Math News July 20th, 2009
Patrick Stein

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

Math News (1 - 25 of about 3129) (xml) (Feedlist)


Long gaps between primesTerence Tao
(17.12.2014 10:25h)

Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and I have just uploaded to the arXiv our paper “Long gaps between primes“. This is a followup work to our two previous papers discussed in this previous post , in which we had simultaneously shown that the maximal gap between primes up to exhibited a lower bound of the shape for some function that went to infinity as ; this improved upon previous work of Rankin and other authors, who established the same bound but with replaced by a constant. Again, see the previous post for a more detailed discussion. In ... [Link]

Hyperbolic SurfacesCharles Siegel
(17.12.2014 08:55h)

Ok, with the hyperbolic plane and its metric and geodesics out of the way, we can start getting into some surface theory. Definition: A hyperbolic surface of genus g is a topological surface of genus g along with a metric that is locally isometric to the hyperbolic plane. Equivalently, it has a Riemannian metric of constant curvature -1. There are some distinguished types of curves on a hyperbolic surface, and I don’t just mean the geodesics though we’ll relate them to geodesics . This is going to be a definition heavy post, but hey, we’re doing a construction, this tends ... [Link]

The Turing movieScott
(17.12.2014 04:36h)

Last week I finally saw The Imitation Game, the movie with Benedict Cumberbatch as Alan Turing. OK, so for those who haven’t yet seen it: should you? Here’s my one paragraph summary: imagine that you told the story of Alan Turing—one greatest triumphs and tragedies of human history, needing no embellishment whatsoever—to someone who only sort-of understood it, and who filled in the gaps with weird fabrications and Hollywood clichés. And imagine that person retold the story to a second person, who understood even less, and that that person retold it to a third, who understood least of all, but ... [Link]

Walter LewinScott
(10.12.2014 17:25h)

Yesterday I heard the sad news that Prof. Walter Lewin, age 78—perhaps the most celebrated physics teacher in MIT’s history—has been stripped of his emeritus status and barred from campus, and all of his physics lectures removed from OpenCourseWare, because an internal investigation found that he had been sexually harassing students online. I don’t know anything about what happened beyond the terse public announcements, but those who do know tell me that the charges were extremely serious, and that “this wasn’t a borderline case.” I’m someone who feels that sexual harassment must never be tolerated, neither here nor anywhere else. ... [Link]

Random matrices have simple spectrumTerence Tao
(08.12.2014 06:01h)

Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple eigenvalues“. Recall that an Hermitian matrix is said to have simple eigenvalues if all of its eigenvalues are distinct. This is a very typical property of matrices to have: for instance, as discussed in this previous post, in the space of all Hermitian matrices, the space of matrices without all eigenvalues simple has codimension three, and for real symmetric cases this space has codimension two. In particular, given any random matrix ensemble of Hermitian or real symmetric matrices with an absolutely continuous distribution, we ... [Link]

Random matrices have simple spectrumTerence Tao
(05.12.2014 03:34h)

Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple eigenvalues“. Recall that an Hermitian matrix is said to have simple eigenvalues if all of its eigenvalues are distinct. This is a very typical property of matrices to have: for instance, as discussed in this previous post, in the space of all Hermitian matrices, the space of matrices without all eigenvalues simple has codimension three, and for real symmetric cases this space has codimension two. In particular, given any random matrix ensemble of Hermitian or real symmetric matrices with an absolutely continuous distribution, we ... [Link]

The hyperbolic planeCharles Siegel
(03.12.2014 20:49h)

So, I know I usually talk about strictly algebraic geometry stuff, but the moduli of curves lives in an interesting place. It’s both an algebraic and an analytic object. So we’re going to start by talking a bit about hyperbolic surfaces, as we work towards a construction of Teichmüller space, which is used to construct the moduli of curves over . We actually have to start with some more basic differential geometric notions that have been neglected, because we never did a “Differential Geometry from the Beginning” series perhaps I should in the future? Chime in in the comments if ... [Link]

A general parity problem obstructionTerence Tao
(01.12.2014 13:43h)

Many problems and results in analytic prime number theory can be formulated in the following general form: given a collection of affine- linear forms , none of which is a multiple of any other, find a number such that a certain property of the linear forms are true. For instance: For the twin prime conjecture, one can use the linear forms , , and the property in question is the assertion that and are both prime. For the even Goldbach conjecture, the claim is similar but one uses the linear forms , for some even integer . For Chen’s theorem, ... [Link]

Analytic prime number theory254A announcement
(01.12.2014 13:43h)

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]

Discretised wave equationsTerence Tao
(01.12.2014 13:43h)

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]

A Confession of Mathematical ErrorsPostBQP Postscripts
(01.12.2014 00:41h)

tl;dr: This post reveals two errors in one of my most-cited papers, and also explains how to fix them. Thanks to Piotr Achinger, Michael Cohen, Greg Kuperberg, Ciaran Lee, Ryan O’Donnell, Julian Rosen, Will Sawin, Cem Say, and others for their contributions to this post. If you look at my Wikipedia page, apparently one of the two things in the world that I’m “known for” along with algebrization is “quantum Turing with postselection.” By this, Wikipedia means my 2004 definition of the complexity class PostBQP—that is, the class of decision problems solvable in bounded-error quantum polynomial time, assuming the ability ... [Link]

Lens of Computation on the SciencesScott
(25.11.2014 23:32h)

This weekend, the Institute for Advanced Study in Princeton hosted a workshop on the “Lens of Computation in the Sciences,” which was organized by Avi Wigderson, and was meant to showcase theoretical computer science’s imperialistic ambitions to transform every other field. I was proud to speak at the workshop, representing CS theory’s designs on physics. But videos of all four of the talks are now available, and all are worth checking out: Computational Phenomena in Biology, by Leslie Valiant Computational Phenomena in Economics, by Tim Roughgarden Computational Phenomena in Social Science, by Jon Kleinberg Computational Phenomena in Physics, by me ... [Link]

Proust, my family and AustraliaKowalski
(25.11.2014 17:10h)

When I was reading Proust, I noted with some amusement the character named simply “Ski” in the first volume of his appearance, a sculptor and amateur musician who is later revealed to be properly called “Viradobetski”, an actual name which was too complicated for the dear Madame Verdurin to try to remember. I just learnt from my better educated brother that Proust s’est inspiré d’Henri Kowalski né en 1841, fils d’un officier polonais émigré en Bretagne. Il était à la fois compositeur de musique et concertiste. or Proust used as model Henri Kowalski, born in 1841, son of a Polish ... [Link]

Kuperberg’s parableScott
(24.11.2014 00:04h)

Recently, longtime friend-of-the-blog Greg Kuperberg wrote a Facebook post that, with Greg’s kind permission, I’m sharing here. A parable about pseudo-skepticism in response to climate science, and science in general. Doctor: You ought to stop smoking, among other reasons because smoking causes lung cancer. Patient: Are you sure? I like to smoke. It also creates jobs. D: Yes, the science is settled. P: All right, if the science is settled, can you tell me when I will get lung cancer if I continue to smoke? D: No, of course not, it’s not that precise. P: Okay, how many cigarettes can ... [Link]

A general parity problem obstructionTerence Tao
(22.11.2014 11:27h)

Many problems and results in analytic prime number theory can be formulated in the following general form: given a collection of affine- linear forms , none of which is a multiple of any other, find a number such that a certain property of the linear forms are true. For instance: For the twin prime conjecture, one can use the linear forms , , and the property in question is the assertion that and are both prime. For the even Goldbach conjecture, the claim is similar but one uses the linear forms , for some even integer . For Chen’s theorem, ... [Link]

Moduli of Riemann SurfacesNew Series
(21.11.2014 03:14h)

Though I’m not quite ready to start next week! I feel that, in the spirit of Jim getting back to the blogging and my continued promises, I’d announce my series now. I’m going to start a detailed series on the moduli of Riemann surfaces, including both topological and geometric aspects. And I figured I’d start it out with a list of references for some of the topics that I’d be covering: Books Algebraic Curves and Riemann Surfaces by Miranda Geometry of Algebraic Curves Volumes I and II by Arbarello, Cornalba, Griffiths and Harris Geometric Invariant Theory by Mumford, Fogarty and ... [Link]

On proof and progress in feminismIzabella Laba
(20.11.2014 04:45h)

The recent allegations against several celebrities have led to a broader conversation on how we, as a society, don’t believe women. In a “he said, she said” situation, we trust the man and assume that the woman is either mistaken or lying. “Taking us seriously” means that we are advised of such and offered an explanation for our dismissal instead of simply being dismissed outright. It’s not only personal bias, conscious or not; there are institutional mechanisms perpetuating this state of affairs. No proof is ever sufficient if it comes from a woman. Should she present multiple affidavits, all signed ... [Link]

ANU hiring postdocsScott Morrison
(20.11.2014 01:06h)

Amnon Neeman has just put up an ad for two postdoctoral positions at the ANU. He says: “The successful applicants should have strong research interests and activities in or related to one of the following fields: Algebraic Geometry, Commutative Algebra, Representation Theory, Algebraic Topology, Algebraic K-Theory. Skills at applying the techniques of triangulated categories to these areas would be a plus.” These are excellent positions — available for up to 3 years, with no teaching requirements, and salaries in the AUD81-89k range. Applications close at the end of January, and I hear Amnon is keen to hire as soon as ... [Link]

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