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

Math News (26 - 50 of about 3129) (xml) (Feedlist)

## ICERM postdoctoral positionsIzabella Laba

(16.11.2014 22:17h)

I’m co-organizing the program Dimension and Dynamics at the Institute for Computational and Experimental Research in Mathematics in Spring 2016. Yes, this means that I hope to participate in the program. Details to follow when they are finalized. ICERM has several postdoctoral positions associated with the program, one for the full academic year and four for the semester. Applications should be submitted via MathJobs. There is also funding for program visitors and workshop participants; the ICERM webpage has more details on that. The program covers various aspects of dimension theory and dynamics, from ergodic theory to hyperbolic dynamics to computation. ... [Link]

## What does the NSA think of academic cryptographers? Recently-declassified document provides cluesScott

(16.11.2014 15:11h)

Brighten Godfrey was one of my officemates when we were grad students at Berkeley. He’s now a highly-successful computer networking professor at the University of Illinois Urbana-Champaign, where he studies the wonderful question of how we could get the latency of the Internet down to the physical limit imposed by the finiteness of the speed of light. Right now, we’re away from that limit by a factor of about 50. Last week, Brighten brought to my attention a remarkable document: a 1994 issue of CryptoLog, an NSA internal newsletter, which was recently declassified with a few redactions. The most interesting ... [Link]

## Der QuantencomputerScott

(14.11.2014 15:16h)

Those of you who read German I don’t might enjoy a joint interview of me and Seth Lloyd about quantum computing, which was conducted in Seth’s office by the journalist Christian Meier, and published in the Swiss newspaper Neue Zürcher Zeitung. Even if you don’t read German, you can just feed the interview into Google Translate, like I did. While the interview covers ground that will be forehead-bangingly familiar to regular readers of this blog, I’m happy with how it turned out; even the slightly-garbled Google Translate output is much better than most quantum computing articles in the English-language press. ... [Link]

## Analytic prime number theory254A announcement

(14.11.2014 12:02h)

In the winter quarter starting January 5 I will be teaching a graduate topics course entitled “An introduction to analytic prime number theory“. As the name suggests, this is a course covering many of the analytic number theory techniques used to study the distribution of the prime numbers . I will list the topics I intend to cover in this course below the fold. As with my previous courses, I will place lecture notes online on my blog in advance of the physical lectures. The type of results about primes that one aspires to prove here is well captured by ... [Link]

## Grothendieck 1928-2014Scott Morrison

(13.11.2014 23:57h)

http://www.liberation.fr/sciences/2014/1~ [Link]

## Which of the following three groups is trivialTest your intuition 24

(13.11.2014 05:54h)

Martin Bridson We have three finitely presented groups A is generated by two generators a and b and one relation B is generated by three generators a, b, c and three relations , . C is generated by four generators a, b, c, d and four relations , , and . Test your intuition: which of the groups A, B or C is trivial Please do not answer this poll if you already knew the answer Take Our Poll function d,c,j {if !d.getElementById j {var pd=d.createElement c ,s;pd.id=j;pd.src='http://s1.wp.com/~ c [0];s.parentNode.insertBefore pd,s ;} else if typeof jQuery !=='undefined' jQuery d.body .trigger ... [Link]

## School Starts at HUJIGil Kalai

(12.11.2014 03:50h)

We are now starting the third week of the academic year at HUJI. As usual, things are very hectic, a lot of activities in the mathematics department, in our sister CS department, around in the campus, and in our combinatorics group. A lot is also happening in other universities around. This semester I am teaching a course on “Social Choice and some Topics from Cooperative Game Theory” in our Federmann Center for the Study of Rationality. I will probably create a page for the course in the near future. During the summer we ran an informal multi-university research activity on ... [Link]

## Interstellar’s dangling wormholesScott

(10.11.2014 20:49h)

Update Nov. 15 : A third of my confusions addressed by reading Kip Thorne’s book! Details at the bottom of this post. On Saturday Dana and I saw Interstellar, the sci-fi blockbuster co-produced by the famous theoretical physicist Kip Thorne who told me about his work on this movie when I met him eight years ago . We had the rare privilege of seeing the movie on the same day that we got to hang out with a real astronaut, Dan Barry, who flew three shuttle missions and did four spacewalks in the 1990s. As the end result of a ... [Link]

## The torsion on CM elliptic curves over prime degree number fieldsJim Stankewicz

(07.11.2014 14:31h)

It’s good to be back! This weekend I’m going to Paris to give a talk in the London-Paris Number Theory seminar so I’m going to give a preview of that talk, based on joint work with Pete Clark and Abbey Bourdon. We will post this onto the arxiv soon. Of course the paper is about the torsion in certain kinds of elliptic curves. The landmark result in this direction is of course Mazur’s theorem, stating that if is any elliptic curve over then the torsion subgroup of is isomorphic to an element of the following: Moreover each group occurs as ... [Link]

## Discretised wave equationsTerence Tao

(06.11.2014 12:40h)

The wave equation is usually expressed in the form where is a function of both time and space , with being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain with some other manifold and replacing the Laplacian with the Laplace-Beltrami operator or adding lower order terms such as a potential, or a coupling with a magnetic field . But for sake of discussion let us work with the classical wave equation on . We will work formally in this post, being unconcerned with issues of convergence, justifying interchange ... [Link]

## Sums of kth powersgowers

(05.11.2014 04:18h)

Recently I was giving a talk at a school in London and had cause to think about sums of kth powers, because I wanted to show people the difference between unenlightening proofs and enlightening ones. My brief was to try to give them some idea of what proofs are and why they are worth bothering about. On the train down, I realized that there is a very simple general argument that can be used to work out formulae for sums of kth powers. After a brief look online, I haven’t found precisely this method, though I’ve found plenty of methods ... [Link]

## A not so well-known theorem of Fouvry, and a challengeKowalski

(04.11.2014 19:17h)

A few weeks ago, as already mentioned, I was in Oxford for the LMS-CMI summer school on bounded gaps between primes. My mini-course on this occasion was devoted to the ideas and results underlying Zhang’s original approach, based on expanding the exponent of distribution of primes in arithmetic progressions to large moduli. In the first lecture, I mentioned a result of Fouvry as a motivation behind the study of other arithmetic functions in arithmetic progressions: roughly speaking, if one can prove that the exponent of distribution of the divisor functions ,…, is strictly larger than , then the same holds ... [Link]

## The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjectureTerence Tao

(01.11.2014 19:23h)

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]

## A trivial generalisation of Cayley’s theoremTerence Tao

(01.11.2014 19:23h)

One of the first basic theorems in group theory is Cayley’s theorem, which links abstract finite groups with concrete finite groups otherwise known as permutation groups . Theorem 1 Cayley’s theorem Let be a group of some finite order . Then is isomorphic to a subgroup of the symmetric group on elements . Furthermore, this subgroup is simply transitive: given two elements of , there is precisely one element of such that . One can therefore think of as a sort of “universal” group that contains up to isomorphism all the possible groups of order . Proof: The group acts ... [Link]

## The “bounded gaps between primes” Polymath project – a retrospectiveTerence Tao

(01.11.2014 19:23h)

The presumably final article arising from the Polymath8 project has now been uploaded to the arXiv as “The “bounded gaps between primes” Polymath project – a retrospective“. This article, submitted to the Newsletter of the European Mathematical Society, consists of personal contributions from ten different participants at varying levels of stage of career, and intensity of participation on their own experiences with the project, and some thoughts as to what lessons to draw for any subsequent Polymath projects. At present, I do not know of any such projects being proposed, but from recent experience I would imagine that some opportunity ... [Link]

## Derived multiplicative functionsTerence Tao

(01.11.2014 19:23h)

Analytic number theory is often concerned with the asymptotic behaviour of various arithmetic functions: functions or from the natural numbers to the real numbers or complex numbers . In this post, we will focus on the purely algebraic properties of these functions, and for reasons that will become clear later, it will be convenient to generalise the notion of an arithmetic function to functions taking values in some abstract commutative ring . In this setting, we can add or multiply two arithmetic functions to obtain further arithmetic functions , and we can also form the Dirichlet convolution by the usual ... [Link]

## 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]

## 0.00023814967230605090687395214144185337601Kowalski

(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]

## 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]

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