A Field Guide to Mathematics

I have decided to upgrade the scope of the MathAndCobb Fund. You can have a look here at the previous announcement. As I explained in my previous post, my content generates “content creator funds” and I decided to give back to the math community, and create my own mini-grant. The MaC Fund was, until now, aimed at graduate students who were going to attend a conference, and the fund provided each student $80-$150. You can see the list of previous recipients and awards here. Now I want to widen the scope of the funding opportunity, and call it the MathAndCobb Fund for Mathematical Exploration, and the latest award was $200.

Or “MaC Fund for ME“, for short.

The fund, as before, will be awarded every two months or so (next deadline will be Feb 15th, and the award will be announced on March 1st). The following are the eligibility criteria:

1) Highschool, college undergrads, or graduate students in mathematics (in a US-based institution) who are applying for travel funds to go to a math conference.
2) Student (as in (1)) who are organizing a math event (e.g., a single event such as a conference, or a recurring event like a math club) and need additional funding for expenses towards the event (e.g., travel, food).
3) Surprise me! People who would like to apply for funding for a math-related reason.

Here are the rules and regulations, and how to apply:

a) Each awardee will receive about 1/4th of my content creator earnings for the previous two months. For example, the next award will be about $200-$250. My earnings fluctuate though, so awards will vary. You get a line in your CV regardless!

b) The awardee (or awardees) agrees to be featured in a TikTok video announcing the award. I will need at the very least a picture of the student (or group), and a description of the event the fund is going towards. Even better, I would love a video by the awardee explaining their project! Even BETTER, a video of a conversation with me about your program and research.

c) Groups traditionally underrepresented in Mathematics are STRONGLY encouraged to apply!

d) Payment will be made via PayPal or Venmo, so the student needs to have an account in one of those platforms.

e) In order to apply, please send an email message to mathandcobb@gmail.com (email subject line: “MaC Fund” + your last name) with the following information:

Your full name
Your Institution and name of advisor if available
Year in your program
What is the funding for and why do you need funds?
Anything else you want to add about you that you want me to consider.

Please share this post widely!
Thank you.

This post will be updated with deadlines to apply for the “MathAndCobb Fund for Mathematical Exploration” and also with a list of recipients.

Deadlines and Previous Recipients:

• July 2026 (Award of $250): Please read the new eligibility criteria! Applications are due June 15th.
• April 2026 (Award of $250): the AMW Chapter at the University of Albany.
• January 2026 (Award of $250): Caylee Spivey (UConn)
• November 2025 (Award of $250): Ameen Hussain (Colorado State University)
• September 2025 (Award of $200): Andrew Canada, Austin Eilers, and Ayden Edwards, founders of CASE (Crosby Association of STEM Excellence).
• May 2025 (Award: a coffee machine for their learning center): The Math Learning Center (University of Wisconsin-Madison).
• March 2025 (Award of $150): Swati (University of South Carolina)
• December 2024 (Award of $150): Bryson Kagy (North Carolina State University)
• October 2024 ($100): Jasmine Camero (Emory University)
• August 2024 ($100): Daniel Flores (Purdue University)
• June 2024 ($80): Vittoria Cristante (Tufts University)

UPDATE: the MaC Fund has been updated! See this post for updated eligibility.

As you probably know, I am on TikTok (@mathandcobb), and I have been regularly posting videos there since January 2021. I am considered a “content creator” and as part of the Content Creator Program, I am rewarded based on the views that my videos get. Not a lot of money! But enough that I have been thinking for a while how to give back to the community (the mathematical community and also the TikTok community of my followers). Thus, I have decided to create a fund to share my earnings with grad students in mathematics! It’s just a drop in the bucket… but every drop helps!

We shall call it the “MaC Fund for Math Grad Student Travel,” or the “MathAndCobb Fund,” or the “MaCTikTok Fund,” or the “MaC Fund” for short.

A list of previous recipients and deadlines can be found in this post.

Here is how it is going to work:

1) Until further notice (e.g., TikTok shuts down in the US), an awardee will be announced every other month, on the first of the month. The first awardee will be announced (on TikTok!) on June 1st. The deadline to be fully considered will be 15 days before the award announcement (so on May 15th in this case).

2) Each awardee will receive about 1/4th of my content creator earnings for the previous two months. For example, the first award will be about $80. My earnings fluctuate though, so awards will vary. You get a line in your CV regardless!

3) Eligibility: grad students in a US-based institution, doing their PhD in Mathematics, and in their third year of grad school or above or, otherwise, already working towards a PhD thesis problem.

4) This is a travel grant, so you need to tell me what conference you are going to that the funds could help with. It is not a lot of money, but I hope every bit helps. There are of course other better funding options to explore!

5) The awardee agrees to be featured in a TikTok video announcing the award. I will need at the very least a picture of the student, and a description of the research they are working on. Even better, I would love a video by the awardee explaining their own research! Even BETTER, a video of a conversation with me about your program and research.

6) Groups traditionally underrepresented in Mathematics are STRONGLY encouraged to apply!

7) Payment will be made via PayPal or Venmo, so the student needs to have an account in one of those platforms.

8) In order to apply, please send an email message to mathandcobb@gmail.com (email subject line: “MaC Fund” + your last name) with the following information:

Your full name
Your PhD Institution and name of advisor if available
Year in your program
What conference are you attending and why?
What are you working on, or learning about, towards your thesis?
Anything else you want to add about you that you want me to consider.

Please share this post widely!
Thank you.

The story “The Day She Proved It” has been accepted to appear at The Mathematical Intelligencer, so now, after a few edits, you can read it over there! Enjoy. My thanks to the Intelligencer editors and referees.

In my TikTok account, you can find a series of (comedy) videos that portray a student taking an incredibly difficult and confusing exam, which I call “the TikTok exam.” Most of the videos are tagged under #thetiktokexam but some of the early ones I didn’t tag with it.

Someone asked me to make the exam available as a PDF, so I’ve uploaded it here! Enjoy.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

• Introduction and advice from others
• What is the point of a recommendation letter anyway?
• Will I ever need a recommendation letter? (YES)
• Who should I ask for a recommendation letter?
• When should I ask for a recommendation letter? (ASAP)
• How should I ask for a letter? (Nicely)
• A sample letter
• Should I remind the letter writers about upcoming deadlines?

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:

• Michael Orrison (Harvey Mudd College).
• Tara Holm (Cornell University).
• Tom Roby (UConn).
• Keith Conrad (UConn; his page is probably the most comprehensive).
• Megumi Harada (McMaster University).
• Ravi Vakil (Stanford University).
• Rob Pollack (Boston University).
• Joseph Silverman (Brown University; also a very thorough page).
• David Zureick-Brown (Emory University).

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!

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.

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.).

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.

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.

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.

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!”

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!

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!

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…

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

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

The prime number 163 is really nice for several reasons. For instance, 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.

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.

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

The set E(Q) of all rational points on an elliptic curve E defined over Q is a finitely generated abelian group (thanks to the Mordell-Weil theorem), so E(Q) has a finite torsion subgroup T(E/Q), and also R(E/Q) rational points of infinite order such that E(Q) is isomorphic to T(E/Q) + 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!

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!

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

12345678910987654321

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

As we mentioned above in the entry for p=2, if q=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!

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.

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

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

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.

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

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

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

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

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.

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.

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!

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

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

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

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

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.

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?

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.