Math News July 20th, 2009
Patrick Stein

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

Math News (1 - 25 of about 2847) (xml) (Feedlist)


Enrichment and the Legendre-Fenchel Transform IThe n-Category Café
(16.04.2014 12:29h)

The Legendre-Fenchel transform, or Fenchel transform, or convex conjugation, is, in its naivest form, a duality between convex functions on a vector space and convex functions on the dual space. It is of central importance in convex optimization theory and in physics it is used to switch between Hamiltonian and Lagrangian perspectives. Suppose that VV is a real vector space and that f:V→[−∞,~ f\colon V\to [-\infty ,+\infty ] is a function then the Fenchel transform is the function f *:V #→[−∞,+∞~ f^{\ast }\colon V^{#}\to [-\infty ,+\infty ] defined on the dual vector space V #V^{#} by f * k ≔sup ... [Link]

writing the paper, and chasing down loose endsPolymath8b, X
(15.04.2014 07:55h)

This is the tenth thread for the Polymath8b project to obtain new bounds for the quantity ; the previous thread may be found here. Numerical progress on these bounds have slowed in recent months, although we have very recently lowered the unconditional bound on from 252 to 246 see the wiki page for more detailed results . While there may still be scope for further improvement particularly with respect to bounds for with , which we have not focused on for a while, it looks like we have reached the point of diminishing returns, and it is time to turn ... [Link]

universo.mathThe n-Category Café
(14.04.2014 19:34h)

A new Spanish language mathematical magazine has been launched: universo.math. Hispanophones should check out the first issue! There are some very interesting looking articles which cover areas from art through politics to research-level mathematics. The editor-in-chief is my mathematical brother Jacob Mostovoy and he wants it to be a mix of Mathematical Intellingencer, Notices of the AMS and the New Yorker, together with less orthodox ingredients; the aim is to keep the quality high. Besides Jacob, the contributors to the first issue that I recognise include Alberto Verjovsky, Ernesto Lupercio and Edward Witten, so universo.math seems to be off to ... [Link]

Is There Anything Beyond Quantum Computing?Scott
(11.04.2014 10:35h)

So I’ve written an article about the above question for PBS’s website—a sort of tl;dr version of my 2005 survey paper NP-Complete Problems and Physical Reality, but updated with new material about the simulation of quantum field theories and about AdS/CFT. Go over there, read the article it’s free , then come back here to talk about it if you like. Thanks so much to Kate Becker for commissioning the article. In other news, there’s a profile of me at MIT News called “The Complexonaut” that some people might find amusing. Oh, and anyone who thinks the main reason to ... [Link]

Eternal sunshine of the progressive mindIzabella Laba
(08.04.2014 19:39h)

Leszek Kolakowski, 2007. Photo: Mariusz Kubik Every now and then, I’m instructed to have more faith in the progressive tendencies of humanity. Racism and sexism, I’m told, are relics of the past, and especially so in science and tech. Progressive, open minded people are against discrimination. Scientists and tech geeks are open minded almost by definition, therefore progressive, therefore against racism and sexism, which therefore are no longer a problem. I should just look around and see how many Chinese and Indian immigrants work in tech and science; clearly, this means that the field is not racist. And if there ... [Link]

The Modular Flow on the Space of LatticesThe n-Category Café
(08.04.2014 01:25h)

Guest post by Bruce Bartlett The following is the greatest math talk I’ve ever watched! Etienne Ghys with pictures and videos by Jos Leys , Knots and Dynamics, ICM Madrid 2006. [See below the fold for some links.] I wasn’t actually at the ICM; I watched the online version a few years ago, and the story has haunted me ever since. Simon and I have been playing around with some of this stuff, so let me share some of my enthusiasm for it! The story I want to tell here is how, via modular flow of lattices in the plane, ... [Link]

On a Topological ToposThe n-Category Café
(07.04.2014 07:08h)

Guest post by Sean Moss In this post I shall discuss the paper “On a Topological Topos” by Peter Johnstone. The basic problem is that algebraic topology needs a “convenient category of spaces” in which to work: the category 𝒯\mathcal{T} of topological spaces has few good categorical properties beyond having all small limits and colimits. Ideally we would like a subcategory, containing most spaces of interest, which is at least cartesian closed, so that there is a useful notion of function space for any pair of objects. A popular choice for a “convenient category” is the full subcategory of 𝒯\mathcal{T} ... [Link]

Waiting for BQP FeverScott
(01.04.2014 16:46h)

Update April 5 : By now, three or four people have written in asking for my reaction to the preprint “Computational solution to quantum foundational problems” by Arkady Bolotin. See here for the inevitable Slashdot discussion, entitled “P vs. NP Problem Linked to the Quantum Nature of the Universe.” It gives me no pleasure to respond to this sort of thing—it would be far better to let papers this gobsmackingly uninformed about the relevant issues fade away in quiet obscurity—but since that no longer seems to be possible in the age of social media, my brief response is here. note: ... [Link]

Big Data PowerThe n-Category Café
(01.04.2014 15:08h)

Guest post by Nils Carqueville and Daniel Murfet My university probably isn’t alone in encouraging mathematicians and computer scientists to embrace the idea of “big data”, or in more sober terminology, “data science”. Here, Nils Carqueville and Daniel Murfet introduce their really excellent article on big data in whole-population surveillance. —TL In recent years Big Data has become an increasingly relevant topic in the economic sector, for intelligence agencies, and for the sciences. A particularly far-reaching development made possible by Big Data is that of unprecedented mass surveillance. As Alexander Beilinson, Stefan Forcey, Tom Leinster and others have pointed out, ... [Link]

Operads and TreesThe n-Category Café
(31.03.2014 10:32h)

Nina Otter is a master’s student in mathematics at ETH Zürich who has just gotten into the PhD program at Oxford. She and I are writing a paper on operads and the tree of life. Anyone who knows about operads knows that they’re related to trees. But I’m hoping someone has proved some precise theorems about this relationship, so that we don’t have to. By operad I’ll always mean a symmetric topological operad. Such a thing has an underlying ‘symmetric collection’, which in turn has an underlying ‘collection’. A collection is just a sequence of topological spaces O nO_n for ... [Link]

The sound you hear is another conjecture in birational geometry dropping like a flyJim Stankewicz
(31.03.2014 00:24h)

These are interesting times to look over the algebraic geometry arxiv postings. Just over a week ago, there was a posting by Tanaka which claimed the minimal model program was false in characteristic two. Then yesterday at the top of the page was a paper by Castravet and Tevelev claiming that the Mori Dream Space conjecture for was false. Then today, there is a paper by Fontanari claiming instead that the Mori Dream Space conjecture is TRUE for the same space, but modded out by the finite group . I’ll keep my remarks brief here, but roughly a Mori Dream ... [Link]

Fourier Series and Flipped ClassroomsThe n-Category Café
(30.03.2014 15:02h)

Term is nearly over, which for me means the end of the 4th year Fourier Analysis course I’ve been teaching for the last couple of years. I was fortunate enough to take over the course from Jim Wright, a genuine expert on the subject, and I inherited a great set of notes from him. But I felt the need to make the course my own, so I’ve been writing my own notes, which I’ve just finished: notes here, plus accompanying problem sheets. They’re mostly about convergence of Fourier series, with a delicious dessert of Fourier analysis on finite abelian groups. ... [Link]

The Cayley-Salmon theorem via classical differential geometryTerence Tao
(29.03.2014 06:34h)

Let be an irreducible polynomial in three variables. As is not algebraically closed, the zero set can split into various components of dimension between and . For instance, if , the zero set is a line; more interestingly, if , then is the union of a line and a surface or the product of an acnodal cubic curve with a line . We will assume that the -dimensional component is non-empty, thus defining a real surface in . In particular, this hypothesis implies that is not just irreducible over , but is in fact absolutely irreducible i.e. irreducible over , ... [Link]

All Hail the distinguished achievement professor!Kowalski
(25.03.2014 11:55h)

Mr. Quomodocumque is probably too modest to mention it himself, so let me be the first mathematics blogger to congratulate Jordan Ellenberg on becoming a Vilas Distinguished Achievement Professor! Which hopefully comes with a lot of free time to visit Switzerland… [Link]

An Exegesis of Yoneda StructuresThe n-Category Café
(24.03.2014 07:01h)

Guest post by Alexander Campbell We want to develop category theory in a general 2-category, in order to both generalise and clarify our understanding of category theory. The key to this endeavour is to express the basic notions of the theory of categories in a natural 2-categorical language. In this way we are continuing a theme present in previous posts from the Kan Extension Seminar, wherein monads and adjunctions were given a 2-categorical setting, and by analogy, in our very first paper, whose purpose was to express basic notions of the theory of sets in a natural categorical language. In ... [Link]

On diplomatsKowalski
(23.03.2014 18:52h)

Quizz: who wrote Il a publié il y a deux ans … un ouvrage relatif au sentiment de l’Infini sur la rive occidentale du lac Victoria-Nyanza et cette année un opuscule moins important, mais conduit d’une plume alerte, parfois même acérée, sur le fusil à répétition dans l’armée bulgare, qui l’ont mis tout à fait hors de pair. or, in translation: He has published two years ago … a book concerning the feeling of Infinity on the occidental shore of the Victoria-Nyanza lake, and this year another booklet, less important but written with a lively, and even piercing, pen, on ... [Link]

This review of Max Tegmark’s book also occurs infinitely often in the decimal expansion of πScott
(23.03.2014 00:27h)

Two months ago, commenter rrtucci asked me what I thought about Max Tegmark and his “Mathematical Universe Hypothesis”: the idea, which Tegmark defends in his recent book Our Mathematical Universe, that physical and mathematical existence are the same thing, and that what we call “the physical world” is simply one more mathematical structure, alongside the dodecahedron and so forth. I replied as follows: …I find Max a fascinating person, a wonderful conference organizer, someone who’s always been extremely nice to me personally, and an absolute master at finding common ground with his intellectual opponents—I’m trying to learn from him, and ... [Link]

Metric entropy analogues of sum set theoryTerence Tao
(20.03.2014 11:30h)

A core foundation of the subject now known as arithmetic combinatorics and particularly the subfield of additive combinatorics are the elementary sum set estimates sometimes known as “Ruzsa calculus” that relate the cardinality of various sum sets and difference sets as well as iterated sumsets such as , , and so forth. Here, are finite non-empty subsets of some additive group classically one took or , but nowadays one usually considers more general additive groups . Some basic estimates in this vein are the following: Lemma 1 Ruzsa covering lemma Let be finite non-empty subsets of . Then may be ... [Link]

Metric entropy analogues of sum set theoryTerence Tao
(20.03.2014 11:27h)

A core foundation of the subject now known as arithmetic combinatorics and particularly the subfield of additive combinatorics are the elementary sum set estimates sometimes known as “Ruzsa calculus” that relate the cardinality of various sum sets and difference sets as well as iterated sumsets such as , , and so forth. Here, are finite non-empty subsets of some additive group classically one took or , but nowadays one usually considers more general additive groups . Some basic estimates in this vein are the following: Lemma 1 Ruzsa covering lemma Let be finite non-empty subsets of . Then may be ... [Link]

The Movie!Why Quantum Computers Cannot Work
(18.03.2014 23:20h)

Here are links to a videotaped lecture in two parts entitled “why quantum computers cannot work” recorded at the Simons Institute for the Theory of Computing on December 2013 and two additional videos: a short talk on topological quantum computers and a twelve minute overview. Here are the links: Overview, Part I, Part II, Topological QC. Why Quantum Computers Cannot Work: Overview and Vision. Why Quantum Computers Cannot Work I: From the “Trivial Flow” to Smoothed Lindblad Evolutions Why Quantum Computers Cannot Work II: Debate, Reasons to Disbelieve, and Experimentation Why Topological Quantum Computers Cannot Work The Geometry of Spacetime ... [Link]

Translating Grothendieck's Biography into EnglishThe n-Category Café
(18.03.2014 17:32h)

Leila Schneps is trying to raise $6,000 for what sounds like a good cause: translating a biography of Grothendieck into English: Translation of Grothendieck Biography. As of this moment she’s raised $350… including $100 of her own money. Leila Schneps writes: Two volumes of a projected 3-volume biography of A. Grothendieck have been completed by Winfried SCHARLAU, and are available in German on Amazon.com through “Books on Demand” as “Wer ist Alexander Grothendieck? Teil 1: Anarchy” and “Wer ist Alexander Grothendieck? Teil 3: Spiritualität”. Grothendieck is aware of Scharlau’s work and even met him during the preparation of the part ... [Link]

Fuzzy Logic and Enriching Over the Category [0,1]The n-Category Café
(15.03.2014 19:21h)

Standard logic involving the truth values ‘true’ and ‘false’ can make it difficult to model some of the fuzziness we use in everyday speech. If you’d bought a bike yesterday then today it would be truthful to say “This bike is new”, but it wouldn’t be truthful so say it in 20 years’ time. However, between now and then there won’t be a specific day on which the statement “This bike is new” suddenly switches from being true to being false. How can you model this situation? One approach to modelling this situation is with fuzzy logic where you allow ... [Link]

Conserved quantities for the surface quasi-geostrophic equationTerence Tao
(15.03.2014 15:04h)

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]

Noether’s theorem, and the conservation laws for the Euler equationsTerence Tao
(15.03.2014 15:04h)

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]

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]

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