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I was greatly saddened to learn that John Conway died yesterday from COVID-19, aged 82.

My own mathematical areas of expertise are somewhat far from Conway’s; I have played for instance with finite simple groups on occasion, but have not studied his work on moonshine and the monster group.  But I have certainly encountered his results every so often in surprising contexts; most recently, when working on the Collatz conjecture, I looked into Conway’s wonderfully preposterous FRACTRAN language, which can encode any Turing machine as an iteration of a Collatz-type map, showing in particular that there are generalisations of the Collatz conjecture that are undecidable in axiomatic frameworks such as ZFC.  [EDIT: also, my belief that the Navier-Stokes equations admit solutions that blow up in finite time is also highly influenced by the ability of Conway’s game of life to generate self-replicating “von Neumann machines“.]

I first met John as an incoming graduate student in Princeton in 1992; indeed, a talk he gave, on “Extreme proofs” (proofs that are in some sense “extreme points” in the “convex hull” of all proofs of a given result), may well have been the first research-level talk I ever attended, and one that set a high standard for all the subsequent talks I went to, with Conway’s ability to tease out deep and interesting mathematics from seemingly frivolous questions making a particular impact on me.  (Some version of this talk eventually became this paper of Conway and Shipman many years later.)

Conway was fond of hanging out in the Princeton graduate lounge at the time of my studies there, often tinkering with some game or device, and often enlisting any nearby graduate students to assist him with some experiment or other.  I have a vague memory of being drafted into holding various lengths of cloth with several other students in order to compute some element of a braid group; on another occasion he challenged me to a board game he recently invented (now known as “Phutball“) with Elwyn Berlekamp and Richard Guy (who, by sad coincidence, both also passed away in the last 12 months).  I still remember being repeatedly obliterated in that game, which was a healthy and needed lesson in humility for me (and several of my fellow graduate students) at the time.  I also recall Conway spending several weeks trying to construct a strange periscope-type device to try to help him visualize four-dimensional objects by giving his eyes vertical parallax in addition to the usual horizontal parallax, although he later told me that the only thing the device made him experience was a headache.

About ten years ago we ran into each other at some large mathematics conference, and lacking any other plans, we had a pleasant dinner together at the conference hotel.  We talked a little bit of math, but mostly the conversation was philosophical.  I regrettably do not remember precisely what we discussed, but it was very refreshing and stimulating to have an extremely frank and heartfelt interaction with someone with Conway’s level of insight and intellectual clarity.

Conway was arguably an extreme point in the convex hull of all mathematicians.  He will very much be missed.

My student, Jaume de Dios, has set up a web site to collect upcoming mathematics seminars from any institution that are open online.  (For instance, it has a talk that I will be giving in an hour.)   There is a form for adding further talks to the site; please feel free to contribute (or make other suggestions) in order to make the seminar list more useful.

UPDATE: Here are some other lists of mathematical seminars online:

Perhaps further links of this type could be added in the comments.  It would perhaps make sense to somehow unify these lists into a single one that can be updated through crowdsourcing.

Just a short post to note that this year’s Abel prize has been awarded jointly to Hillel Furstenberg and Grigory Margulis for “for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics”.  I was not involved in the decision making process of the Abel committee this year, but I certainly feel that the contributions of both mathematicians are worthy of the prize.  Certainly both mathematicians have influenced my own work (for instance, Furstenberg’s proof of Szemeredi’s theorem ended up being a key influence in my result with Ben Green that the primes contain arbitrarily long arithmetic progressions); see for instance these blog posts mentioning Furstenberg, and these blog posts mentioning Margulis.

As part of social distancing efforts to slow down the spread of the novel coronavirus, several universities have now transitioned, or begun transitioning, to online teaching models.  (My home university of UCLA has not yet done so, but is certainly considering the option.  UPDATE: we are transitioning.)  As a consequence, I thought it might be an appropriate time to start a discussion on the pros and cons of various technologies for giving talks and lectures online, particularly in the context of mathematical talks where there may be special considerations coming for instance for the need to do mathematical computations on a blackboard or equivalent.  My own institution is for instance recommending the use of Zoom for lectures and Respondus for giving finals, and has a limited number of classrooms set up for high quality video and audio casting, as well as a platform for discussion forums and course materials for each class.  For smaller meetings, such as one-on-one meetings with graduate students, one can of course improvise using off-the-shelf tools such as Skype.  I would be interested in knowing what other options are available and what success lecturers have had with them.

The same goes for giving mathematical talks.  I learned recently (from Jordan Ellenberg) that Rachel Preis has recently launched a “virtual math seminar on open conjectures in number theory in arithmetic geometry” (VaNTAGe) that is run using the BlueJeans platform.  And for many years there has been a regular joint math seminar between UC Berkeley, U. Paris-Nord, U. Zurich, and U. Bonn (see e.g., this calendar), and nowadays many mathematical institutes stream their talks or at least videotape them to place them online later.  Our own department does not have a dedicated lecture hall for videocasting, so I would be interested in knowing of any successful ways to improvise such casting with more portable technology. (Skype in principle could work here, but I have found this to be clunky even for smaller meetings involving just a handful of partcipants.)

EDIT: in addition to lectures and talks, it would also be topical to discuss online options for office hours, midterms, and final exams.

The National Academies of Sciences, Engineering, and Medicine have initiated a project on “Illustrating the Impact of the Mathematical Sciences“, in which various media will be produced to showcase how mathematics impacts the modern world.  (I am serving on the committee for creating this media, which has been an interesting experience; the first time for instance that I have had to seriously interact with graphic designers.)  One of the first products is a “webinar” series on the ten topics our committee have chosen to focus on, that is currently running weekly on Tuesdays.  Last week I moderated the first such webinar, titled “From Solving to Seeing”, in which Profs. Gunther Uhlmann and Anna Gilbert presented ways in which inverse problems, compressed sensing, and other modern mathematical techniques have been used to obtain images (such as MRI images) that would not otherwise be accessible.  Next week I will moderate another webinar, titled “Abstract Geometry, Concrete Impact”, in which Profs. Katherine Stange and Jordan Ellenberg will discuss how modern abstract geometries are used in modern applications such as cryptography.  The full list of webinars and the latest information on the speakers can be found at this website.  (Past webinars can be viewed directly from the web site; live webinars require a (free) registration, and offer the ability to submit text questions to the speakers via the moderator.)

We are currently in the process of designing posters (and possibly even a more interactive online resource) for each of the ten topics listed in the webinars; hopefully these will be available in a few months.

I have just returned from Basel, Switzerland, on the occasion of the awarding of the 2019 Ostrowski prize to Assaf Naor.  I was invited to give the laudatio for Assaf’s work, which I have uploaded here.  I also gave a public lecture (intended at the high school student level) at the University of Basel entitled “The Notorious Collatz conjecture”; I have uploaded the slides for that here. (Note that the slides here are somewhat unpolished as I was not initially planning to make them public until I was recently requested to do so.  In particular I do not have full attribution for some of the images used in the slides.)

Basel has historically been home to a number of very prominent mathematicians, most notably Jacob Bernoulli, whose headstone I saw at the Basel Minster,

and also Leonhard Euler, for which I could not find a formal memorial, but I did at least see a hotel bearing his name:

I just heard the news that Louis Nirenberg died a few days ago, aged 94.  Nirenberg made a vast number of contributions to analysis and PDE (and his work has come up repeatedly on my own blog); I wrote about his beautiful moving planes argument with Gidas and Ni to establish symmetry of ground states in this post on the occasion of him receiving the Chern medal, and on how his extremely useful interpolation inequality with Gagliardo (generalising a previous inequality of Ladyzhenskaya) can be viewed as an amplification of the usual Sobolev inequality in this post.  Another fundamentally useful inequality of Nirenberg is the John-Nirenberg inequality established with Fritz John: if a (locally integrable) function $f: {\bf R} \to {\bf R}$ (which for simplicity of exposition we place in one dimension) obeys the bounded mean oscillation property

$\displaystyle \frac{1}{|I|} \int_I |f(x)-f_I|\ dx \leq A \quad (1)$

for all intervals $I$, where $f_I := \frac{1}{|I|} \int_I f$ is the average value of $f$ on $I$, then one has exponentially good large deviation estimates

$\displaystyle \frac{1}{|I|} |\{ x \in I: |f(x)-f_I| \geq \lambda A \}| \leq \exp( - c \lambda ) \quad (2)$

for all $\lambda>0$ and some absolute constant $c$.  This can be compared with Markov’s inequality, which only gives the far weaker decay

$\displaystyle \frac{1}{|I|} |\{ x \in I: |f(x)-f_I| \geq \lambda A \}| \leq \frac{1}{\lambda}. \quad (3)$

The point is that (1) is assumed to hold not just for a given interval $I$, but also all subintervals of $I$, and this is a much more powerful hypothesis, allowing one for instance to use the standard Calderon-Zygmund technique of stopping time arguments to “amplify” (3) to (2).  Basically, for any given interval $I$, one can use (1) and repeated halving of the interval $I$ until significant deviation from the mean is encountered to locate some disjoint exceptional subintervals $J$ where $f_J$ deviates from $f_I$ by $O(A)$, with the total measure of the $J$ being a small fraction of that of $I$ (thanks to a variant of (3)), and with $f$ staying within $O(A)$ of $f_I$ at almost every point of $I$ outside of these exceptional intervals.  One can then establish (2) by an induction on $\lambda$.  (There are other proofs of this inequality also, e.g., one can use Bellman functions, as discussed in this old set of notes of mine.)   Informally, the John-Nirenberg inequality asserts that functions of bounded mean oscillation are “almost as good” as bounded functions, in that they almost always stay within a bounded distance from their mean, and in fact the space BMO of functions of bounded mean oscillation ends up being superior to the space $L^\infty$ of bounded measurable functions for many harmonic analysis purposes (among other things, the space is more stable with respect to singular integral operators).

I met Louis a few times in my career; even in his later years when he was wheelchair-bound, he would often come to conferences and talks, and ask very insightful questions at the end of the lecture (even when it looked like he was asleep during much of the actual talk!).  I have a vague memory of him asking me some questions in one of the early talks I gave as a postdoc; I unfortunately do not remember exactly what the topic was (some sort of PDE, I think), but I was struck by how kindly the questions were posed, and how patiently he would listen to my excited chattering about my own work.

Just a short post to announce that nominations are now open for the Maryam Mirzakhani New Frontiers Prize, which is a newly announced annual \$50,000 award from the Breakthrough Prize Foundation presented to early-career, women mathematicians who have completed their PhDs within the past two years, and recognizes outstanding research achievement.  (I will be serving on the prize committee.)  Nominations for this (and other breakthrough prizes) can be made at this page.

In the fall quarter (starting Sep 27) I will be teaching a graduate course on analytic prime number theory.  This will be similar to a graduate course I taught in 2015, and in particular will reuse several of the lecture notes from that course, though it will also incorporate some new material (and omit some material covered in the previous course, to compensate).  I anticipate covering the following topics:

1. Elementary multiplicative number theory
2. Complex-analytic multiplicative number theory
3. The entropy decrement argument
4. Bounds for exponential sums
5. Zero density theorems
6. Halasz’s theorem and the Matomaki-Radziwill theorem
7. The circle method
8. (If time permits) Chowla’s conjecture and the Erdos discrepancy problem

Lecture notes for topics 3, 6, and 8 will be forthcoming.

The AMS and MAA have recently published (and made available online) a collection of essays entitled “Living Proof: Stories of Resilience Along the Mathematical Journey”.  Each author contributes a story of how they encountered some internal or external difficulty in advancing their mathematical career, and how they were able to deal with such difficulties.  I myself have contributed one of these essays; I was initially somewhat surprised when I was approached for a contribution, as my career trajectory has been somewhat of an outlier, and I have been very fortunate to not experience to the same extent many of the obstacles that other contributors write about in this text.    Nevertheless there was a turning point in my career that I write about here during my graduate years, when I found that the improvised and poorly disciplined study habits that were able to get me into graduate school due to an over-reliance on raw mathematical ability were completely inadequate to handle the graduate qualifying exam.  With a combination of an astute advisor and some sheer luck, I was able to pass the exam and finally develop a more sustainable approach to learning and doing mathematics, but it could easily have gone quite differently.  (My 20 25-year old writeup of this examination, complete with spelling errors, may be found here.)