Math News July 20th, 2009
Patrick Stein

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

Math News (26 - 50 of about 2848) (xml) (Feedlist)


Conserved quantities for the Euler equationsTerence Tao
(15.03.2014 15:04h)

The Euler equations for incompressible inviscid fluids may be written as where is the velocity field, and is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that is bounded. We will take the dimension to be at least two, with the three-dimensional case being of course especially interesting. The Euler equations are the inviscid limit of the Navier-Stokes equations; as discussed in my previous post, one potential route to establishing finite time blowup for the latter equations when is to be able to construct “computers” solving the Euler equations, which generate smaller replicas ... [Link]

Large quadratic programsPolymath8b, IX
(15.03.2014 15:04h)

This is the ninth thread for the Polymath8b project to obtain new bounds for the quantity either for small values of in particular or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page. The focus is now on bounding unconditionally in particular, without resorting to the Elliott-Halberstam conjecture or its generalisations . We can bound whenever one can find a symmetric square-integrable function supported on the simplex such that 4 \int_{{\cal R}_{k}} F t_1,\dots,t_k ^2\ dt_1 \dots dt_{k-1} dt_k. ' title='\displaystyle > 4 \int_{{\cal R}_{k}} F ... [Link]

A few analysis resourcesgowers
(12.03.2014 15:33h)

This will be my final post associated with the Analysis I course, for which the last lecture was yesterday. It’s possible that I’ll write further relevant posts in the nearish future, but it’s also possible that I won’t. This one is a short one to draw attention to other material that can be found on the web that may help you to learn the course material. It will be an incomplete list: further suggestions would be welcome in the comments below. A good way to test your basic knowledge of some of the course would be to do a short ... [Link]

Sunset, Garry PointIzabella Laba
(09.03.2014 16:41h)

Because it’s been a while since I posted a photo. The weather has not been cooperating recently, so instead here’s one from the archives. [Link]

Review of the Elements of 2-CategoriesThe n-Category Café
(09.03.2014 09:20h)

Guest post by Dimitri Zaganidis First of all, I would like to thank Emily for organizing the Kan extension seminar. It is a pleasure to be part of it. I want also to thank my advisor Kathryn Hess and my office mate Martina Rovelli for their revisions. In the fifth installment of the Kan Extension Seminar we read the paper “Review of the Elements of 2-categories” by G.M Kelly and Ross Street. This article was published in the Proceedings of the Sydney Category Theory Seminar, and its purpose is to “serve as a common introduction to the authors’ paper in ... [Link]

The Scientific Case for PâScott
(07.03.2014 09:46h)

Out there in the wider world—OK, OK, among Luboš Motl, and a few others who comment on this blog—there appears to be a widespread opinion that P≠NP is just “a fashionable dogma of the so-called experts,” something that’s no more likely to be true than false. The doubters can even point to at least one accomplished complexity theorist, Dick Lipton, who publicly advocates agnosticism about whether P=NP. Of course, not all the doubters reach their doubts the same way. For Lipton, the thinking is probably something like: as scientists, we should be rigorously open-minded, and constantly question even the most ... [Link]

Obedience trainingIzabella Laba
(07.03.2014 05:33h)

This happy and cheerful period on my blog would not be complete without some mention of Anton Makarenko’s “Pedagogical poem,” aka “The Road to Life.” Full text available here. Makarenko, in case you don’t know, was one of the founders of Soviet pedagogy, best known for his work at the Gorky colony and the Dzerzhynsky colony, and for the book in question. The Soviet government was chaotic and disorganized through much of the 1920s, with multiple factions and doctrines competing for dominance. On a practical level, the Bolsheviks had little if any experience with actual governance and running of the ... [Link]

Conserved quantities for the surface quasi-geostrophic equationTerence Tao
(07.03.2014 03:25h)

As in the previous post, all computations here are at the formal level only. In the previous blog post, the Euler equations for inviscid incompressible fluid flow were interpreted in a Lagrangian fashion, and then Noether’s theorem invoked to derive the known conservation laws for these equations. In a bit more detail: starting with Lagrangian space and Eulerian space , we let be the space of volume-preserving, orientation-preserving maps from Lagrangian space to Eulerian space. Given a curve , we can define the Lagrangian velocity field as the time derivative of , and the Eulerian velocity field . The volume-preserving ... [Link]

Operads of Finite GroupsThe n-Category Café
(05.03.2014 18:18h)

Guest post by Nick Gurski I have been thinking about various sorts of operads with my PhD student Alex Corner, and have become interested in the following very concrete question: what are examples of operads in the category of finite groups under the cartesian product? I don’t know any really interesting examples, but maybe you do! After the break I will explain why I got interested in this question, and tell you about some examples that I do know. Alex and I started off thinking about various sorts of things you might do with operads in Cat\mathbf{Cat}, and were eventually ... [Link]

The many principles of conservation of numberDavid Speyer
(04.03.2014 19:28h)

In algebraic geometry, we like to make statements like: “two conics meet at points”, “a degree four plane curve has bitangents”, “given four lines in three space, there are lines that meet all of them”. In each of these, we are saying that, as some parameter the conics, the degree four curve, the lines changes, the number of solutions to some equation stays constant. The “principle of conservation of number” refers to various theorems which make this precise. In my experience, students in algebraic geometry tend to pick up the rough idea but remain hazy on the details, most likely ... [Link]

Noether’s theorem, and the conservation laws for the Euler equationsTerence Tao
(03.03.2014 17:08h)

Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. Rigorous justification of the conservation laws and other identities arising from the formal manipulations below can usually be established in an a posteriori fashion once the identities are in hand, without the need to rigorously justify the manipulations used to come up with these identities . It is a remarkable fact in the theory of differential equations that many of the ordinary and partial differential equations that are of interest particularly in ... [Link]

Recent papers by Susskind and Tao illustrate the long reach of computationScott
(02.03.2014 23:55h)

Most of the time, I’m a crabby, cantankerous ogre, whose only real passion in life is using this blog to shoot down the wrong ideas of others. But alas, try as I might to maintain my reputation as a pure bundle of seething negativity, sometimes events transpire that pierce my crusty exterior. Maybe it’s because I’m in Berkeley now, visiting the new Simons Institute for Theory of Computing during its special semester on Hamiltonian complexity. And it’s tough to keep up my acerbic East Coast skepticism of everything new in the face of all this friggin’ sunshine. Speaking of which, ... [Link]

Levon Khachatrian’s Memorial Conference in YerevanGil Kalai
(02.03.2014 21:41h)

Workshop announcement The National Academy of Sciences of Armenia together American University of Armenia are organizing a memorial workshop on extremal combinatorics, cryptography and coding theory dedicated to the 60th anniversary of the mathematician Levon Khachatrian. Professor Khachatrian started his academic career at the Institute of Informatics and Automation of National Academy of Sciences. From 1991 until the end of his short life in 2002 he spent at University of Bielefeld, Germany where Khachatrian’s talent flourished working with Professor Rudolf Ahlswede. Professor Khachatrian’s most remarkable results include solutions of problems dating back over 40 years in extremal combinatorics posed by ... [Link]

Should Mathematicians Cooperate with GCHQ?The n-Category Café
(02.03.2014 20:39h)

I’ve just submitted a piece for the new Opinions section of the monthly LMS Newsletter: Should mathematicians cooperate with GCHQ? Update: now available p.34 . The LMS is the London Mathematical Society, which is the UK’s national mathematical society. My piece should appear in the April edition of the newsletter, and you can read it below. Here’s the story. Since November, I’ve been corresponding with people at the LMS, trying to find out what connections there are between it and GCHQ. Getting the answer took nearly three months and a fair bit of pushing. In the process, I made some ... [Link]

How do the power-series definitions of sin and cos relate to their geometrical interpretations?gowers
(02.03.2014 16:26h)

I hope that most of you have either asked yourselves this question explicitly, or at least felt a vague sense of unease about how the definitions I gave in lectures, namely and relate to things like the opposite, adjacent and hypotenuse. Using the power-series definitions, we proved several facts about trigonometric functions, such as the addition formulae, their derivatives, and the fact that they are periodic. But we didn’t quite get to the stage of proving that if and is the angle that the line from to makes with the line from to , then and . So how does ... [Link]

More things of the day s Kowalski
(01.03.2014 20:46h)

1 Today’s Word of The Day in the OED: afanc, which we learn is In Welsh mythology: an aquatic monster. Also: an otter or beaver identified as such a monster. Maybe the Welsh otters, like their rugbymen, are particularly fierce? 2 Yesterday’s Google doodle, in Switzerland at least, celebrated the 57th birthday of Gaston Lagaffe I’ve heard that Gaston Billard is mostly unknown to the US or English, leaving many people with no reaction to the mention of the contrats de Mesmaeker Contrats or to the interjection Rogntudju!. This lack of enlightenment is a clear illustration of the superiority of ... [Link]

Mathematical Research Community on Cluster Algebras in Utah this summerDavid Speyer
(26.02.2014 18:46h)

This June 8 to 14, there will be a week long gathering in Snowbird, Utah for young mathematicians working on cluster algebras. The target audience here are either current graduate students, or people with Ph. D. in the last 3 or so years, who would be ready to start working on problems in cluster algebras. The hope is to spend a lot of time getting collaborations and projects going during the week. The organizers are Michael Gekhtman, Mark Gross, Gregg Musiker, Gordana Todorov and me. We still have room for a number more applicants, so we would like to encourage ... [Link]

Conserved quantities for the Euler equationsTerence Tao
(26.02.2014 06:08h)

The Euler equations for incompressible inviscid fluids may be written as where is the velocity field, and is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that is bounded. We will take the dimension to be at least two, with the three-dimensional case being of course especially interesting. The Euler equations are the inviscid limit of the Navier-Stokes equations; as discussed in my previous post, one potential route to establishing finite time blowup for the latter equations when is to be able to construct “computers” solving the Euler equations, which generate smaller replicas ... [Link]

The many ways of affinenessKowalski
(25.02.2014 11:30h)

Last Saturday, the OED Word of the Day was affineur. Now, I know very well what an affineur is my favorite is Jean d’Alos, and I especially like his renowned Tome de Bordeaux, the excellence of which can probably be confirmed by Mr. Quomodocumque , but for a few seconds I had in mind the picture of a fearsome algebraic geometer busily transforming all projective varieties into affine ones. I looked at the adjacent words in the OED; there is quite a list of them involving affine-ness in some way listed here with dates of first use, as recorded in ... [Link]

Australian Research Council journal listScott Morrison
(24.02.2014 11:11h)

This post may only be of interest to Australian mathematicians; sorry! Summary: A number of mathematics journals e.g. Quantum Topology, Forum of Mathematics Sigma and Pi, and probably many others , are not listed on the new official journal list in Australia. Please, help identify missing journals, and submit feedback via http://jacci.arc.gov.au/. Every few years the Australian Research Council updates their “official list of journals”. One might wonder why it’s necessary to have such a list, but nevertheless it is there, and it is important that it is accurate because the research outputs of Australian mathematicians are essentially filtered by ... [Link]

Differentiating power seriesgowers
(22.02.2014 17:17h)

I’m writing this post as a way of preparing for a lecture. I want to discuss the result that a power series is differentiable inside its circle of convergence, and the derivative is given by the obvious formula . In other words, inside the circle of convergence we can think of a power series as like a polynomial of degree for the purposes of differentiation. A preliminary question about this is why it is not more or less obvious. After all, writing , we have the following facts. Writing , we have that . For each , . If we ... [Link]

Large quadratic programsPolymath8b, IX
(22.02.2014 05:23h)

This is the ninth thread for the Polymath8b project to obtain new bounds for the quantity either for small values of in particular or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page. The focus is now on bounding unconditionally in particular, without resorting to the Elliott-Halberstam conjecture or its generalisations . We can bound whenever one can find a symmetric square-integrable function supported on the simplex such that 4 \int_{{\cal R}_{k}} F t_1,\dots,t_k ^2\ dt_1 \dots dt_{k-1} dt_k. ' title='\displaystyle > 4 \int_{{\cal R}_{k}} F ... [Link]

Metric Spaces, Generalized Logic, and Closed CategoriesThe n-Category Café
(21.02.2014 02:17h)

Guest post by Tom Avery Before getting started, I’d like to thank Emily for organizing the seminar, as well as all the other participants. It’s been a lot of fun so far! I’d also like to thank my supervisor Tom Leinster for some very helpful suggestions when writing this post. In the fourth instalment of the Kan Extension Seminar we’re looking at Lawvere’s paper “Metric spaces, generalized logic, and closed categories”. This is the paper that introduced the surprising description of metric spaces as categories enriched over a certain monoidal category ℝ\mathbb{R}. A lot of people find this very striking ... [Link]

ConferencesKowalski
(17.02.2014 20:41h)

Here are two forthcoming conferences that I am co-organizing with Philippe Michel this year: 1 Quite soon, the traditional Number Theory Days the eleventh edition of this yearly two-day meeting that alternates between EPF Zürich and ETH Lausanne , will be held in Zürich on March 7 and 8; the web page is available, with the schedule and the titles of the talks; the speakers this year are Raf Cluckers who is also giving a Nachdiplomvorlesung at FIM on the topic of motivic integration and applications , Lillian Pierce, Trevor Wooley, Tamar Ziegler and Tamar is probably happily surprised not ... [Link]

This will be counting butKowalski
(16.02.2014 16:05h)

For the first time ever, I have temporarily reached the top of the culture pecking order in France, since I was able to go see “Einstein on the Beach” during its run at the Théâtre du Châtelet in January. This was clearly “the” évènement to attend even two months or so before, buying two good contiguous seats was almost impossible , and the reactions of the audience suggested that many people were there because of the “the” instead of their desire to see the évènement, and were correspondingly a bit nonplussed by the work. Thus the two people sitting on ... [Link]

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