Math News July 20th, 2009
Patrick Stein

An aggregation of some of the math blogs that I follow.

Math News (26 - 50 of about 3111) (xml) (Feedlist)

Kloosterman pathsKowalski
(29.10.2014 17:09h)

It was almost twenty years ago that I started drawing and looking at the graphs of Kloosterman sums at least, that’s a likely date; I don’t remember when it began, but I put some drawings as whimsical illustrations — see, e.g., page 26 — in my PhD thesis to enliven it, and that was around 1997–1998 . Kloosterman path This year, by a lucky coincidence, I can finally say something definite about these graphs, and indeed something quite interesting. What happened is that Will Sawin was visiting Switzerland in June to talk to Philippe Michel and me about some very ... [Link]

The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjectureTerence Tao
(28.10.2014 15:22h)

I’ve just uploaded to the arXiv my paper “The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture“. As the title suggests, this paper links together the Elliott-Halberstam conjecture from sieve theory with the conjecture of Vinogradov concerning the least quadratic nonresidue of a prime . Vinogradov established the bound and conjectured that for any fixed 0}' title='{\varepsilon>0}' class='latex' />. Unconditionally, the best result so far up to logarithmic factors that holds for all primes is by Burgess, who showed that for any fixed 0}' title='{\varepsilon>0}' class='latex' />. See this previous post for a proof of these bounds. In this ... [Link]

(27.10.2014 17:38h)

Yesterday my younger son was playing dice; the game involved throwing 6 dices simultaneously, and he threw a complete set 1, 2, 3, 4, 5, 6, twice in a row! Is that a millenial-style coincidence worth cosmic pronouncements? Actually, not that much: since the dices are indistinguishable, the probability of a single throw of this type is so about one and a half percent. And for two, assuming independence, we get a probability or a bit more than one chance in five throusand. This is small, but not extraordinarily so. The dices are thrown from a cup, so the independence ... [Link]

A Banach algebra proof of the prime number theoremTerence Tao
(25.10.2014 18:52h)

The prime number theorem can be expressed as the assertion as , where is the von Mangoldt function. It is a basic result in analytic number theory, but requires a bit of effort to prove. One “elementary” proof of this theorem proceeds through the Selberg symmetry formula where the second von Mangoldt function is defined by the formula or equivalently We are avoiding the use of the symbol here to denote Dirichlet convolution, as we will need this symbol to denote ordinary convolution shortly. For the convenience of the reader, we give a proof of the Selberg symmetry formula below ... [Link]

MaplesIzabella Laba
(19.10.2014 18:31h)


ICM2014 — Bhargava, Gentry, Sandersgowers
(19.10.2014 06:22h)

On my last day at the ICM I ended up going to fewer talks. As on the previous two days the first plenary lecture was not to be missed — it was Maryam Mirzakhani — so despite my mounting tiredness I set my alarm appropriately. I was a little surprised when I got there by just how empty it was, until eventually I saw that on the screens at the front it said that the lecture was cancelled because of her Fields medallist’s lecture the following Tuesday. I belonged to the small minority that had not noticed this, partly because ... [Link]

ICM2014 — Kollár, Conlon, Katz, Krivelevich, Milnorgowers
(19.10.2014 06:22h)

As the ICM recedes further into the past, these posts start to feel less and less fresh. I’ve had an enforced break from them as over the course of three days I drove my family from the south of France back to Cambridge. So I think I’ll try to do what I originally intended to do with all these posts, and be quite a lot briefer about each talk. As I’ve already mentioned, Day 3 started with Jim Arthur’s excellent lecture on the Langlands programme. In a comment on that post, somebody questioned my use of “Jim” rather than “James”. ... [Link]

ICM2014 — Jim Arthur plenary lecturegowers
(19.10.2014 06:22h)

The main other thing I did on day two of the congress was go to a reception in the evening hosted by the French Embassy. It was less formal than that makes it sound, and as I circulated I met a number of people I hadn’t seen for quite a while, as well as others I had got to know at the congress itself. The French ambassador, who was disconcertingly young looking, gave a speech, as did Artur Avila as you know, Avila, like Ngo four years ago, is French , and one other person, whose name I’ve annoyingly forgotten. ... [Link]

ICM2014 — Barak, Guralnick, Browngowers
(19.10.2014 06:22h)

Here’s a little puzzle to get this post started. Of the fourteen 21st-century Fields medallists if you count Perelman , seven — Lafforgue, Voevodsky, Tao, Werner, Smirnov, Avila and Mirzakhani — have something interesting in common that the others lack. What is this property? That’s a fairly easy question, so let’s follow it up with another one: how surprised should we be about this? Is there unconscious bias towards mathematicians with this property? Of this year’s 21 plenary lecturers, the only one with the property was Mirzakhani, and out of the 20 plenary lecturers in 2010, the only one with ... [Link]

ICM2014 — Emmanuel Candès plenary lecturegowers
(19.10.2014 06:22h)

If you are a mathematical researcher, do you ever stop to ask yourself what the point is of all your research? Do you worry that the world could get along just fine without it? One person who doesn’t lose any sleep over doubts like this is Emmanuel Candès, who gave the second plenary lecture I went to. He began by talking a little about the motivation for the kinds of problems he was going to discuss, which one could summarize as follows: his research is worthwhile because it helps save the lives of children. More precisely, it used to be ... [Link]

ICM2014 — Ian Agol plenary lecturegowers
(19.10.2014 06:22h)

On the second day of the congress I hauled myself out of bed in time, I hoped, to have a shower and find some breakfast before the first plenary lecture of the congress started at 9am. The previous day in the evening I had chanced upon a large underground shopping mall directly underneath the conference centre, so I thought I’d see if I could find some kind of café there. However, at 8:30 in the morning it was more or less deserted, and I found myself wandering down very long empty passages, constantly looking at my watch and worrying that ... [Link]

ICM2014 — Khot laudatiogowers
(19.10.2014 06:22h)

After McMullen’s laudatio on Mirzakhani, it was time for Sanjeev Arora to talk about the work of the Nevanlinna prize winner Subhash Khot. It was also the time that a significant proportion of the audience decided that enough was enough and left the room. The same thing happened in Hyderabad four years ago, and on both occasions I was fairly shocked: I think it shows a striking disrespect, not so much for the speaker and prizewinner, though there is that aspect too, as for theoretical computer science in general. It seems to say, “Right, that’s the interesting prizes over — ... [Link]

ICM2014 — Mirzakhani laudatiogowers
(19.10.2014 06:22h)

I’m going to try the same exercise with Curt McMullen’s talk about Mirzakhani’s work that I did with Ofer Zeitouni’s about Hairer: that is, I’ll begin by seeing what I can remember if I don’t look at my notes. However, I remember disoncertingly little, and what I do remember is somewhat impressionistic. The most concrete thing I remember without being 100% sure I’ve got it right is that one of Mirzakhani’s major results concerns counting closed geodesics in Riemann surfaces. A geodesic is roughly speaking a curve that feels like a straight line to an inhabitant of the surface. Another ... [Link]

ICM2014 — Hairer laudatiogowers
(19.10.2014 06:22h)

I haven’t kept up anything like the frequency of posts at this ICM that I managed at the last one. There are at least three reasons for this. One is that I was in the middle of writing up a result, so I devoted some of my rare free moments to that. Another is that the schedule was so full of good talks that I hardly skipped any sessions. And the third is that on the last day I was taken ill: I won’t go into too much detail, but let’s say that what I had sort of rhymed with ... [Link]

ICM2014 — Bhargava laudatiogowers
(19.10.2014 06:22h)

I ended up writing more than I expected to about Avila. I’ll try not to fall into the same trap with Bhargava, not because there isn’t lots to write about him, but simply because if I keep writing at this length then by the time I get on to some of the talks I’ve been to subsequently I’ll have forgotten about them. Dick Gross also gave an excellent talk. He began with some of the basic theory of binary quadratic forms over the integers, that is, expressions of the form . One assumes that they are primitive meaning that , ... [Link]

ICM2014 — Avila laudatiogowers
(19.10.2014 06:22h)

As I said in my previous post, I don’t think I’m going to try all that hard to explain the work of the prizewinners, since it has been very well explained in other places except that much more attention has gone to the Fields medallists than to the Nevanlinna prize winner — maybe I’ll try to redress the balance a little bit there . Instead, I’d just like to mention a few things that I found interesting or amusing during the laudationes. The first one was an excellent talk by Etienne Ghys on the work of Artur Avila. The only ... [Link]

ICM2014 — opening ceremonygowers
(19.10.2014 06:22h)

I’d forgotten just how full the first day of an ICM is. First, you need to turn up early for the opening ceremony, so you end up sitting around for an hour and half or so before it even starts. Then there’s the ceremony itself, which lasts a couple of hours. Then in the afternoon you have talks about the four Fields Medallists and the Nevanlinna Prize winner, with virtually no breaks. Then after a massive ten minutes, the Nevanlinna Prize winner talks about his in this case own work, about which you have just heard, but in a bit ... [Link]

ICM2014 — first impressionsgowers
(19.10.2014 06:22h)

I’m writing this at 6:22am in my hotel room in Seoul, which is in a hotel that is right next to the conference centre, to the point where you don’t have to go out of doors to get from one to the other. I’ve just had a good night’s sleep, even though in French time which is what I was used to until the day before yesterday — if that concept still means anything my entire night has been during the day time, and now is about the time I’d be thinking of going to bed. I feel a bit ... [Link]

ICM2014 — introductory postgowers
(19.10.2014 06:22h)

Four years ago I blogged from the ICM in Hyderabad. The posts are amongst the most popular I have written — my statistics show that some of them are still being read quite regularly even now. Right now I’m sitting in Charles de Gaulle airport waiting to board a plane to Seoul, where I will be attending this year’s ICM, or rather, as I did last time, attending the first half of it. I’m not sure I’ll have the time or energy to write quite as much about ICM2014 as I did about ICM2010, but I’ll do what I can. ... [Link]

(19.10.2014 06:22h)

The title of this post is a nod to Terry Tao’s four mini-polymath discussions, in which IMO questions were solved collaboratively online. As the beginning of what I hope will be a long exercise in gathering data about how humans solve these kinds of problems, I decided to have a go at one of this year’s IMO problems, with the idea of writing down my thoughts as I went along. Because I was doing that and doing it directly into a LaTeX file rather than using paper and pen , I took quite a long time to solve the problem: ... [Link]

ECM2016 — your chance to influence the programmegowers
(19.10.2014 06:22h)

UPDATE: I HAVE NOW GONE BACK TO MODERATING COMMENTS ONLY IF THEY ARE FROM PEOPLE WHO HAVE NOT HAD A COMMENT ACCEPTED IN THE PAST. SO IF YOU HAVE A SUGGESTION TO MAKE FOR AN ECM2016 SPEAKER, PLEASE EMAIL A MEMBER OF THE COMMITTEE DIRECTLY RATHER THAN COMMENTING HERE. Just before I start this post, let me say that I do still intend to write a couple of follow-up posts to my previous one about journal prices. But I’ve been busy with a number of other things, so it may still take a little while. This post is about the ... [Link]

A few quick announcementsScott
(14.10.2014 20:06h)

I gave a new survey talk at Yale, entitled “When Exactly Do Quantum Computers Provide a Speedup?” Here are the PowerPoint slides. Thanks so much to Rob Schoelkopf for inviting me, and to everyone else at Yale for an awesome visit. Aephraim Steinberg asks me to announce that the call for nominations for the 2015 John Stewart Bell Prize is now available. Ronitt Rubinfeld asks me to remind people that the STOC’2015 submission deadline is November 4. Here’s the call for papers. Likewise, Jeff Kinne asks me to remind people that the Complexity’2015 submission deadline is November 26. Here’s the ... [Link]

Diversity and mathematicsIzabella Laba
(14.10.2014 19:42h)

Mother Jones, last year: According to a new psychology paper, our political passions can even undermine our very basic reasoning skills. More specifically, the study finds that people who are otherwise very good at math may totally flunk a problem that they would otherwise probably be able to solve, simply because giving the right answer goes against their political beliefs. I was reminded of it while reading the article “Does Diversity Trump Ability? An Example of the Misuse of Mathematics in the Social Sciences” in the Notices of the AMS. The author, Abigail Thompson, takes on a well known and ... [Link]

Additive limitsTerence Tao
(13.10.2014 06:25h)


A lecture by NogaGil Kalai
(10.10.2014 00:15h)

Noga with Uri Feige among various other heroes A few weeks ago I devoted a post to the 240-summit conference for Péter Frankl, Zoltán Füredi, Ervin Győri and János Pach, and today I will bring you the slides of Noga Alon’s lecture in the meeting. Noga is my genious twin academic brother – we both were graduate students under the supervision of Micha A. Perles in the same years and we both went to MIT as postocs in fall 1983. The lecture starts with briefly mentioning four results by the birthday boys related to combinatorics and geometry and continues with ... [Link]

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