## The problem with combinatorics textbooks

Every now and then I think about writing a graduate textbook in Combinatorics, based on some topics courses I have taught. I scan my extensive lecture notes, think about how much time it would take, and whether there is even a demand for this kind of effort. Five minutes later I would always remember that YOLO, deeply exhale and won’t think about it for a while.

**What’s wrong with Combinatorics?**

To illustrate the difficulty, let me begin with two quotes which contradict each other in the most illuminating way. First, from the Foreword by **Richard Stanley** on (his former student) Miklós Bóna’s “*A Walk Through Combinatorics*” textbook:

The subject of combinatorics is so vast that the author of a textbook faces a difficult decision as to what topics to include. There is no more-or-less canonical corpus as in such other subjects as number theory and complex variable theory. [here]

Second, from the Preface by **Kyle Petersen** (and Stanley’s academic descendant) in his elegant “*Inquiry-Based Enumerative* *Combinatorics*” textbook:

Combinatorics is a very broad subject, so the difficulty in writing about the subject is not what to include, but rather what to exclude. Which hundred problems should we choose? [here]

Now that this is all clear, you can probably insert your own joke about importance of teaching inclusion-exclusion. But I think the issue is a bit deeper than that.

I’ve been thinking about this when updating my “*What is Combinatorics*” quotation page (see also my old blog post on this). You can see a complete divergence of points of view on how to answer this question. Some make the definition or description to be very broad (sometimes even ridiculously broad), some relatively narrow, some are overly positive, while others are revoltingly negative. And some basically give up and say, in effect “it is what it is”. This may seem puzzling, but if you concentrate on the narrow definitions and ignore the rest, a picture emerges.

Clearly, these people are not talking about the same area. They are talking about sub-areas of Combinatorics that they know well, that they happen to learn or work on, and that they happen to like or dislike. Somebody made a choice what part of Combinatorics to teach them. They made a choice what further parts of Combinatorics to learn. These choices are increasingly country or culture dependent, and became formative in people’s mind. And they project their views of these parts of Combinatorics on the whole field.

So my point is — there is no right answer to “*What is Combinatorics?*“, in a sense that all these opinions are biased to some degree by personal education and experience. Combinatorics is just too broad of a category to describe. It’s a bit like asking “*what is good food?*” — the answers would be either broad and bland, or interesting but very culture-specific.

**Courses and textbooks**

How should one resolve the issue raised above? I think the answer is simple. Stop claiming that *Combinatorics*, or worse, *Discrete Mathematics*, is one subject. That’s not true and hasn’t been true for a while. I talked about this in my “Unity of Combinatorics” book review. Combinatorics is comprised of many sub-areas, see the *Wikipedia article* I discussed here (long ago). Just accept it.

As a consequence, you should never teach a “Combinatorics” course. **Never!** Especially to graduate students, but to undergraduates as well. Teach courses in any and all of these subjects: *Enumerative Combinatorics*, *Graph Theory*, *Probabilistic Combinatorics*, *Discrete Geometry*, *Algebraic Combinatorics*, *Arithmetic Combinatorics*, etc. Whether introductory or advanced versions of these courses, there is plenty of material for each such course.

Stop using these broad “a little bit about everything” combinatorics textbooks which also tend to be bulky, expensive and shallow. It just doesn’t make sense to teach both the *five color theorem* and the *Catalan numbers* (see also here) in the same course. In fact, this is a disservice to both the students and the area. Different students want to know about different aspects of Combinatorics. Thus, if you are teaching the same numbered undergraduate course every semester you can just split it into two or three, and fix different syllabi in advance. The students will sort themselves out and chose courses they are most interested in.

**My own teaching**

At UCLA, with the help of the Department, we split one Combinatorics course into two titled “Graph Theory” and “Enumerative Combinatorics”. They are broader, in fact, than the titles suggest — see Math 180 and Math 184 here. The former turned out to be quite a bit more popular among many applied math and non-math majors, especially those interested in CS, engineering, data science, etc., but also from social sciences. Math majors tend to know a lot of this material and flock to the latter course. I am not saying you should do the same — this is just an example of what *can *be done.

I remember going through a long list of undergraduate combinatorics textbooks a few years ago, and found surprisingly little choice for the enumerative/algebraic courses. Of the ones I liked, let me single out Bóna’s “*Introduction to Enumerative and Analytic Combinatorics“* and Stanley’s “*Algebraic Combinatorics*“. We now use both at UCLA. There are also many good *Graph Theory* course textbooks of all levels, of course.

Similarly, for graduate courses, make sure you make the subject relatively narrow and clearly defined. Like a topics class, except accessible to beginning graduate students. Low entry barrier is an advantage Combinatorics has over other areas, so use it. To give examples from my own teaching, see unedited notes from my graduate courses:

*Combinatorics of posets* (Fall 2020)

*Combinatorics and Probability on groups* (Spring 2020)

*Algebraic Combinatorics* (Winter 2019)

*Discrete and Polyhedral Geometry* (Fall 2018) This is based on my book. See also videos of selected topics (in Russian).

Combinatorics of Integer Sequences (Fall 2016)

*Combinatorics of Words *(Fall 2014)

*Tilings* (Winter 2013, lecture-by-lecture refs only)

#### In summary

In my experience, the more specific you make the combinatorics course the more interesting it is to the students. Don’t be afraid that the course would appear be too narrow or too advanced. That’s a stigma from the past. You create a good course and the students will quickly figure it out. They do have their own FB and other chat groups, and spread the news much faster than you could imagine…

Unfortunately, there is often no good textbook to cover what you want. So you might have to work a little harder harder to scout the material from papers, monographs, etc. In the internet era this is easier than ever. In fact, many extensive lecture notes are already available on the web. Eventually, all the appropriate textbooks will be written. As I mentioned before, this is one of the very few silver linings of the pandemic…

**P.S. ** (July 8, 2021) I should have mentioned that in addition to “a little bit about everything” textbooks, there are also “a lot about everything” doorstopper size volumes. I sort of don’t think of them as textbooks at all, more like mixtures of a reference guide, encyclopedia and teacher’s manual. Since even the thought of teaching from such books overwhelms the senses, I don’t expect them to be widely adopted.

Having said that, these voluminous textbooks can be incredibly valuable to both the students and the instructor as a source of interesting supplementary material. Let me single out an excellent recent “*Combinatorial Mathematics*” by Doug West written in the same clear and concise style as his earlier “*Introduction to Graph Theory*“. Priced modestly (for 991 pages), I recommend it as “further reading” for all combinatorics courses, even though I strongly disagree with the second sentence of the Preface, per my earlier blog post.

## How to tell a good mathematical story

As I mentioned in my previous blog post, I was asked to contribute to to the **Early Career Collection** in the *Notices of the AMS*. The paper is not up on their website yet, but I already submitted the proofs. So if you can’t wait — the short article is **available here**. I admit that it takes a bit of a chutzpah to teach people how to write, so take it as you will.

Like my previous “*how to write*” article (see also my blog post), this article is mildly opinionated, but hopefully not overly so to remain useful. It is again aimed at a novice writer. There is a major difference between the way *fiction *is written vs. *math*, and I am trying to capture it somehow. To give you some flavor, here is a quote:

What kind of a story?Imagine a non-technical and non-detailed version of the abstract of your paper. It should be short, to the point, and straightforward enough to be atweet, yet interesting enough for one person towantto tell it, and for the listener curious enough to be asking for details. Sounds difficult if not impossible? You are probably thinking that way, because distilled products always lack flavor compared to the real thing. I hear you, but let me give you some examples.Take Aesop’s fable “

” written over 2500 years ago. The story would be “The Tortoise and the HareA creature born with a gift procrastinated one day, and was overtaken by a very diligent creature born with a severe handicap.” The names of these animals and the manner in which one lost to another are less relevant to the point, so the story is very dry. But there are enough hints to make some readers curious to look up the full story.Now take “

”, the original 1984 movie. The story here is (spoiler alert! ) “The TerminatorA man and a machine come from another world to fight in this world over the future of the other world; the man kills the machine but dies at the end.” If you are like me, you probably have many questions about the details, which are in many ways much more exciting than the dry story above. But you see my point – this story is a bit like an extended tag line, yet interesting enough to be discussed even if you know the ending.

## It could have been worse! Academic lessons of 2020

Well, this year sure was interesting, and not in a good way. Back in 2015, I wrote a blog post discussing how *video talks* are here to stay, and how we should all agree to start giving them and embrace watching them, whether we like it or not. I was right about that, I suppose. OTOH, I sort of envisioned a gradual acceptance of this practice, not the shock therapy of a phase transition. So, what happened? It’s time to summarize the lessons and roll out some new predictions.

**Note:** this post is about the *academic life* which is undergoing some changes. The changes in real life are much more profound, but are well discussed elsewhere.

#### Teaching

This was probably the bleakest part of the academic life, much commented upon by the media. Good thing there is more to academia than teaching, no matter what the ignorant critics think. I personally haven’t heard anyone saying post-March 2020, that online education is an improvement. If you are like me, you probably spent much more time preparing and delivering your lectures. The quality probably suffered a little. The students probably didn’t learn as much. Neither party probably enjoyed the experience too much. They also probably cheated quite a bit more. Oh, well…

Let’s count the * silver linings*.

*First*, it will all be over some time next year. At UCLA, not before the end of Summer. Maybe in the Fall…

*Second*, it could’ve been worse. Much worse. Depending on the year, we would have different issues. Back in 1990, we would all be furloughed for a year living off our savings. In 2000, most families had just one personal computer (and no smartphones, obviously). Let the implications of that sink in. But even in 2010 we would have had giant technical issues teaching on Skype (right?) by pointing our laptop cameras on blackboards with dismal effect. The infrastructure which allows good quality streaming was also not widespread (people were still using Redbox, remember?)

*Third*, the online technology somewhat mitigated the total disaster of studying in the pandemic time. Students who are stuck in faraway countries or busy with family life can watch stored videos of lectures at their convenience. Educational and grading software allows students to submit homeworks and exams online, and instructors to grade them. Many other small things not worth listing, but worth being thankful for.

*Fourth*, the accelerated embrace of the educational technology could be a good thing long term, even when things go back to normal. No more emails with scanned late homeworks, no more canceled/moved office hours while away at conferences. This can all help us become better at teaching.

*Finally*, a long declared “death of MOOCs” is no longer controversial. As a long time (closeted) opponent to online education, I am overjoyed that MOOCs are no longer viewed as a positive experience for university students, more like something to suffer through. Here in CA we learned this awhile ago, as the eagerness of the current Gov. Newsom (back then Lt. Gov.) to embrace online courses did not work out well at all. Back in 2013, he said that the whole UC system needs to embrace online education, pronto: “*If this doesn’t wake up the U.C. [..] I don’t know what will.*” Well, now you know, Governor! I guess, in 2020, I don’t have to hide my feelings on this anymore…

#### Research

I always thought that mathematicians can work from anywhere with a good WiFi connection. True, but not really – this year was a mixed experience as lonely introverts largely prospered research wise, while busy family people and extraverts clearly suffered. Some day we will know how much has research suffered in 2020, but for me personally it wasn’t bad at all (see e.g. some of my results described in my previous blog post).

#### Seminars

I am not even sure we should be using the same word to describe research seminars during the pandemic, as the experience of giving and watching math lectures online are so drastically different compared to what we are used to. Let’s count the differences, which are both positive and negative.

- The personal interactions suffer. Online people are much more shy to interrupt, follow up with questions after the talk, etc. The usual pre- or post-seminar meals allow the speaker to meet the (often junior) colleagues who might be more open to ask questions in an informal setting. This is all bad.
- Being online, the seminar opened to a worldwide audience. This is just terrific as people from remote locations across the globe now have the same access to seminars at leading universities. What arXiv did to math papers, covid did to math seminars.
- Again, being online, the seminars are no longer restricting themselves to local speaks or having to make travel arrangements to out of town speakers. Some UCLA seminars this year had many European speakers, something which would be prohibitively expensive just last year.
- Many seminars are now recorded with videos and slides posted online, like we do at the UCLA Combinatorics and LA Combinatorics and Complexity seminars I am co-organizing. The viewers can watch them later, can fast forward, come back and re-watch them, etc. All the good features of watching videos I extolled back in 2015. This is all good.
- On a minor negative side, the audience is no longer stable as it varies from seminar to seminar, further diminishing personal interactions and making level of the audience somewhat unpredictable and hard to aim for.
- As a seminar organizer, I make it a personal quest to encourage people to turn on their cameras at the seminars by saying hello only to those whose faces I see. When the speaker doesn’t see the faces, whether they are nodding or quizzing, they are clueless whether the they are being clear, being too fast or too slow, etc. Stopping to ask for questions no longer works well, especially if the seminar is being recorded. This invariably leads to worse presentations as the speakers can misjudge the audience reactions.
- Unfortunately, not everyone is capable of handling technology challenges equally well. I have seen remarkably well presented talks, as well as some of extremely poor quality talks. The ability to mute yourself and hide behind your avatar is the only saving grace in such cases.
- Even the true haters of online educations are now at least semi-on-board. Back in May, I wrote to Chris Schaberg dubbed by the insufferable Rebecca Schuman as “
*vehemently opposed to the practice*“. He replied that he is no longer*that opposed*to teaching online, and that he is now in a “it’s really complicated!” camp. Small miracles…

#### Conferences

The changes in conferences are largely positive. Unfortunately, some conferences from the Spring and Summer of 2020 were canceled and moved, somewhat optimistically, to 2021. Looking back, they should all have been held in the online format, which opens them to participants from around the world. Let’s count upsides and downsides:

- No need for travel, long time commitments and financial expenses. Some conferences continue charging fees for online participation. This seems weird to me. I realize that some conferences are vehicles to support various research centers and societies. Whatever, this is unsustainable as online conferences will likely survive the pandemic. These organizations should figure out some other income sources or die.
- The conferences are now truly global, so the emphasis is purely on mathematical areas than on the geographic proximity. This suggests that the (until recently) very popular AMS meetings should probably die, making AMS even more of a publisher than it is now. I am especially looking forward to the death of “joint meetings” in January which in my opinion outlived their usefulness as some kind of
*math extravaganza*events bringing everyone together. In fact, Zoom simply can’t bring five thousand people together, just forget about it… - The conferences are now open to people in other areas. This might seem minor — they were always open. However, given the time/money constraints, a mathematician is likely to go only to conferences in their area. Besides, since they rarely get invited to speak at conferences in other areas, travel to such conferences is even harder to justify. This often leads to
*groupthink*as the same people meet year after year at conferences on narrow subjects. Now that this is no longer an obstacle, we might see more interactions between the fields. - On a negative side, the best kind of conferences are small informal workshops (think of Oberwolfach, AIM, Banff, etc.), where the lectures are advanced and the interactions are intense. I miss those and hope they come back as they are really irreplaceable in the only setting. If all goes well, these are the only conferences which should definitely survive and even expand in numbers perhaps.

#### Books and journals

A short summary is that in math, everything should be electronic, instantly downloadable and completely free. Cut off from libraries, thousands of mathematicians were instantly left to the perils of their university library’s electronic subscriptions and their personal book collections. Some fared better than others, in part thanks to the arXiv, non-free journals offering old issues free to download, and some ethically dubious foreign websites.

I have been writing about my copyleft views for a long time (see here, there and most recently there). It gets more and more depressing every time. Just when you think there is some hope, the resilience of paid publishing and reluctance to change by the community is keeping the unfortunate status quo. You would think everyone would be screaming about the lack of access to books/journals, but I guess everyone is busy doing something else. Still, there are some lessons worth noting.

*You really must have all your papers freely available online*. Yes, copyrighted or not, the publishers are ok with authors posting their papers on their personal website. They are not ok when*others*are posting your papers on their websites, so the free access to your papers is on you and your coauthors (if any). Unless you have already done so, do this asap! Yes, this applies even to papers accessible online by subscription to selected libraries. For example, many libraries including all of UC system no longer have access to Elsevier journals. Please help both us and yourself! How hard is it to put the paper on the arXiv or your personal website? If people like Noga Alon and Richard Stanley found time to put hundreds of their papers online, so can you. I make a point of emailing to people asking them to do that every time I come across a reference which I cannot access. They rarely do, and usually just email me the paper. Oh, well, at least I tried…- Learn to use databases like MathSciNet and Zentralblatt. Maintain your own website by adding the slides, video links as well as all your papers. Make sure to clean up and keep up to date your Google Scholar profile. When left unattended it can get overrun with random papers by other people, random non-research files you authored, separate items for same paper, etc. Deal with all that – it’s easy and takes just a few minutes (also, some people judge them). When people are struggling trying to do research from home, every bit of help counts.
- If you are signing a book contract, be nice to online readers. Make sure you keep the right to display a public copy on your website. We all owe a great deal of gratitude to authors who did this. Here is my favorite, now supplemented with high quality free online lectures. Be like that! Don’t be like one author (who will remain unnamed) who refused to email me a copy of a short 5 page section from his recent book. I wanted to teach the section in my graduate class on
*posets*this Fall. Instead, the author suggested I buy a paper copy. His loss — I ended up teaching some other material instead. Later on, I discovered that the book is already available on one of those ethically compromised websites. He was fighting a battle he already lost!

#### Home computing

Different people can take different conclusions from 2020, but I don’t think anyone would argue the importance of having good home computing. There is a refreshing variety of ways in which people do this, and it’s unclear to me what is the optimal set up. With a vaccine on the horizon, people might be reluctant to further invest into new computing equipment (or video cameras, lights, whiteboard, etc.), but the holiday break is actually a good time to marinate on what worked out well and what didn’t.

Read your evaluations and take them to heart. Make changes when you see there are problems. I know, it’s unfair, your department might never compensate you for all this stuff. Still, it’s a small price to pay for having a safe academic job in the time of widespread anxiety.

#### Predictions for the future

*Very briefly*: I think online seminars and conferences are here to stay. Local seminars and small workshops will also survive. The enormous*AMS meetings*and expensive*Theory CS meetings*will play with the format, but eventually turn online for good or die untimely death.- Online teaching will remain being offered by every undergraduate math program to reach out to students across the spectrum of personal circumstances. A small minority of courses, but still. Maybe one section of each calculus, linear algebra, intro probability, discrete math, etc. Some faculty might actually prefer this format to stay away from office one semester. Perhaps, in place of a sabbatical, they can ask for permission to spend a semester some other campus, maybe in another state or country, while they continue teaching, holding seminars, supervising students, etc. This could be a perk of academic life to compete with the “
*remote work*” that many businesses are starting to offer on a permanent basis. Universities would have to redefine what they mean by “residence” requirement for both faculty and students. - More university libraries will play hardball and unsubscribe from major for-profit publishers. This would again sound hopeful, but not gain a snowball effect for at least the next 10 years.
- There will be some standardization of online teaching requirements across the country. Online cheating will remain widespread. Courts will repeatedly rule that business and institutions can discount or completely ignore all 2020 grades as unreliable in large part because of the cheating scandals.

#### Final recommendations

- Be nice to your junior colleagues. In the winner-take-all no-limits online era, the established and well-known mathematicians get invited over and over, while their junior colleagues get overlooked, just in time when they really need help (job market might be tough this year). So please go out of your way to invite them to give talks at your seminars. Help them with papers and application materials. At least reply to their emails! Yes, even small things count…
- Do more organizing if you are in position to do so. In the absence of physical contact, many people are too shy and shell-shocked to reach out. Seminars, conferences, workshops, etc. make academic life seem somewhat normal and the breaks definitely allow for more interactions. Given the apparent abundance of online events one my be forgiven to think that no more is needed. But more locally focused online events are actually important to help your communities. These can prove critical until everything is back to normal.

### Good luck everybody! Hope 2021 will be better for us all!

## Take an interview!

We all agree that Math is a *human endeavor*, yet we know so preciously little about mathematicians as humans working in mathematics. Our papers tend to have preciously few quotable sentences outside of the dry mathematical context. In fact, most introductions are filled with passages of the form “X introduced the celebrated tool *pqr*, which over the next 20 years was refined by A, B and C, and most recently was used by D to prove *Z’s conjecture*“. It is such a weak tea to convey contributions of six people in one short sentence, yet we all do this nonetheless.

In my “*How to write a clear math paper*” article accompanying this blog post, I argue that at least the first paragraph or the first subsection of a long paper can be human and aimed at humans. That is the place where one has freedom to be eloquent, inspiring, congratulatory, prescient, revelatory and quotable. I still believe that, but now I have a new suggestion, see the title of this blog post.

#### The art of autobiographies

These days many great scientists remain active into very old age, and rarely want or have time to write an autobiography. Good for them, bad for us. Psychologically this is understandable — it feels a little *epitaphish*, so they would much rather have someone else do that. But then their real voice and honest thoughts on life and math are lost, and can never be recorded. There is blogging, of course, but that’s clearly not for everyone.

There are some notable exceptions to this, of course. When I was in High School, reading autobiographies of Richard Feynman, Stan Ulam and Norbert Wiener was a pure joy, a window into a new world. The autobiоgraphy by Sofya Kovalevskaya was short on mathematical stories, but was so well written I think I finished the whole thing in one sitting. G.H. Hardy’s “*Apology*” is written in different style, but clearly self-revealing; while I personally disagree with much of his general point, I can see why the book continues to be read and inspire passionate debates.

More recently, I read William Tutte, “*Graph Theory As I Have Known It*“, which is mostly mathematical, but with a lot of personal stories delivered in an authoritative voice. It’s a remarkable book, I can’t praise it enough. Another one of my favorites is Steven Krantz, “*Mathematical Apocrypha*” and its followup, which are written in the first person, in a pleasant light rumor mill style. Many stories in these near-autobiographies were a common knowledge decades ago (even if some were urban legends), but are often the only way for us to learn *now* how it was back then.

On the opposite end of the spectrum there is L.S. Pontryagin’s autobiography (in Russian), which is full of wild rumors, vile accusations, and banal antisemitism. The book is a giant self-own, yet I couldn’t stop myself from hate-reading the whole thing just so I could hear all these interesting old stories from horse’s mouth.

Lately, the autobiographies I’ve been reading are getting less and less personal, with little more than background blurbs about each paper. Here are those by George Lusztig and Richard Stanley. It’s an unusual genre, and I applaud the authors for taking time to write these. But these condensed CV-like auto-bios clearly leave a lot of room for stories and details.

#### Why an interview?

Because a skillful interviewer can help a mathematician reveal personal stories, mathematical and metamathematical beliefs, and even general views (including controversial ones). Basically, reveal the humanity of a person that otherwise remains guarded behind endless Definition-Lemma-Theorem constructions.

Another reason to interview a person is to **honor** her or his contributions to mathematics. In the aftermath of my previous blog post, I got a lot of contradictory push-back. Some would say “*I am shocked, shocked, to find that there is corruption going on. I have submitted to many invited issues, served as a guest editor for others and saw none of that! So you must be wrong, wrong, wrong*.” Obviously, I am combining several POVs, satirizing and paraphrasing for the effect.

Others would say “*Yes, you are right, some journals are not great so my junior coauthors do suffer, the refereeing is not always rigorous, the invited authors are often not selected very broadly, but what can I do? The only way I can imagine to honor a person is by a math article in an invited issue of a peer review journal, so we must continue this practice*” (same disclaimer as above). Yeah, ok the imaginary dude, that’s just self-serving with a pretense of being generous and self-sacrificing. (Yes, my straw man fighting skill are unparalleled).

In fact, there are many ways to honor a person. You can give a talk about that person’s contributions, write a survey or a biographical article, organize a celebratory conference, or if you don’t want to be bothered simply add a dedication in the beginning of the next article you publish. Or, better yet, **interview the honoree**. Obviously, do this some time soon, while this person is alive, and make sure to put the interview online for everyone to read or hear.

#### How to do an interview?

Oh, you know, via Zoom, for example. The technical aspects are really trivial these days. With permission, you can record the audio/video by pushing one button. The very same Zoom (or Apple, Google, Amazon, Microsoft, etc.) have good speech-to-text programs which will typeset the whole interview for you, modulo some light editing (especially of math terminology). Again, with a couple of clicks, you can publish the video or the audio on YouTube, the text on your own website or any social media. Done. Really, it’s *that* easy!

#### Examples

I have many favorites, in fact. One superb **video collection** is done by the Simons Institute. I already blogged here about terrific interviews with László Lovász and Endre Szemerédi. The interviewer for both is Avi Wigderson, who is obviously extremely knowledgeable of the subject. He asked many pointed and interesting questions, yet leaving the interviewees plenty of space to develop and expand on their their answers. The videos are then well edited and broken into short watchable pieces.

Another interesting collection of video interviews is made by CIRM (in both English and French). See also general video collections, some of which have rather extensive and professionally made interviews with a number of notable mathematicians and scientists. Let me single out the *Web of Stories*, which include lengthy fascinating interviews with Michael Atiyah, Freeman Dyson, Don Knuth, Marvin Minsky, and many others.

I already wrote about how to watch a math video talk (some advice may be dated). Here it’s even easier. At the time of the pandemic, when you are Zoom fatigued — put these on your big screen TV and watch them as documentaries with as much or as little attention as you like. I bet you will find them more enlightening than the news, Netflix or other alternatives.

Authorized biography **books **are less frequent, obviously, but they do exist. One notable recent example is “*Genius At Play: The Curious Mind of John Horton Conway*” by Siobhan Roberts which is based on many direct conversations. Let me also single out perhaps lesser known “*Creative Minds, Charmed Lives*” by Yu Kiang Leong, which has a number of interesting interviews with excellent mathematicians, many of the them not on other lists. For example, on my “*What is Combinatorics*” page, I quote extensively from his interview with Béla Bollobás, but in fact the whole interview is worth reading.

Finally, there is a truly remarkable collection of **audio **interviews by Eugene Dynkin with leading mathematicians of his era, spanning from 1970s to 2010s (some in English, some in Russian). The collection was digitized using Flash which died about five years ago, rendering the collection unusable. When preparing this post I was going to use this example as a cautionary tale, but to my surprise someone made it possible to download them in .mp3. Enjoy! Listening to these conversations is just delightful.

#### Final thoughts

Remember, you don’t have to be a professional interviewer to do a good job. Consider two most recent interviews with Noga Alon and Richard Stanley by Toufik Mansour, both published at *ECA*. By employing a simple trick of asking the same well prepared questions, he allows the reader to compare and contrast the answers, and make their own judgement on which ones they like or agree with the most. Some answers are also quite revealing, e.g. Stanley saying he occasionally thinks about the RH (who knew?), or Alon’s strong belief that “*mathematics should be considered as one unit*” (i.e. without the area divisions). The problems they consider to be important are also rather telling.

Let me mention that in the digital era, even the amateur long forgotten interviews can later be found and proved useful. For example, I concluded my “*History of Catalan numbers*” with a quote from an obscure Richard Stanley’s interview to the MIT undergraduate newspaper. There, he was discussing the origins of his Catalan numbers exercise which is now a book. Richard later wrote to me in astonishment as he actually completely forgot he gave that interview.

**So, happy watching, listening, and reading all the interviews! Hope you take some interviews yourself for all of us to enjoy!**

**P.S.** (Added Dec 3, 2020) At my urging, Bruce Rothschild has typed up a brief “*History of Combinatorics at UCLA*“. I only added hyperlinks to it, to clarify the personalities Bruce is talking about (thus, all link mistakes are mine).

**P.P.S.** (Added Feb 6, 2021) At my request, the editors of *ECA* clarified their interview process (as of today, they have posted nine of them). Their interviews are conducted over email and are essentially replies to the nearly identical sets of questions. The responses are edited for clarity and undergo several rounds of approval by the interviewee. This practice is short of what one would traditionally describe as a *journalistic interview* (e.g., there are no uncomfortable questions), and is more akin to *writing a puff piece*. Still, we strongly support this initiative by the *ECA* as the first systematic effort to put combinatorialists on record. Hopefully, with passage of time others types of interviews will also emerge from various sources.

## How to write math papers clearly

Writing a mathematical paper is both an act of recording mathematical content and a means of communication of one’s work. In contrast with other types of writing, the style of math papers is incredibly rigid and resistant to even modest innovation. As a result, both goals suffer, sometimes immeasurably. The * clarity* suffers the most, which affects everyone in the field.

Over the years, I have been giving advice to my students and postdocs on how to write clearly. I collected them all in ** these notes.** Please consider reading them and passing them to your students and colleagues.

Below I include one subsection dealing with different reference styles and what each version really means. This is somewhat subjective, of course. Enjoy!

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**4.2. How to cite a single paper.** The citation rules are almost as complicated as Chinese honorifics, with an added disadvantage of never being discussed anywhere. Below we go through the (incomplete) list of possible ways in the decreasing level of citation importance and/or proof reliability.

(1) “*Roth proved Murakami’s conjecture in* [Roth].” Clear.

(2) “*Roth proved Murakami’s conjecture *[Roth].” Roth proved the conjecture, possibly in a different paper, but this is likely a definitive version of the proof.

(3) “*Roth proved Murakami’s conjecture, see* [Roth].” Roth proved the conjecture, but [Roth] can be anything from the original paper to the followup, to some kind of survey Roth wrote. Very occasionally you have “*see* [Melville]”, but that usually means that Roth’s proof is unpublished or otherwise unavailable (say, it was given at a lecture, and Roth can’t be bothered to write it up), and Melville was the first to publish Roth’s proof, possibly without permission, but with attribution and perhaps filling some minor gaps.

(4) “*Roth proved Murakami’s conjecture* [Roth], *see also* [Woolf].” Apparently Woolf also made an important contribution, perhaps extending it to greater generality, or fixing some major gaps or errors in [Roth].

(5) “*Roth proved Murakami’s conjecture in* [Roth] (*see also* [Woolf]).” Looks like [Woolf] has a complete proof of Roth, possibly fixing some minor errors in [Roth].

(6) “*Roth proved Murakami’s conjecture* (*see* [Woolf]).” Here [Woolf] is a definitive version of the proof, e.g. the standard monograph on the subject.

(7) “*Roth proved Murakami’s conjecture, see e.g. * [Faulkner, Fitzgerald, Frost].” The result is important enough to be cited and its validity confirmed in several books/surveys. If there ever was a controversy whether Roth’s argument is an actual proof, it was resolved in Roth’s favor. Still, the original proof may have been too long, incomplete or simply presented in an old fashioned way, or published in an inaccessible conference proceedings, so here are sources with a better or more recent exposition. Or, more likely, the author was too lazy to look for the right reference, so overcompensated with three random textbooks on the subject.

(8) “*Roth proved Murakami’s conjecture* (*see e.g.* [Faulkner, Fitzgerald, Frost]).” The result is probably classical or at least very well known. Here are books/surveys which all probably have statements and/or proofs. Neither the author nor the reader will ever bother to check.

(9) “*Roth proved Murakami’s conjecture.*^{7} Footnote 7: *See* [Mailer].” Most likely, the author never actually read [Mailer], nor has access to that paper. Or, perhaps, [Mailer] states that Roth proved the conjecture, but includes neither a proof nor a reference. The author cannot

verify the claim independently and is visibly annoyed by the ambiguity, but felt obliged to credit Roth for the benefit of the reader, or to avoid the wrath of Roth.

(10) “*Roth proved Murakami’s conjecture.*^{7} Footnote 7: *Love letter from H. Fielding to J. Austen, dated December 16, 1975.*” This means that the letter likely exists and contains the whole proof or at least an outline of the proof. The author may or may not have seen it. Googling will probably either turn up the letter or a public discussion about what’s in it, and why it is not available.

(11) “*Roth proved Murakami’s conjecture.*^{7} Footnote 7: *Personal communication.*” This means Roth has sent the author an email (or said over beer), claiming to have a proof. Or perhaps Roth’s student accidentally mentioned this while answering a question after the talk. The proof

may or may not be correct and the paper may or may not be forthcoming.

(12) “*Roth claims to have proved Murakami’s conjecture in* [Roth].” Paper [Roth] has a well known gap which was never fixed even though Roth insists on it to be fixable; the author would rather avoid going on record about this, but anything is possible after some wine at a banquet. Another possibility is that [Roth] is completely erroneous as explained elsewhere, but Roth’s

work is too famous not to be mentioned; in that case there is often a followup sentence clarifying the matter, sometimes in parentheses as in “(*see, however,* [Atwood])”. Or, perhaps, [Roth] is a 3 page note published in *Doklady Acad. Sci. USSR* back in the 1970s, containing a very brief outline of the proof, and despite considerable effort nobody has yet to give a complete proof of its Lemma 2; there wouldn’t be any followup to this sentence then, but the author would be happy to clarify things by email.

UPDATE 1. (Nov 1, 2017): There is now a video of the MSRI talk I gave based on the article.

UPDATE 2. (Mar 13, 2018): The paper was published in the *Journal of Humanistic Mathematics*. Apparently it’s now number 5 on “Most Popular Papers” list. Number 1 is “My Sets and Sexuality”, of course.

UPDATE 3. (March 4, 2021): I wrote a followup paper and a blog post titled “How to tell a good mathematical story“, with a somewhat different emphasis.