## How I chose Enumerative Combinatorics

Apologies for not writing anything for awhile. After Feb 24, the *math *part of the “*life and math*” slogan lost a bit of relevance, while the actual events were stupefying to the point when I had nothing to say about the *life *part. Now that the shock subsided, let me break the silence by telling an old personal story which is neither relevant to anything happening right now nor a lesson to anyone. Sometimes a story is just a story…

#### My field

As the readers of this blog know, I am a * Combinatorialist*. Not a “proud one”. Just “a combinatorialist”. To paraphrase a military slogan “there are many fields like this one, but this one is mine”. While I’ve been defending my field for years, writing about its struggles, and often defining it, it’s not because this field is more important than others. Rather, because it’s so frequently misunderstood.

In fact, I *have *worked in other (mostly adjacent) fields, but that was usually because I was curious. Curious what’s going on in other areas, curious if they had tools to help me with my problems. Curious if they had problems that could use my tools. I would go to seminars in other fields, read papers, travel to conferences, make friends. Occasionally this strategy paid off and I would publish something in another field. Much more often nothing ever came out of that. It was fun regardless.

Anyway, I wanted to work in combinatorics for as long as I can remember, since I was about 15 or so. There is something inherently discrete about the way I see the world, so much that having additional structure is just obstructing the view. Here is how Gian-Carlo Rota famously put it:

Combinatorics is an honest subject. […] You either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. [

Los Alamos Science, 1985]

I agree. Also, I really like to count. When prompted, I always say “*I work in Combinatorics*” even if sometimes I really don’t. But in truth, the field is much too large and not unified, so when asked to be more specific (this rarely happens) I say “*Enumerative Combinatorics*“. What follows is a short story of how I made the choice.

#### Family vacation

When I was about 18, Andrey Zelevinsky (ז״ל) introduced me and Alex Postnikov to Israel Gelfand and asked what should we be reading if we want to do combinatorics. Unlike most leading mathematicians in Russia, Gelfand had a surprisingly positive view on the subject (see e.g. his quotes here). He suggested we both read Macdonald’s book, which was translated into Russian by Zelevinsky himself just a few years earlier. The book was extremely informative but dry as a fig and left little room for creativity. I read a large chunk of it and concluded that if this is what modern combinatorics looks like, I want to have nothing to do with it. Alex had a very different impression, I think.

Next year, my extended family decided to have a vacation on a Russian “river cruise”. I remember a small passenger boat which left Moscow river terminal, navigated a succession of small rivers until it reached Volga. From there, the boat had a smooth gliding all the way to the Caspian Sea. The vacation was about three weeks of a hot Summer torture with the only relief served by mouth-watering fresh watermelons.

What made it worse, there was absolutely nothing to see. Much of the way Volga is enormously wide, sometimes as wide as the English channel. Most of the time you couldn’t even see the river banks. The cities distinguished themselves only by an assortment of Lenin statues, but were unremarkable otherwise. Volgograd was an exception with its very impressive tallest statue in Europe, roughly as tall as the Statue of Liberty when combined with its pedestal. Impressive for sure, but not worth the trip. Long story short, the whole cruise vacation was dreadfully boring.

#### One good book can make a difference

While most of my relatives occupied themselves by reading crime novels or playing cards, I was reading a math book, the only book I brought with me. This was Stanley’s *Enumerative Combinatorics* (vol. 1) whose Russian translation came out just a few months earlier. So I read it cover-to-cover, but doing only the easiest exercises just to make sure I understand what’s going on. That book changed everything…

Midway through, when I was reading about linear extensions of posets in Ch. 3 with their obvious connections to *standard Young tableaux* and the hook-length formula (which I already knew by then), I had an idea. From Macdonald’s book, I remembered the *q*-analogue of #SYT via the “*charge*“, a statistics introduced by Lascoux and Schützenberger to give a combinatorial interpretation of *Kostka polynomials*, and which works even for skew Young diagram shapes. I figured that skew shapes are generic enough, and there should be a generalization of the charge to all posets. After several long days filled with some tedious calculations by hand, I came up with both the statement and the proof of the *q-*analogue of the number of linear extensions.

I wrote the proof neatly in my notebook with a clear intent to publish my “remarkable discovery”, and continued reading. In Ch. 4, all of a sudden, I read the “*P-partition theory*” that I just invented by myself. It came with various applications and connections to other problems, and was presented so well, much nicer than I would have!

After some extreme disappointment, I learned from the notes that the P-partition theory was a large portion of Stanley’s own Ph.D. thesis, which he wrote *before I was born*. For a few hours, I did nothing but meditate, staring at the vast water surrounding me and ignoring my relatives who couldn’t care less what I was doing anyway. I was trying to think if there is a lesson in this fiasco.

It pays to be positive and self-assure, I suppose, in a way that only a teenager can be. That day I concluded that I am clearly doing something right, definitely smarter than everyone else even if born a little too late. More importantly, I figured that Enumerative Combinatorics done “Stanley-style” is really the right area for me…

#### Epilogue

I stopped thinking that I am smarter than everyone else within weeks, as soon as I learned more math. I no longer believe I was born too late. I did end up doing a lot of Enumerative Combinatorics. Much later I became Richard Stanley’s postdoc for a short time and a colleague at MIT for a long time. Even now, I continue writing papers on the numbers of linear extensions and standard Young tableaux. Occasionally, I also discuss their *q-*analogues (like in my most recent paper). I still care and it’s still the right area for me…

Some years later I realized that historically, the “charge” and Stanley’s q-statistics were not independent in a sense that both are generalizations of the *major index* by Percy MacMahon. In his revision of vol. 1, Stanley moved the P-partition theory up to Ch. 3, where it belongs IMO. In 2001, he received the Steele’s Prize for Mathematical Exposition for the book that changed everything…

## 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.

## Why you shouldn’t be too pessimistic

In our math research we make countless choices. We chose a problem to work on, decide whether its claim is true or false, what tools to use, what earlier papers to study which might prove useful, who to collaborate with, which computer experiments might be helpful, etc. Choices, choices, choices… Most our choices are private. Others are public. This blog is about wrong public choices that I made misjudging some conjectures by being overly pessimistic.

#### The meaning of conjectures

As I have written before, conjectures are crucial to the developments of mathematics and to my own work in particular. The concept itself is difficult, however. While traditionally conjectures are viewed as some sort of “*unproven laws of nature*“, that comparison is widely misleading as many conjectures are descriptive rather than quantitative. To understand this, note the stark contrast with experimental physics, as many mathematical conjectures are not particularly testable yet remain quite interesting. For example, if someone conjectures there are infinitely many *Fermat primes*, the only way to dissuade such person is to actually disprove the claim.

There is also an important social aspect of conjecture making. For a person who poses a conjecture, there is a certain clairvoyance respected by other people in the area. Predictions are never easy, especially of a precise technical nature, so some bravery or self-assuredness is required. Note that social capital is spent every time a conjecture is posed. In fact, a lot of it is lost when it’s refuted, you come out even if it’s proved relatively quickly, and you gain only if the conjecture becomes popular or proved possibly many years later. There is also a “*boy who cried wolf*” aspect for people who make too many conjectures of dubious quality — people will just tune out.

Now, for the person working on a conjecture, there is also a *betting aspect* one cannot ignore. As in, are you sure you are working in the right direction? Perhaps, the conjecture is simply *false *and you are wasting your time… I wrote about this all before in the post linked above, and the life/career implications on the solver are obvious. The success in solving a well known conjecture is often regarded much higher than a comparable result nobody asked about. This may seem unfair, and there is a bit of celebrity culture here. Thinks about it this way — two lead actors can have similar acting skills, but the one who is a star will usually attract a much larger audience…

#### Stories of conjectures

Not unlike what happens to papers and mathematical results, conjectures also have stories worth telling, even if these stories are rarely discussed at length. In fact, these “** conjecture stories**” fall into a few types. This is a little bit similar to the “

*types of scientific papers*” meme, but more detailed. Let me list a few scenarios, from the least to the most mathematically helpful:

**(1)** * Wishful thinking*. Say, you are working on a major open problem. You realize that a famous conjecture

**A**follows from a combination of three conjectures

**B**,

**C**and

**D**whose sole motivation is their applications to

**A**. Some of these smaller conjectures are beyond the existing technology in the area and cannot be checked computationally beyond a few special cases. You then declare that this to be your “

*program*” and prove a small special case of

**C**. Somebody points out that

**D**is trivially false. You shrug, replace it with a weaker

**D’**which suffices for your program but is harder to disprove. Somebody writes a long state of the art paper disproving

**D’**. You shrug again and suggest an even weaker conjecture

**D”**. Everyone else shrugs and moves on.

**(2)** ** Reconfirming long held beliefs**. You are working in a major field of study aiming to prove a famous open problem

**A**. Over the years you proved a number of special cases of

**A**and became one the leaders of the area. You are very optimistic about

**A**discussing it in numerous talks and papers. Suddenly

**A**is disproved in some esoteric situations, undermining the motivation of much of your older and ongoing work. So you propose a weaker conjecture

**A’**as a replacement for

**A**in an effort to salvage both the field and your reputation. This makes happy everyone in the area and they completely ignore the disproof of

**A**from this point on, pretending it’s completely irrelevant. Meanwhile, they replace

**A**with

**A’**in all subsequent papers and beamer talk slides.

**(3)** ** Accidental discovery.** In your ongoing work you stumble at a coincidence. It seem, all objects of a certain kind have some additional property making them “

*nice*“. You are clueless why would that be true, since being

*nice*belongs to another area

**X**. Being

*nice*is also too abstract to be checked easily on a computer. You consult a colleague working in

**X**whether this is obvious/plausible/can be proved and receive No/Yes/Maybe answers to these three questions. You are either unable to prove the property or uninterested in problem, or don’t know much about

**X**. So you mention it in the

*Final Remarks*section of your latest paper in vain hope somebody reads it. For a few years, every time you meet somebody working in

**X**you mention to them your “nice conjecture”, so much that people laugh at you behind your back.

** (4) Strong computational evidence.** You are doing computer experiments related to your work. Suddenly certain numbers appear to have an unexpectedly nice formula or a generating function. You check with OEIS and the sequence is there indeed, but not with the meaning you wanted. You use the “

*scientific method*” to get a few more terms and they indeed support your conjectural formula. Convinced this is not an instance of the “

*strong law of small numbers*“, you state the formula as a conjecture.

**(5) Being contrarian. ** You think deeply about famous conjecture

**A**. Not only your realize that there is no way one can approach

**A**in full generality, but also that it contradicts some intuition you have about the area. However,

**A**was stated by a very influential person

*N*and many people believe in

**A**proving it in a number of small special cases. You want to state a

**non-A**conjecture, but realize the inevitable PR disaster of people directly comparing you to

*N*. So you either state that you don’t believe in

**A**, or that you believe in a conjecture

**B**which is either slightly stronger or slightly weaker than

**non-A**, hoping the history will prove you right.

**(6) Being inspirational.** You think deeply about the area and realize that there is a fundamental principle underlying certain structures in your work. Formalizing this principle requires a great deal of effort and results in a conjecture

**A**. The conjecture leads to a large body of work by many people, even some counterexamples in esoteric situations, leading to various fixes such as

**A’**. But at that point

**A’**is no longer the goal but more of a direction in which people work proving a number of

**A**-related results.

Obviously, there are many other possible stories, while some stories are are a mixture of several of these.

#### Why do I care? Why now?

In the past few years I’ve been collecting references to my papers which solve or make some progress towards my conjectures and open problems, putting links to them on my research page. Turns out, over the years I made a lot of those. Even more surprisingly, there are quite a few papers which address them. Here is a small sampler, in random order:

**(1)** Scott Sheffield proved my *ribbon tilings *conjecture.

**(2)** Alex Lubotzky proved my conjecture on *random generation* of a finite group.

**(3)** Our generalized *loop-erased random walk* conjecture (joint with Igor Gorodezky) was recently proved by Heng Guo and Mark Jerrum.

**(4)** Our *Young tableau bijections* conjecture (joint with Ernesto Vallejo) was resolved by André Henriques and Joel Kamnitzer.

**(5)** My *size Ramsey numbers* conjecture led to a series of papers, and was completely resolved only recently by Nemanja Draganić, Michael Krivelevich and Rajko Nenadov.

**(6)** One of my *partition bijection* problems was resolved by Byungchan Kim.

The reason I started collecting these links is kind of interesting. I was very impressed with George Lusztig and Richard Stanley‘s lengthy writeups about their collected papers that I mentioned in this blog post. While I don’t mean to compare myself to these giants, I figured the casual reader might want to know if a conjecture in some paper had been resolved. Thus the links on my website. I recommend others also do this, as a navigational tool.

#### What gives?

Well, looks like none of my conjectures have been disproved yet. That’s a good news, I suppose. However, by going over my past research work I did discover that on three occasions when I was thinking about other people’s conjectures, I was much too negative. This is probably the result of my general inclination towards “*negative thinking*“, but each story is worth telling.

**( i)** Many years ago, I spent some time thinking about

*Babai’s conjecture*which states that there are universal constants

*C*,

*c*>0, such that for every simple group

*G*and a generating set

*S*, the diameter of the

*Cayley graph*Cay(

*G,S*) is at most

*C*(log |

*G*|)

^{c}. There has been a great deal of work on this problem, see e.g. this paper by Sean Eberhard and Urban Jezernik which has an overview and references.

Now, I was thinking about the case of the symmetric group trying to apply *arithmetic combinatorics* ideas and going nowhere. In my frustration, in a talk I gave (Galway, 2009), I wrote on the slides that “there is much less hope” to resolve Babai’s conjecture for *A _{n} *than for simple groups of Lie type or bounded rank. Now, strictly speaking that judgement was correct, but much too gloomy. Soon after, Ákos Seress and Harald Helfgott

**a remarkable quasi-polynomial upper bound in this case. To my embarrassment, they referenced my slides as a validation of the importance of their work.**

*proved*Of course, Babai’s conjecture is very far from being resolved for *A _{n}*. In fact, it is possible that the diameter is always

*O*(

*n*

^{2}). We just have no idea. For simple groups of Lie type or large rank the existing worst case diameter bounds are exponential and much too weak compared to the desired bound. As Eberhard and Jezernik amusingly wrote in the paper linked above, “

*we are still exponentially stupid*“…

**( ii)** When he was my postdoc at UCLA, Alejandro Morales told me about a curious conjecture in this paper (Conjecture 5.1), which claimed that the number of certain nonsingular matrices over the finite field

**F**

*is polynomial in*

_{q}*q*with positive coefficients. He and coauthors proved the conjecture is some special cases, but it was wide open in full generality.

Now, I thought about this type of problems before and was very skeptical. I spent a few days working on the problem to see if any of my tools can disprove it, and failed miserably. But in my stubbornness I remained negative and suggested to Alejandro that he should drop the problem, or at least stop trying to prove rather than disprove the conjecture. I was wrong to do that.

Luckily, Alejandro ignored my suggestion and soon after ** proved **the polynomial part of the conjecture together with Joel Lewis. Their proof is quite elegant and uses certain recurrences coming from the

*rook theory*. These recurrences also allow a fast computation of these polynomials. Consequently, the authors made a number of computer experiments and

*the positivity of coefficients part of the conjecture. So the moral is not to be so negative. Sometimes you need to prove a positive result first before moving to the dark side.*

**disproved****( iii)** The final story is about the beautiful

*Benjamini conjecture*in probabilistic combinatorics. Roughly speaking, it says that for every finite vertex transitive graph

*G*on

*n*vertices and diameter

*O*(

*n*/log

*n*) the critical percolation constant

*p*

_{c}<1. More precisely, the conjecture claims that there is

*p*<1-ε, such that a

*p*-percolation on

*G*has a connected component of size >

*n*/2 with probability at least δ, where constants ε, δ>0 depend on the constant implied by the

*O*(*) notation, but not on

*n*. Here by “

*p*-percolation” we mean a random subgraph of

*G*with probability

*p*of keeping and 1-

*p*of deleting an edge, independently for all edges of

*G*.

Now, Itai Benjamini is a fantastic conjecture maker of the best kind, whose conjectures are both insightful and well motivated. Despite the somewhat technical claim, this conjecture is quite remarkable as it suggested a finite version of the “*p*_{c}<1″ phenomenon for infinite groups of superlinear growth. The latter is the famous *Benjamini–Schramm conjecture* (1996), which was recently ** proved **in a remarkable breakthrough by Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi, Franco Severo and Ariel Yadin. While I always believed in that conjecture and even proved a tiny special case of it, finite versions tend to be much harder in my experience.

In any event, I thought a bit about the Benjamini conjecture and talked to Itai about it. He convinced me to work on it. Together with Chis Malon, we wrote a paper proving the claim for some Cayley graphs of abelian and some more general classes of groups. Despite our best efforts, we could not prove the conjecture even for Cayley graphs of abelian groups in full generality. Benjamini noted that the conjecture is tight for products of two cyclic groups, but that justification did not sit well with me. There seemed to be no obvious way to prove the conjecture even for the Cayley graph of *S _{n}* generated by a transposition and a long cycle, despite the very small

*O*(

*n*

^{2}) diameter. So we wrote in the introduction: “In this paper we present a number of positive results toward this unexpected, and, perhaps, overly optimistic conjecture.”

As it turns out, it was us who were being overly pessimistic, even if we never actually stated that we believe the conjecture is false. Most recently, in an amazing development, Tom Hutchcroft and Matthew Tointon **proved **a slightly weaker version of the conjecture by adapting the methods of Duminil-Copin et al. They assume the *O*(*n*/(log* n*)^{c}) upper bound on the diameter which they prove is sufficient, for some universal constant *c*>1. They also extend our approach with Malon to prove the conjecture for all Cayley graphs of abelian groups. So while the Benjamini conjecture is not completely resolved, my objections to it are no longer valid.

#### Final words on this

All in all, it looks like I was never formally wrong even if I was a little dour occasionally (*Yay*!?). Turns out, some conjectures are actually true or at least likely to hold. While I continue to maintain that not enough effort is spent on trying to disprove the conjectures, it is very exciting when they are proved. * Congratulations* to Harald, Alejandro, Joel, Tom and Matthew, and posthumous congratulations to Ákos for their terrific achievements!

## The Unity of Combinatorics

I just finished my very first * book review* for the

*Notices of the AMS*. The authors are Ezra Brown and Richard Guy, and the book title is the same as the blog post. I had mixed feelings when I accepted the assignment to write this. I knew this would take a lot of work (I was wrong — it took a

*huge*amount of work). But the reason I accepted is because I strongly suspected that there is

*“unity of combinatorics”, so I wanted to be proved wrong. Here is how the book begins:*

**no**One reason why Combinatorics has been slow to become accepted as part of mainstream Mathematics is the common belief that it consists of a bag of isolated tricks, a number of areas: [very long list – IP] with little or no connection between them. We shall see that they have numerous threads weaving them together into a beautifully patterned tapestry.

Having read the book, I continue to maintain that there is no unity. The book review became a balancing act — how do you write a somewhat positive review if you don’t believe into the mission of the book? Here is the first paragraph of the portion of the review where I touch upon themes very familiar to readers of this blog:

As I see it, the whole idea of combinatorics as a “

slow to become accepted” field feels like a throwback to the long forgotten era. This attitude was unfair but reasonably common back in 1970, outright insulting and relatively uncommon in 1995, and was utterly preposterous in 2020.

After a lengthy explanation I conclude:

To finish this line of thought, it gives me no pleasure to conclude that the case for the unity of combinatorics is too weak to be taken seriously. Perhaps, the unity of mathematics as a whole is an easier claim to establish, as evident from [Stanley’s] quotes. On the other hand, this lack of unity is not necessarily a bad thing, as we would be amiss without the rich diversity of cultures, languages, open problems, tools and applications of different areas.

Enjoy the full review! And please comment on the post with your own views on this alleged “unity”.

P.S. A large part of the book is freely downloadable. I made this website for the curious reader.

**Remark** (ADDED April 17, 2021)

Ezra “Bud” Brown gave a talk on the book illustrating many of the connections I discuss in the review. This was at a memorial conference celebrating Richard Guy’s legacy. I was not aware of the video until now. Watch the whole talk.

## 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.