we are able to move on! Only to advance a couple of steps until we find the next roadblock, which is invariably more ominous and intimidating than the previous one. Oh well.

Try Math Overflow?

Sometimes, nothing works, and after months of work, there is little juice left in our research engines. This is particularly true at the end of the Spring semester, when we are exhausted from all our teaching, committees, department meetings, hiring, refereeing, and our other various academic duties. There is only one foolproof way to reload our research energy tanks: a good conference. It is time to set aside our project to travel somewhere exotic, like Pittsburgh, PA, or Oaxaca, Mexico, and get together with all our friends and colleagues for a few days of math, coffee, laughs, food, and commiserating (as I write this, we are in Covid-19 social isolation mode, though, so we hope in-person conferences do take place in Summer 2020). In a good conference, the math energy is electric, contagious, the talks are invigorating, and we travel back home ready to fight the good fight and prove our next lemma towards our next great theorem. We might be just as stuck as we were before, but our energy and enthusiasm are renewed. We are back in the game, baby!

Group photo for CTNT 2018, UConn, Storrs, CT.

Grand Finale

After months of progress and setbacks, and further progress, we reach the glorious moment when our first complete draft is ready. We send it to our closest allies for comment and wait. After a few days, some of them reply with feedback, additional references to check, things to verify, etc. We fix these issues, and (with a fair amount of trepidation) it is time to share with the extended network: we post our pre-print in the arXiv. Two or three days later, a paper of our own creation appears in the math daily Subj-class mailing message that our colleagues world-wide will receive, so that they too can inspect our paper. More feedback arrives from across the globe, and after all the necessary changes and updates, our paper is ready to leave the nest, and it is sent to a journal’s editor, who in turn will send it to a referee… The refereeing process is a beast of its own that will be the subject of another post, so we will not dwell on that here. It suffices to say that after a few iterations of rejections, reports, and revisions, our proud and joy, the paper we have working on for months, most likely years, is accepted for publication in a peer-reviewed journal, and we celebrate the birth of our latest research baby, only for a few minutes, because we are very busy with another two or three projects that are begging for more of our attention.

Examination Day

What if we pushed proctoring methods to their most draconian extremes?

Peter was patiently sitting in the waiting area of the examination room. The antechamber was rectangular, uncomfortably small, with white walls and floors, bright LED recessed lights in the unusually low ceiling, without any windows, two chairs, and just two identical metallic doors that faced each other: the one he used to come in, and the door that presumably led to the examination room itself. Either door was flanked by a security agent in what could only be described as heavy riot gear. The so-called educational enforcement agents took turns staring at Peter and at the only other student who was in the waiting area. Peter noticed the equipment that hang ominously from the belt of the security guards: a small pistol, a rubberized baton, a black bottle which he supposed to be pepper spray, and a device that he surmised to be a taser. One of the guards noticed that Peter was looking at his assortment of weapons, and ran a hand over some of them, in a threatening gesture indicating that he was ready to use them if need be.

The awkward silence in the room was only perturbed by the monotonous humming of the ventilation system, and the fast breathing of his fellow student. Peter looked at him for a moment, and as soon as the student looked up, the security guard yelled “NO EYE CONTACT OF ANY SORT!” so Peter diverted his eyes to look at the floor and keep waiting.

A few minutes later, an LED light on top of the door to the examination room changed color, turning from red to green, and a mechanical sound announced that the gate was now unlocked. An unfriendly computerized voice called “Mr. Lukov” and Peter stood up. The security agent opened the door, pointed at Peter, said “your turn”, and then pointed at the entrance. For just a split second, Peter saw the frightened face of the student he was leaving behind, and walked through to the next chamber.

The next room, as it turned out, was not the examination room. Or, at least, Peter surely hoped this was not the examination room, because it was tiny, almost as small as an old fashioned phone booth (which Peter had only seen in old movies). As soon as he entered, the agent closed the door behind him, and Peter stood in the minimalist room with a quizzical expression on his face. The door in front of him was locked. So he waited. The unfriendly computer voice announced “scanning for electronic devices” and the ceiling LED lights in the room turned blue. Mechanical sounds indicated that some device was working its way up and down inside the walls, so Peter waited some more. The instructions he had received prior to the examination were clear, “NO WEAPONS OR ELECTRONIC DEVICES ALLOWED,” so he was not carrying his phone or laptop. In fact, he was not carrying anything with him into the exam room. Nevertheless, all of a sudden an alarm started blaring, the room lights turned red, and the voice declared “ELECTRONIC DEVICE DETECTED” in its most unfriendly tone yet. Peter, in a panic, patted his pants looking for his phone, but he did not have it. The door behind him opened once again, and the agent yanked him out of the tiny scanning room, and yelled in his face “WHAT IS THAT” pointing at his wrist. Peter went livid as he realized that he had completely forgotten about his smart watch. He took it off, and the agent disposed of it in what looked like a garbage chute. Peter managed to lock eyes for a moment with the other student, who now looked pale as a ghost, ready to start crying.

Unceremoniously, the agent pushed Peter back into the scanning room, and closed and locked the door behind him. Once again the lights turned blue, the voice announced “scanning for electronic devices,” but this time there was no alarm. “Scanning complete,” it said, and the door in front of him unlocked. A new educational enforcement agent rudely ushered him into the next chamber. In this small room, there was a table and another person dressed in a lab coat. On the table, there was a machine which resembled the ones at the optometrist office. “Please sit down and place your chin here,” the lab coat person said. Peter obeyed, and the machine moved until two lenses were almost touching his eyes. The computerized voice commanded “proceed with retina identification” and the machine began scanning his eyes with a pair of red light beams.

“Identification of subject is complete. Mr. Peter J. Lukov. Proceed with baseline cardiac signal collection.”

The lab coat technician moved the retina scanner away, and wrapped a wireless band around each of Peter’s arms. A computer screen immediately started displaying a few graphs that seemed to follow Peter’s heartbeat and blood pressure. The technician typed some information into a database, and after a while, said “state your name and last name, for the record.”

“Peter Lukov,” Peter said.

“Middle name.”

“Jay,” said Peter.

“I said middle name, not middle initial,” retorted the lab coat, annoyed.

“It is Jay, J, A, Y,” clarified Peter without humor in his voice. The lab coat looked at him for a second to consider if Peter was mocking him.

“Your mother’s maiden name,” lab coat continued.

“Frey,” Peter replied.

“Your first pet’s name,” lab coat said in a monotone voice, looking at his keyboard.

“Falkor,” and an image of his beloved fluffy puppy flashed through Peter’s mind.

“Have you ever cheated in an examination,” lab coat said in an accusatory tone, looking straight into Peter’s eyes.

“Hmmm, ah, hmm, hm, … no,” Peter hesitated.

The graphs on the computer screen went wild. The lights in the room turned red. The educational enforcement agent put a hand on Peter’s shoulder and squeezed it, painfully, but the lab coat raised a hand signaling to stop.

“Mr. Lukov, let me give you one more chance. Have you ever cheated in an examination.”

“Yes, it was long ago though…” Peter began, but the lab coat raised his hand again asking him to stop. The lights turned blinding white once again. The agent let go of Peter’s shoulder. “We know all about it, Mr. Lukov, it is in your educational records,” the lab coat clarified.

After a few more questions, the technician asked Peter to stand, and the agent motioned him towards the next door. Peter tried to remove the bands around his arms, but the lab coat warned him “leave them on until the end of the examination.”

After leaving the lab coat behind in his small room, Peter found himself in a vast hangar, so large that he thought one could comfortably fit several 747 planes if need be. Many doors had opened at the same time, and many students, just like him, entered the examination area, escorted by educational enforcement agents. There were desks as far as the eye could see, and students were quietly taking their exams. The agent moved his head indicating that Peter should follow him, and brought him to a desk. The agent waited until Peter was sitting down, and then marched back towards the wall with the long row of doors.

The desk had a computer terminal on it, which became alive as soon as Peter sat down. A splash screen full of cheerful colors read “Welcome to your exam, student.” And then a message said “look directly to the camera for five seconds.” Peter found the small camera embedded at the top of the computer screen and counted to five in his head. An awful photograph of Peter in the hangar appeared on the screen (he looked haggard after a long night of studying for this test), with a message “Welcome, Peter J. Lukov!” The screen changed once again, now to a black screen with “Read instructions carefully” written in red font.

  • You have exactly 1 hour for this exam.
  • The computer will turn off exactly after 1 hour, and all unsaved work will be lost.

Peter noticed that a clock had already started its countdown at the bottom right corner of the screen. 59:59, 59:58, 59:57,… so he hurried to read the rest of the ten commandments:

  • You shall not disturb any other student taking an exam in the Jane and Edward Gantry Examination Center.
  • NEVER take your hands off of the keyboard during the exam. If you do, the computer will shut down and all progress will be lost.
  • Do not blink more than 5 times per minute. Excessive blinking will result in loss of identification credentials, and you will have to be subject to an additional retina scan.
  • Do not look anywhere except your computer screen during the exam. If the camera detects that your eyes are not on the screen or keyboard for more than 3 seconds at a time, the computer will shut down and all progress will be lost.
  • Speaking (or any other sound whatsoever) is NOT allowed in the examination room. ANY sound of 10 dB (decibels) or higher will result in a computer shut down.
  • STAY CALM: an excessive heart rate and/or blood pressure beyond your baseline will be interpreted as an attempt to cheat in this exam, and will result in an immediate computer shut down.
  • No questions are allowed during the exam.
  • Press any key to proceed to your exam.

Peter pressed the space bar, the screen displayed a cheerful “Good luck!” and then it displayed the first question of his Calculus 1 exam. On the bottom right corner of the screen the counter kept decreasing (58:32, 58:31, 58:30,…). On the top right corner Peter could see himself in a copy of the video feed. On the top left corner, a message blinked “Exam in Progress. Video Recording in Progress.” Finally, on the bottom left corner, a graph displayed his heart rate and blood pressure, which reminded him of the tight bands still wrapped around his arms.

Peter took a deep but silent breath, and began working on his exam. The first question was like nothing he had ever seen before in any of the homework exercises, so he fought the urge to physically scratch his head in puzzlement. With his hands still on the keyboard, he noticed the heart rate was climbing, so he took another silent deep breath, and started doing some calculations. After a few minutes (54:12, 54:11, 54:10,…), he had an answer and pressed the “SUBMIT” button on the keyboard (which was where “Enter” was supposed to be).

A new problem appeared on the screen, and he had just started working on it, when a loud sneeze came from a student a few rows in front of him. The student sneezed, and immediately cried “OH NO, NO!” as his computer shut down. In no time, an educational enforcement agent appeared out of nowhere ready to escort the student out of the examination room. The student yelled “fuck this shit!” and the unfriendly computerized voice boomed through the hangar “MR. TRAVIS JORGENSSEN, YOUR ACADEMIC MISCONDUCT OFFENSE (FOUL LANGUAGE) HAS BEEN REPORTED TO THE UNIVERSITY EDUCATIONAL AUTHORITIES.”

Peter fought back the irresistible urge to look at the student who was being escorted out of the room. And then he realized that his own nose was now itchy, and he panicked at the thought that a sneeze may follow. His blood pressure graph spiked, and a red warning sign that read “Stay Calm!” started blinking on his screen. He blinked twice, forced himself not to blink a third time, and concentrated once again on his exam. A new sneeze was heard far inside the hangar, and Peter could hear another student being forced to leave the room. He sighed and a mic symbol appeared on his screen, which seemed to measure the decibels of his sigh. At 7dB, the sigh was allowed, and Peter carried on.

He completed the second problem and pressed SUBMIT. A note appeared on the screen: “Just a friendly reminder that your answers are being automatically checked against online sources and other student solutions. Cheating and/or plagiarism will not be tolerated.”

“Thanks for the encouragement,” Peter thought, and stared at the third problem that had just appeared on the screen. Only now he realized that a banner on the lower part of the screen had an animated advertisement for a well-known soda company, which he found extremely annoying and distracting but, as the administrators said, “those ads pay for your affordable tuition,” even though tuition had never been affordable, and Peter would graduate with a daunting student loan.

The third problem had a glaring typo. One of the coefficients that appeared in an equation was “342y9” which made absolutely no sense. Peter looked around the screen for a “HELP” button, but none was to be found. He considered raising his hand, like in the good old days, but that would mean his hand leaving the keyboard, and the computer would shut down. “And didn’t the instructions say no questions allowed? But this was a typo!” Once again he fought back the urge to scratch his head, which was actually itchy, and tried to think what to do.

47:25, 47:24, 47:23,…

He had to make a decision now, so he figured that whoever typed the question was trying to reach a number near the letter “y” so the digit must be 6 or 7. The number 6 seemed to be closer to the letter “y” and Peter solved the problem with the coefficient “34269” in place of the typo. Peter would later find out that the digit was supposed to be a 0.

He pressed SUBMIT and the computer announced “There are 15 remaining questions in this exam. at this pace, you will only complete 12 of them.” Peter, enraged, picked up the pace, and his heart rate went right up to the allowable limit.

A few minutes later, he was absorbed in thought, when he heard a loud thud. Out of the corner of his eye, he could see that the student next to him had passed out. An educational enforcement agent took his sweet time to approach the unconscious student, and with a delicate kick with his boot, the agent seemed to determine that the medical condition did not require emergency care. The agent patiently waited standing next to Peter for two other agents who also arrived at a leisurely pace. Among the three of them, they awkwardly lifted the student by a leg, a leg, and an arm, and carried her away towards a door marked with a red cross.

Peter felt powerless and anxious, and his eyes watered a little. A warning came up on the screen: “Retinas are undetectable!” so Peter lowered his head for a couple of seconds to wipe his face with the sleeve of his hoody, without his hands ever leaving the keyboard.

Peter completed question after question, including those on topics that their inept professor had not covered, but they were on the syllabus, so they were responsible to learn on their own.

“Five minutes remaining, please wrap it up,” the computer warned, “and do not forget to save your results.”

By his own count, Peter had an additional six questions to complete, so he hurried up. Some students were already leaving, though according to their body expressions, they did not seem happy with their performance in the test. Seeing students leave, however, made Peter very anxious, and both his heart rate and blood pressure started to spike. Peter saved his progress, and continued furiously working, until the counter read 3, 2, (saved progress one last time), 1.

The screen went black for a moment, and then simply said “Enjoy your day!”

The computerized voice boomed “LEAVE THROUGH THE NEAREST EXIT” throughout the hangar and Peter, exhausted, walked outside, happy to get some natural sunlight on his face. He was about to walk away, when a familiar painful grip squeezed his shoulder. An educational enforcement agent was standing behind him. “You forgot to remove your arm bands,” and pulled them from Peter’s arms. “Have a nice day,” he said to Peter, and the agent returned to the hangar.

How to Referee a (Math) Paper

Refereeing papers is one of those essential skills of a professional research mathematician that you are never taught how to do but, one day, you will find a referee request in your inbox. Suddenly, you are thrown into the world of refereeing without any proper training. In this blog post, I will give my own take on how to referee (math) papers.

You can run but you can’t hide. Eventually, an editor will find you and send you a referee request.

Now what?

If you do not have any previous experience on what to do next, this is a blog post for you. Also, read this piece by Arend Bayer on “Writing, and reading, referee reports,” and this piece on “What makes a good PRIMUS review.” Finally, this paper by Bjorn Poonen on “Practical suggestions for mathematical writing” can be a great source of feedback for authors.

Why do we referee papers?

Refereeing papers is a service that mathematicians provide to the community. Math papers can be long and complicated, and the refereeing process gives you the opportunity to have other research mathematicians read your paper carefully for correctness and for suggestions, before it is published. It is a hard job, it can take many, many hours, and it is unpaid. But we publish papers, and others referee our papers so, in turn, we return the favor by refereeing papers by others.

This blog post is not about the math publication system, which deserves an entire different entry. Here I will limit myself to the task of refereeing a paper, and we will leave the editorial commentary on journals, predatory journals, “publish or perish,” the tenure system and the need to publish, etc., for another time.

When do mathematicians start refereeing?

Before you accept a referee job, there are several important factors to consider. But before we go into these factors, let us first address the question “when should you start taking on referee jobs?” Or, more generally, “who should be a referee?”

The most important qualification in order to be a referee is that you need to be an “expert” in the topic of the paper, which in this case means that (i) you have enough background to follow and digest the arguments and techniques used in the paper under review, and (ii) you are familiar with the literature on the subject, enough to know how this result fits in the published record. If you are invited to referee, then the editor believes you are sufficiently qualified to write a review of the paper, so now it is up to you to decide if you are a good fit for the job.

In light of all this, typically, mathematicians start refereeing after (a) they graduate with a PhD, and (b) they have published at least one paper. And the first paper you are asked to referee is probably related to your thesis, or to the topics of your first papers.

Note that some grad students are asked to referee sometimes… I do not think this is, in general, a good idea or fair to the student or the author. Perhaps a better idea would be for the PhD advisor and grad student to collaborate on a review, which would provide a training opportunity in refereeing papers, but I am also not sure this is a great idea either, since someone else’s paper and career is on the line.

Should I accept the referee job?

When you receive a referee request, you will be able to see a copy of the paper so you can decide if you can accept the refereeing job at this time. Go ahead, have a look, and then consider the following factors:

  • You are under no obligation whatsoever to referee papers. As I mentioned above, this is an unpaid job, so you can always politely decline an invitation to referee. However, if you are a research mathematician that publishes papers, then you should consider reviewing papers as part of your service that keeps the community going.
  • Is this a journal you know about and we should be refereeing for? Please be aware that there are many publications out there that are “fake” or dishonest, so only accept referee jobs from reputable journals.
  • Is there a conflict of interests that disqualifies you for this job? If you cannot be an impartial referee then you should not accept the job. Simply let the editor know, and bow out. Here is a list of common conflicts: the paper is by your advisor, one of your students, a close collaborator, a family member, a close friend or a person you have a personal conflict with, the paper’s results are very much like a paper you are writing yourself, etc. If you think there might be a conflict of interests, there is probably a conflict. You can always consult with the editor of the journal, and let them decide. Note that some fields are really small, so there are a lot of connections that may be unavoidable. In summary: if for whatever reason, you think you will not be an impartial referee, then please reject the job.
  • Do you have time for this job? As I mentioned above, refereeing is a hard job and to do it well, it takes time (probably many hours). Ask the editor when the referee job is needed by, and if you cannot possibly have it ready by their deadline or soon after, then let them know you are too busy to take on this job at this time. Please keep in mind that some mathematicians’ careers, particularly graduate students and postdocs, are in the balance here, and timely refereeing can make a huge difference in their next job search.
  • Are you refereeing too much? If you accept too many jobs, then it might jeopardize your time for your own research. As a general rule, I referee about twice or three times as many papers as I submit to journals. Why? Many journals require two different referees, so I figure that two people kindly took the time to referee my paper, so I need to give back to the community two refereeing jobs for every paper I publish myself. If I have already accepted a refereeing job(s), and I am too busy with it, I will simply let the editors know that I am not available at this time to write a good and timely report.
  • Are you a good fit for the job? Have a look at the paper and try to get a sense of the topic, and the techniques used in the proofs. If you are unfamiliar with them, then you may not be the “expert” they are looking for, and it may take you an enormous time to familiarize yourself with the techniques and the literature on the subject, so you should be honest with the editor and simply say that this is far from your area of expertise, and you are not a good fit. Refereeing is not the time to learn a new area, when someone else’s career is on the line. Of course, the paper is most likely brand new research, so you will learn a lot reading and reviewing the paper! But it shouldn’t be too far afield.

What kind of a referee job am I being asked to do?

Typically, editors will ask for one of two types of referee jobs: a quick opinion, or a full referee report. In a quick opinion, you are only asked to evaluate if the paper is a good fit for the journal, and the results are interesting enough for the refereeing process to continue ahead. Typically, this opinion is not even shared with the authors, so it is an internal editorial process, and the editors just want a quick note back from you (one paragraph or two) about the paper with your first impressions (see below for more comments about how to evaluate the fit of a paper).

If the editor is asking you for a long-form referee report, then read on.

Should I reject a paper right away?

Assuming you have answered yes to all the questions above, then it is time to get started: accept the job, download the paper, and start lightly browsing its contents.

The first decision you need to make is if the paper should be rejected right away because, in your opinion, the paper is not a good fit for the journal. This is a hard call to make, so you can ask the editor for more information on what kind of papers they are looking to publish. Another good idea is to go through the journal’s archives, and look for other papers in the same area that they have recently published. Is the paper under review, in principle, at about the same level or above than recent papers that have appeared in the same journal? If so, then go ahead with the job. If the paper is clearly not a fit, if the result is known, if the combination of results and techniques are not strong enough for the journal, if the paper needs a huge amount of work,… then reply to the editors with a rejection. The sooner the better, and if you can, please offer a quick opinion, and suggestions of better journal fits.

Please do not (ever!) be mean when you reject a paper, or if you write a quick opinion. Harsh words are completely unnecessary. Just be professional, and imagine you are the one at the receiving end of the rejection letter. Be honest and direct, but always try to offer some constructive criticism.

If you are in the middle of refereeing and you find a big problem with a proof, then stop right away, and consider for a while if that’s a mistake that cannot be salvaged. If so, you may need to reject the paper on those grounds. Or at least ask the authors for clarification.

By the way, never contact the authors directly. All communication should go through the editor and the online editorial system. The anonymous nature of refereeing ensures that referees can be impartial and honest.

How much time should I spend refereeing this thing??

Refereeing can take many hours, and if the paper is long, it can be months of work. Make sure the editors have given you a deadline that is reasonable so that you don’t have to put everything else aside to review the paper. Let the editors know what is a manageable deadline to have a report ready.

That said, once you start refereeing, if the job is taking longer that you imagined, then there might be other factors to consider. If the ref job is taking too long because something came up, you might want to let the editors know so they can reassign the job if needed. If the ref job is taking way too long because the paper is just not well written, or the arguments are confusing, or you are spending too much time fixing small steps of their proofs… then consider rejecting the paper on those grounds.

Note that a rejection is not necessarily a death sentence for the paper. Most journals offer sending back the paper to the authors for “light revisions” or “major revisions.” If you don’t want to quite reject the paper, but you think that it needs a great amount of work before it is ready for you to review it again, you can send it back with an initial set of general comments indicating what the authors would need to do for you to consider it. For example, you can ask the authors to restructure the paper, to add more detail in proofs, to add more results in a certain direction that seems to be conspicuously missing from the paper, etc.

What am I actually looking for while refereeing?

You are now in the thick of it, reading the paper, it looks like a good fit, and the paper seems worth looking at in detail. What now? What is an editor and an author actually looking for?

The amount of detail and time you put into a report is a personal choice. The bare minimum amount of work a referee needs to do is to check that all the arguments are mathematically correct. In other words, make sure the proofs are correct, and the theorems are stated correctly. However, most of us go an extra mile, and give feedback to improve the paper in several ways.

  • Should I worry about grammar and sentence structure? This is optional, because it can be a very time consuming job to go into this level of detail. I do care about this, and I can’t let it go, so I will go into all sorts of grammar comments, but that’s just me. The key is that I wan the paper to be readable, and easily understandable by others, so if bad sentence structure is getting in the way of the math, then I will definitely comment on it and suggest alternative sentences that would be male a clearer, easier to digest argument.
  • Should I check every piece of math line by line? This is tricky. You need to check that the arguments are mathematically sound, so you need to go into enough detail to ascertain as much. If you are not checking certain arguments in the paper (e.g., because they are standard, or not main point of the paper) then let the editor know, or simply write it in the referee report.
  • Should I provide suggestions? Yes!! Absolutely. The reason you are doing this job is because you are an expert in the field. You are the target audience! So any suggestions you may have, are very much welcome, and that’s the kind of referee report that enriches the refereeing experience and improves papers. You can offer references, alternative proofs, short cuts, examples, or any other kind of suggestion that you think would improve the quality of the paper (particularly if it improves its readability). However, you cannot expect that the authors will overhaul the paper with your suggestions… after all, it is their paper and you are not a coauthor.
  • Should I be tough? No. Do not, in any way, write comments that can be construed as offensive. You should be an impartial, professional, honest and direct referee. So if things are missing, or if there are glaring mistakes, simply point them out in a plain way, and let them deal with the mistakes. If your comment is going to read like “the authors should know that…” then remove that comment and think of a way to point the problem out in a neutral way.
  • I am in a pissy mood. Should I referee at this time? No. It will not go well. You will be annoyed by every single little thing, and you might end up rejecting the paper for some minor thing. Step away, relax, watch a movie, go for a walk, sleep on it, and when you are back in a constructive mood, go back to the paper and keep going.
  • Should I be really nice? You do not have to go out of your way to be complimentary to the authors, but (negative) referee reports can be hard pills to swallow, particularly for early stage mathematicians. So I try to sound encouraging about the good parts, and offer constructive criticism and ideas whenever possible. The key is to strike a balance so that your report is useful.
  • I found a mistake. Should I reject the paper? Not yet. How big of a mistake is it? Is it a simple error that can be fixed? Offer a solution (though you are not obligated to do so). Is it a complicated issue that you cannot fix yourself in a reasonable amount of time? Write it down in the report, and let them deal with it (this may be a minor or major revision, depending on the size of the gap in the proof). Is it a catastrophic error? Then, yes, contact the editor, let them know there is a serious issue with the paper, and reject it.
  • Should I evaluate the overall quality of the paper? Yes. This is a very hard thing to do, but yes, absolutely, the editor will want to know your overall impression after you have looked at the entire paper. First, I gain a first impression of the paper, enough to decide whether the paper is a good fit for the paper and I am going ahead with the process. And then I wait until I have read the paper in detail to decide on an overall impression of the paper.
  • How do I actually referee? That’s your personal choice, but I print a hardcopy of the paper, and write all my comments on the paper itself and margins, so that when I am ready to write, I go comment by comment and expand on it in the report.

The referee report

It is time to write all your comments and feedback on the actual report. Consider adding the following components to your reports:

  • NOT anywhere in the report: your name, affiliation, email address. Make sure the report is anonymous and that you are not writing things in a way that will easily identify yourself.
  • Title and authors of the paper under review.
  • Journal where the paper is submitted (this is mostly for my records, because sometimes you get to referee the same paper twice for different journals!).
  • Overview: a summary of the results of the paper, so the editor and authors know that you have actually read the paper. It is also a place to state the main results in your opinion, which may differ from the results that the authors think are the main results! This section is a neutral zone, however, so you are just stating results without colorful commentary.
  • Recommendation: a narrative of the strengths and weaknesses of the paper, in your expert opinion, which concludes with a recommendation for the editors: reject, accept, minor revision, major revision, etc. You can include here big items that the authors need to address before the paper is accepted, and general comments about the paper.
  • Detailed comments: this is an itemized list of comments. Please include pages and lines and theorem numbers that you are referring to, so that the authors know what you are exactly talking about.
  • Conclusion: any other general comments that may improve the paper, or thoughts about the paper itself.

Once you are done, send the anonymous referee report to the editors, in their preferred contact method, probably through their online editorial system.

What happens then?

After the report is sent back to the editors, the editorial team may be waiting for other referees to also send their reports. Once they have all their reports, they will make a decision. If they ask the authors for a revision, they might ask you to look at the paper one more time. I usually accept because it is efficient, since I am already familiar with the paper, but again, it is your call if you are available or busy at this time.

Finally, thanks for taking time to do a great job refereeing papers! Authors definitely appreciate the hard work of a referee.

How to Request a (Math) Recommendation Letter

Table of Contents

Introduction and advice from others

At some point in your career (earlier than you think!) you will need a recommendation letter. Perhaps you are applying for an REU, grad school, a conference, a job,… and you need one or more people to vouch for you. Who should you ask and how do you ask for a letter? You are essentially asking someone “hey how do you really feel about me, and would you be willing to put it in writing?” and that’s a daunting thought. The goal of this post is to help you navigate this delicate situation.

The first piece of good news is that all of us have asked for letters of recommendation in the past, and we are all grateful to those who wrote letters for us, which opened many doors. Surely, I have a long list of people to thank for their letters over the years! So we are well aware of the power and the need for strong recommendation letters, and we want to do our best to help others advance their careers. In light of this, we expect that we will be asked to write recommendation letters, and many of us consider letter-writing as part of our mathematical community good citizenship.

The second good piece of news, particularly for someone like me writing a post about this topic, is that there is a lot of great advice already available online on how to ask for recommendation letters. Here is a few such pages, in no particular order. They will give you an idea of what most people are looking for before writing a letter of recommendation:

Notice that many of these people give credit to each other for the advice in their pages. I thank all of them for putting all of their advice in a public place!

Sport GIFs & random things: Spiderman pointing meme | Spiderman meme, Meme  template, Cartoon memes

The summary of all the advice below (TL;DR): ask mathematicians that know you well, and have a reason to think highly of your potential. Ask for a letter well in advance. Give a lot of detailed information to your letter writers (about you, deadlines, requirements, etc). Be patient.

What is the point of a recommendation letter anyway?

Why do we require recommendation letters in the first place? From my point of view, letters of recommendation are meant to assure that (a) the candidate is a good fit for the position or program they are applying for, and (b) the candidate has the appropriate preparation to thrive in the position or program. In addition, many programs are overwhelmed by a large number of applicants, so the letters help the selection committees while deciding what candidates will benefit the most from participating in the program. The letters of recommendation are rarely the only data point of an application, but a good recommendation letter can add a new dimension to a candidate’s file that would not fit well in other parts of an application (CV, cover letter, statements, etc.).

Will I ever need a recommendation letter? (YES)

Yes. You will absolutely need a recommendation letter at some point. Therefore, it does not hurt to think strategically and ahead of time: who will be my letter writers when the time comes? Please read the rest of the document with this question in mind, and try to network and build professional relationships with mathematicians who will later be in a position to write a strong letter for you. I also wrote about networking in this previous post about applying for jobs in math.

Who should I ask for a recommendation letter?

My first piece of advice is that you will probably need more than one letter, so you should think of who all your letter writers will be, before you start asking for letters. Each letter could discuss a different dimension of your background, so choose a variety of letter writers that can speak to a number of different parts of your application. Also: talk to someone (e.g., your advisor) about who you plan to ask for letters, and discuss your options.

The best recommenders are people that:

  • know you well, from a mathematical point of view;
  • expect that you will ask for a letter of recommendation at some point. In other words, it should not come as a surprise that you are asking for a letter of recommendation from said person; and
  • are in a position to write a great letter about you. This means that the writer knows you well, through a mathematical experience that demonstrates your great potential.

Here are some examples of people who would be good options for a letter:

  • Your academic advisor. If you have had a good relationship with your advisor, they probably know you best, and they know about your strengths, and your struggles. An advisor can give a holistic point of view that most others are not typically capable of.
  • A professor you took a (math) class from (and did very well in it). A professor can speak about your work ethic, the quality of your work, and about your enthusiasm for the material. If you did not do great in their class, or if this was a very large class and you had little interaction with the professor, then they are probably not the best choice for letter writing. Ideally, this would be a math professor, but professors from other subjects can also write for you.
  • A mathematician you worked/collaborated with. If you were part of an REU, an independent study, a senior thesis, a research project, and you worked with alongside another mathematician on some sort of mathematical project, then they are probably a good choice for a letter writer. As in the previous bullet point, they can speak about your work ethic, the quality of your work, and about your enthusiasm for the material. In addition, they can speak about your potential as a colleague or member of a mathematical community or department.
  • A mathematician that knows about your skills and your background for other reasons. Perhaps you have been in contact with a mathematician for other reasons than those mentioned above. As long as they know you relatively well, from a mathematical point of view, and there is no clear conflict of interests in their relation to you, then they can be a good candidate for letter writer.

The people who are the wrong choices are the complement of those described above:

  • People who barely know you, no matter how well-known they are in their field, are not good choices. They will not have much to say about you, and they will probably reject writing a letter for you anyway. If they did write a letter, it would be an impersonal letter at best, and it may do more harm than good.
  • People who are at your own level. For example, if you are a student, do not ask another student to write a letter for you. Even if they know you well, letter readers are looking for impartial observers that are capable of summarizing your potential from a higher ground. For example, for tenure cases, all (or most) letters should come from professors with tenure or above.
  • People who are family, friends, or close relations. If the letter writer cannot be an impartial judge of your potential, then the letter itself is worthless.
  • People who are known to be harsh letter writers. There is no need to risk it. If you know that a person is a tough critic, then ask a letter from someone else.
  • People who you are not sure they can write a great letter about you. If you have doubts about whether the would-be letter writer knows you well enough, then they do not know you well enough, and you should ask someone else.

When should I ask for a recommendation letter? (ASAP)

This one is easy: as soon as humanly possible. That early? Yes. You can let them know months in advance (“at some point I will ask you for a letter of recommendation, if you don’t mind”) for generic letters, and as soon as you can when it is game time and you know you will need a letter for a program, grant proposal, job, or what have you.

At the very least, try to give a letter writer two weeks notice, but keep in mind that many people require 3 weeks or more to write a (good) letter. It also depends on the type of letter: an email message with a paragraph about the candidate can be a quick job, but a letter of recommendation for a job search can be very time consuming and the letter writer might need a month or more.

“What?? Who takes a month to write a letter?” you might say. Keep in mind that mathematicians receive many such requests, so it is not the case that the letter takes a full month to write, but that the person might need a month to find the time to piece all the parts of the letter together for you. Professors can be very busy, particularly during letter writing season, so please be understanding that they will need as much advance notice as you can give them to be able to write the best letter possible for you.

Sometimes, however, you find out about an opportunity and need a letter in short notice, and that’s fine – it happens. Then, let the potential letter writer know that this is the case, and let them decide whether they can produce a letter in a short period of time.

How should I ask for a letter? (Nicely)

This one is an easy one as well: nicely! Seriously. Just ask nicely. Most frequently, people will ask for a letter of recommendation by email. Ideally, you would ask them first in person… though there are not a lot of in-person opportunities these days (I am writing this during a pandemic). But the truth is that asking for a letter of recommendation is a hard thing to do, and email is a good medium to put together a well-thought out request for a big ask. Also, an email message gives the recommender time to think about it and respond once they have made up their mind about whether they are an appropriate person to write a letter for you.

Here are the things you need to include in your initial message when you request a letter of recommendation:

  • An introduction to remind the person of who you are, and how they know you (be specific). If you don’t think they’ll remember who you are after a brief introduction… then they are probably not a good choice. For example, “Dear Professor X, you were my professor in real analysis last semester, and I really enjoyed your class. …”
  • Briefly, why do you need a letter? For example, “… The reason I am writing to you is that I am in the process of getting ready to apply for REU programs. …”
  • A nice request to write a letter for you. Include a date when a letter is due. It is important to ask for a strong letter of recommendation. For example, “… and I was hoping to ask you if you could write a strong letter of recommendation for me. The first deadline for a letter is on Jan 1st. …”
  • Be as specific as possible about the program you are applying for, what is needed, and when it is needed. For example, “… I am applying to this program in particular, [URL], which has a deadline of Jan 1st for letters of recommendation. Letters need to be sent by email to this address [email address] with the subject line “Letter of Rec. for [student’s name]”.
  • Attach to your message basic documents that almost all letter writers will want to see before writing a letter, or even when considering writing a letter. For example, “… I have attached to this message my CV, an unofficial copy of my transcripts, and a draft of my statement of purpose. …”
  • Feel free to add details about you that you would like writers to highlight in the letters. If you have received an award, or have relevant experience for the program you are applying for, or you have published a paper on the subject, etc., then let the writers know, in case they want to mention this in their letter. For example, “… I have been doing an independent study on the same topic of the REU with Professor Y, so I think this program would be greatly beneficial for my career. … ”
  • Ask them what do THEY need from you. First check their website, to see if they have a section precisely about writing letters of recommendation. Otherwise, ask them to tell you what would they need to write a letter for you. For example, “… If you are willing to write a letter for me, let me know if there is anything else you need from me.”
  • Thank them for their time, and wait. For example, “… I would really appreciate it if you could find the time to write a letter for me. Thank you!”

A sample letter

Here is the message once again, in one piece:

Dear Professor X,

You were my professor in real analysis last semester, and I really enjoyed your class. The reason I am writing to you is that I am in the process of getting ready to apply for REU programs and I was hoping to ask you if you could write a strong letter of recommendation for me. The first deadline for a letter is on Jan 1st.

I am applying to this program in particular, [URL], which has a deadline of Jan 1st for letters of recommendation. Letters need to be sent by email to this address [email address] with the subject line “Letter of Rec. for [student’s name]”.

I have attached to this message my CV, an unofficial copy of my transcripts, and a draft of my statement of purpose. I have been doing an independent study on the same topic of the REU with Professor Y, so I think this program would be greatly beneficial for my career.

If you are willing to write a letter for me, please let me know if there is any other information you need from me. I would really appreciate it if you could find the time to write a letter for me.

Thank you!

Should I remind the letter writers about upcoming deadlines?

As a deadline approaches, you might be wondering if your letter writers have submitted their letters on time. Luckily, many online application system will tell you if the letters have been submitted, so you don’t have to bother your writers.

Your initial message requesting a letter of recommendation should be very informative, with a breakdown of all the deadlines, so the letter writers should have all the information they need, and most of the time the letters will arrive on time. If you want, you can ask your writers early on if they would like reminders. Or if you see that a deadline is approaching and a letter has not been submitted, you may want to send a friendly email reminder about the deadline. That is usually fine, and many times welcome, because during busy times some of these things can be forgotten in a pile of other things to do. But avoid pestering your writers with too many messages and reminders!

Good luck!

My Favorite Prime Numbers

This blog post is based on a Twitter thread on the same topic.

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…

p=2

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=mn-1 is a prime number, for some natural number m, then either q=2 or m=2. Primes of the form q=2n-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 2n 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/pnZ)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

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.

p=163

The prime number 163 is really nice for several reasons. For instance, epi*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 x2-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.

p=1093 and 3511

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 p2 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.

p=4001 and 4003

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!

p=11, 37, 389, 5077, (117223), and 19047851

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) + ZR(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!

p=65537

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 Fn = 22^n+1, known as Fermat numbers, are always prime. The first few Fermat numbers F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 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 F5 = 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!

p=12345678910987654321

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.

p=282,589,933 − 1

As we mentioned above in the entry for p=2, if q=mn-1 is a prime number, for some natural number m, then either q=2 or m=2. Moreover, if q=2n-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, 211-1 = 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 M51 = 282,589,933 − 1. It is worth noting the mind-blowing fact that M51 has 24,862,048 digits.

A really cool fact about Mersenne primes is their relationship to even perfect numbers: if 2p-1 is prime, then 2p-1(2p-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 282,589,932 * (282,589,933 − 1) … a perfect number with 49,724,095 digits!

List-O-Mania

This blog post is an excerpt of a new upcoming book that I am writing. Feedback and suggestions are welcome!

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.

Fibonacci’s challenge

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.

Statues of Leonardo Pisano, “Fibonacci,” (left) and the Holy Roman Emperor, Frederick II (right). Images source: Wikipedia Commons. Authors: Hans-Peter Postel (left), and Wolfgang Rieger (right).

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:

  1. 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.
  2. 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.
  3. 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.

Hilbert’s Problems

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).

David Hilbert (1862-1943). Image source: Wikimedia Commons.

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.

  1. 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.)
  2. Prove that the axioms of arithmetic are consistent. (Resolved in 1936.)
  3. 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.)
  4. Construct all metrics where lines are geodesics. (Partial progress depending on the interpretation of the problem.)
  5. Are continuous groups automatically differential groups? (Resolved in 1953, but an interpretation of this problem, the Hilbert-Smith conjecture, is still open.)
  6. Mathematical treatment of the axioms of physics. (Partially resolved.)
  7. Is a^b transcendental, for an algebraic number a =/= 0,1, and an irrational algebraic number b? (Resolved in 1934.)
  8. Problems on Prime Numbers. These include the Riemann hypothesis, Goldbach’s conjecture, and the twin prime conjecture. (All three are unresolved.)
  9. Find the most general law of the reciprocity theorem in any algebraic number field. (Partially resolved.)
  10. Find an algorithm to determine whether a given polynomial diophantine equation with integer coefficients has an integer solution. (Resolved in 1970.)
  11. Solving quadratic forms with algebraic numerical coefficients. (Resolved in 1924.)
  12. Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field. (Unresolved.)
  13. Solve 7th degree equations using algebraic (variant: continuous) functions of two parameters. (Unresolved.)
  14. Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? (Resolved in 1959.)
  15. Rigorous foundation of Schubert’s enumerative calculus. (Partially resolved.)
  16. Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. (Unresolved.)
  17. Express a non-negative rational function as quotient of sums of squares. (Resolved in 1927.)
  18. (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.)
  19. Are the solutions of regular problems in the calculus of variations always necessarily analytic? (Resolved in 1957.)
  20. Do all variational problems with certain boundary conditions have solutions? (Resolved during the course of the 20th century.)
  21. Proof of the existence of linear differential equations having a prescribed monodromy group. (Partially resolved.)
  22. Uniformization of analytic relations by means of automorphic functions. (Partially resolved.)
  23. 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.

Landau’s Problems

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.

Edmund Landau (1877-1938). Image source: Overwolfach Photo Collection.
  1. Goldbach’s conjecture: can every even integer greater than 2 be written as the sum of two primes? (Unresolved.)
  2. Twin prime conjecture: are there infinitely many primes p such that p + 2 is prime? (Unresolved.)
  3. Legendre’s conjecture: does there always exist at least one prime between consecutive perfect squares? (Unresolved.)
  4. 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!

Yitang Zhang (1955-). Image source: Wikipedia Commons.

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).

The Weil Conjectures

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:

  1. Zeta functions are rational.
  2. Zeta functions satisfy a functional equation and Poincaré duality.
  3. Zeta functions satisfy an analog of the Riemann hypothesis.
  4. 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.

Smale’s Problems

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:

  1. The Riemann hypothesis. (Unresolved.)
  2. The Poincaré conjecture. (Resolved in 2003.)
  3. P versus NP. (Unresolved.)
  4. Shub-Smale tau-conjecture on the integer zeros of a polynomial of one variable. (Unresolved.)
  5. Can one decide if a diophantine equation f(x,y) = 0 has an integer solution in exponential time? (Unresolved.)
  6. 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.)
  7. The Thomson problem on minimizing the distribution of N points on a 2-sphere. (Unresolved.)
  8. Extend the mathematical model of general equilibrium theory to include price adjustments. (Unresolved.)
  9. The linear programming problem. (Unresolved.)
  10. Pugh’s closing lemma. (Partially resolved in 2016.)
  11. Is one-dimensional dynamics generally hyperbolic? (This problem had two parts, the first part is unresolved, and the second part is resolved.)
  12. 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.)
  13. 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.)
  14. Do the properties of the Lorenz attractor exhibit that of a strange attractor? (Resolved in 2002.)
  15. Navier-Stokes existence and smoothness. (Unresolved.)
  16. The jacobian conjecture. (Unresolved.)
  17. Solving polynomial equations in polynomial time in the average case. (Resolved in 2016.)
  18. 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?

Millennium Prize Problems

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.

  1. The Birch and Swinnerton-Dyer conjecture. (Unresolved.)
  2. The Hodge conjecture. (Unresolved.)
  3. Navier-Stokes existence and smoothness. (Unresolved.)
  4. The P versus NP problem. (Unresolved.)
  5. The Poincaré conjecture. (Resolved in 2003.)
  6. The Riemann hypothesis. (Unresolved.)
  7. 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.

Clopen Concept

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.

Away

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.

Andrew Wiles speaking at the Andrew Wiles Building, Oxford, UK. Image taken by the author.

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.

My Search For Jobs, Round #3: The Final Boss Fight

In my previous posts, I described my first job search during the last year of my PhD at BU, and then my second search while I was a visiting professor at Colby College for a year. In this post, I describe my “final boss fight” in search of a tenure-track position in mathematics. In fact, just like my favorite heroes in the Geonosis arena, there were three final bosses awaiting me at the end of the road, with three corresponding tenure-track interviews (each one of them with an unexpected twist), which I’ll try my best to summarize in this post.

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.

The first paragraph of my 2007 research statement.

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!

A train station in San Diego.

Interlude: A Rant about the Joint Math Meetings

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.

The author in San Diego, with some math friends, a toddler, and an engineer (January 2008).

Small Colleges #1 and #2

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.

On-Campus Interview #1

Final Boss Fight #1: NEXU

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.

On-Campus Interview #3

Final Boss Fight #2: REEK.

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.

The UConn Interviews

Final Boss Fight #3: ACKLAY.

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.

The San Diego Convention Center.

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.

My Search For Jobs, Round #2: Looking for a Postdoc

This post is Part 2 of a 3-part series of posts on my own twists and turns searching for academic jobs in mathematics. This post is about my second round on the job hunt, a few months after graduating, while I was 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.

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.

Spreadsheet of jobs in the Fall of 2004

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.

Time stands still when you are waiting for a message 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 nothing about Galois representations? Then what?”

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.

A silly picture to immortalize 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.

Bonus pic of my saltwater aquarium that I kept while living in Ithaca (all coral and fish were born locally).

My Search for Jobs, Round #1: Thesis and Jobs, with a Twist

This post is Part 1 of a series of posts on my own twists and turns searching for academic jobs in mathematics. This post is about my first round on the job hunt, during my last year in grad school, and it is, as many other such accounts, a roller-coaster of proud accomplishments and soul-crushing rejection letters.

Defend the Thesis and Apply for Jobs, with a Twist

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.

Colby College, during a recent visit in November 2019.

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.