tag:blogger.com,1999:blog-69126032879302404512016-08-30T17:03:37.052+00:00Noncommutative GeometryArupnoreply@blogger.comBlogger110125tag:blogger.com,1999:blog-6912603287930240451.post-45745669549529875042016-07-24T18:43:00.000+00:002016-07-24T18:43:17.791+00:00A motivic product formulaThe classical product formula for number fields is a fundamental tool in arithmetic. In 1993, Pierre Colmez published a truly inspired generalization of this to the case of Grothendieck's motives. In turn, this spring Urs Hartl and Rajneesh Kumar Singh put an equally inspired manuscript on the arXiv devoted to translating Colmez into the theory of Drinfeld modules and the like. Underneath the David Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-61700710282770695192016-06-14T01:43:00.002+00:002016-06-14T01:43:49.563+00:00What is a functional equation?Like all number theorists I am fascinated (to say the least) with the functional equation of
classical L-series. Years ago, I came up with a simple characterization of functional equations basically using only complex conjugation. This point being that, via a canonical change of variables (going back to Riemann), such L-series are, up to a nonzero scalar, given by real power series with the David Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-29671668693068999052016-03-26T23:33:00.002+00:002016-03-26T23:41:53.496+00:00An indirect consequence of the famous Lucas congruence...So, in the course of function field arithmetic, one runs into the binomial coefficients (like one does most everywhere in mathematics); or rather the coefficients modulo a prime p. The primary result about binomial coefficients modulo p is of course the congruence of Lucas. In function field arithmetic
one seems to be unable to avoid the group obtained by permuting p-adic (or q-adic) coefficientsDavid Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-40827014384257900312015-11-22T20:51:00.000+00:002015-12-11T16:41:57.120+00:00Review of "Arithmetic of characteristic p special L-values" by B. Anglès and L. TaelmanBruno Anglès and Lenny Taelman have published a profound study of finite characteristic special values as listed in the title. This appeared in Proc. London Math. Soc. (3) 110 (2015) 1000-1032. The analogy with classical theory is clear throughout the paper, as well as some potentially interesting differences; these results bode very well for the future of the subject. I have written a long David Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-45942890402385263212015-10-30T19:42:00.001+00:002015-10-31T00:29:48.212+00:00C. Armana's Formula for the coefficients of $h$
(full pdf is at https://drive.google.com/file/d/0BwCbLZazAtweOC01dFgyQjBmUlU/view?usp=sharing)
David Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-776336522303127822015-08-11T19:53:00.001+00:002015-08-11T19:53:45.708+00:00Grand Unification in the Spectral Pati-Salam Model
Last week we (Chamseddine-Connes-van Suijlekom) posted a preprint on grand unification in the spectral Pati–Salam model which I summarize here (and here).
The paper builds on two recent discoveries in the noncommutative geometry approach to particle physics: we showed
how to obtain inner fluctuations of the metric without having to
assume the order one condition on the Dirac operator. Walterhttp://www.blogger.com/profile/18443611486145891072noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-42366136832363614902015-07-18T22:26:00.001+00:002015-07-18T22:26:54.789+00:00Uffe HaagerupUffe Haagerup was a wonderful man, with a perfect kindness and openness of mind, and a mathematician of incredible power and insight.
His whole career is a succession of amazing achievements and of decisive and extremely influential contributions to the field of operator algebras, C*-algebras and von Neumann algebras.
His first work (1973-80) concerned the theory of weights and more generally Alain Conneshttps://plus.google.com/109079449189755445238noreply@blogger.com1tag:blogger.com,1999:blog-6912603287930240451.post-62039386980839997682015-07-18T22:25:00.000+00:002015-07-18T22:27:51.872+00:00Daniel KastlerDaniel Kastler played for many many years a key role as a leading Mathematical Physicist in developing Algebraic Quantum Field Theory. He laid the foundations of the subject as the famous
"Haag-Kastler" axioms in his joint paper with Rudolf Haag in 1964. He gathered around him, in Bandol, a whole international school of mathematicians and physicists. With the devoted help of his beloved wife Alain Conneshttps://plus.google.com/109079449189755445238noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-20334999426213638662015-07-18T20:50:00.000+00:002015-07-18T20:50:02.605+00:00Two great lossesIt is with incommensurable sadness that we learned of the death of two great figures of the fields of operator algebras and mathematical physics.
Daniel Kastler died on July 4-th in his house in Bandol.
Uffe Haagerup died on July 5-th in a tragic accident while swimming near his summer house in Denmark.
I will write on each of them separately.
Alain Conneshttps://plus.google.com/109079449189755445238noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-61655473105086659312015-01-02T17:45:00.000+00:002015-01-02T17:45:42.093+00:00QUANTA OF GEOMETRYThis is a short update on the post called "particles in quantum gravity",
there were interesting comments and rather than answering them in the
blog i just want to point to a long and detailed talk which I gave in the
Hausdorff Institute in Bonn in December and which is now available on YouTube.
In any case this is a good occasion to wish you all a
HAPPY NEW YEAR 2015!Alain Conneshttps://plus.google.com/109079449189755445238noreply@blogger.com3tag:blogger.com,1999:blog-6912603287930240451.post-26142917610457443002014-12-12T02:27:00.002+00:002014-12-17T22:09:33.320+00:00The Digit PrincipleDavid Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-33588629303698001002014-11-09T16:36:00.000+00:002014-11-09T16:36:53.317+00:00PARTICLES IN QUANTUM GRAVITYThe purpose of this post is to explain a recent discovery that we did with my two physicists collaborators Ali Chamseddine and Slava Mukhanov. We wrote a long paper Geometry and the Quantum: Basics which we put on the arXiv, but somehow I feel the urge to explain the result in non-technical terms.
The subject is the notion of particle in Quantum Gravity. In particle physics there is a well ACnoreply@blogger.com7tag:blogger.com,1999:blog-6912603287930240451.post-78634279450593853382014-08-26T12:39:00.000+00:002014-08-26T12:39:38.352+00:00Differentiation and the missing Kummer congruenceDavid Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-81869375565450036792014-08-13T03:04:00.000+00:002014-08-13T03:04:53.944+00:00Fields Medals 2014: Maryam Mirzakhani, Martin Hairer, Manjul Bhargava, Artur AvilaCongratulations to all 2014 Fields medalists! Very well deserved and also really nice to see a woman wining a Fields medal for the first time ever (and of course I am specially delighted that she has the same undergraduate alma mater, Sharif University, as I! Quanta magazine has a coverage of all four winners Avila, Bhargava, Hairer, Mirzakhani.
It was a bit unusual to see the results Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-33611532029976756552014-07-05T19:03:00.000+00:002014-08-05T14:49:54.872+00:00Lectures on VideoI would like to draw your attention to the following lectures just posted on youtube
1. Alain Connes: Arithmetic Site
Update: and a related interview where some of the relevant ideas in topos theory and the impact of Grothendieck is discussed.
2. Ali Chamseddine: Spectral Geometric Unification
Masoud Khalkhalihttp://www.blogger.com/profile/03769072750559219167noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-31223015150320708072014-06-05T13:30:00.000+00:002014-06-05T13:30:40.745+00:00Announcement book "Noncommutative Geometry and Particle Physics" by Walter van Suijlekom
My book "Noncommutative Geometry and Particle Physics" is due to appear this summer with Springer:
This textbook provides an introduction to noncommutative geometry and presents a
number of its recent applications to particle physics. It is intended
for graduate students in mathematics/theoretical physics who are new to
the field of noncommutative geometry, as well as for researchers in
Walterhttp://www.blogger.com/profile/18443611486145891072noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-66959374225491687652014-06-04T13:11:00.001+00:002014-06-04T13:11:54.254+00:00Quotes from Alain Connes' lecture on the spectral Standard Model at Radboud University Nijmegen
Alain Connes: “Change of paradigm unit of length” (video)
Alain Connes: “What is a noncommutative space and its group of symmetries” (video)
Alain Connes: “Spectral action, Yang-Mills theory” (video)
Alain Connes: “Derivation of the Standard Model from noncommutative geometry” (video)
Walterhttp://www.blogger.com/profile/18443611486145891072noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-2787960844198708032014-03-18T18:46:00.002+00:002014-03-18T20:20:44.765+00:00Review of a paper by Gebhard Boeckle and the group S_(q)
So this post is a bit of an experiment. My friends at Math Reviews recently sent me a really interesting Math. Z. paper by Gebhard Boeckle. I spent some time reviewing it and found it contained very interesting results and calculations that pointed, yet again, to some possible underlying action of the group $S_{(q)}$ that I have discussed in other posts here. If you combine it with the new David Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-39033544028717876472014-02-04T01:53:00.002+00:002014-02-18T12:54:32.575+00:00zeta zeroes AND gamma polesThe arithmetic of function fields over finite fields has always been a ``looking-glass'' window into the standard arithmetic of number fields, varieties, motives etc.; sort of ``life based on silicon'' as opposed to the classical ``carbon-based'' complex-valued constructions. It has constantly amazed me, and frankly given me great pleasure, to see the way that analogies always seem to work out inDavid Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-89329906137579732222013-09-08T18:22:00.002+00:002013-09-08T18:22:56.125+00:00Trimester program on Non-commutative Geometry and its Applications
From September-December 2014 there will be a trimester program on Non-commutative Geometry and its Applications at the Hausdorff Research Institute for Mathematics.
There will be four workshops during the trimester:
September 15-19, Non-commutative geometry's interactions with mathematics.
September 22-26, Quantum physics and non-commutative geometry.
November 24-28, Number theory and Walterhttp://www.blogger.com/profile/18443611486145891072noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-27534619671158227072013-09-05T01:38:00.001+00:002013-09-05T01:38:57.437+00:00Analytic continuation in the blogosphere....Hi. For those interested, I have started another blog at http://dmgoss.wordpress.com/ to cover items that are probably not appropriate (too technical, specialized, etc.) for this wonderful blog.... Best, DavidDavid Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0tag:blogger.com,1999:blog-6912603287930240451.post-66285866793377086192013-08-23T07:55:00.002+00:002013-08-23T07:55:25.004+00:00Website Noncommutative Geometry and Particle Physics
A new website on noncommutative geometry has been created, connected to the workshop Noncommutative Geometry and Particle Physics organized at the Lorentz Centre in Leiden in October 2013. As this type of workshop only allows for a limited number of participants, this website will form the virtual portal for a wider audience.
It will contain updates during the workshop, documents Walterhttp://www.blogger.com/profile/18443611486145891072noreply@blogger.com1tag:blogger.com,1999:blog-6912603287930240451.post-66734665544122675632013-07-09T19:58:00.003+00:002013-07-16T17:08:37.374+00:00A-expansionsAs I have written about before, the integers Z play a dual role in arithmetic. On the one hand, they are obviously scalars in terms of the fields of definitions of varieties etc.; yet, on the other hand, they are also operators, as in the associated Z-action on multiplicative groups (or the groups of rational points of abelian varieties etc.). This is absolutely so basic that we do not notice it David Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com4tag:blogger.com,1999:blog-6912603287930240451.post-69687938996964052432013-04-19T01:37:00.001+00:002013-04-23T15:24:46.419+00:00Wronski, Vandermonde, and Moore!This post is based on a recent letter by Matt Papanikolas outlining some results he has discovered whilst writing a (highly anticipated!) monograph on $L$-values in finite characteristic. In staring at Matt's letter, I realized that he allowed one to relate the big 3 matrices (Wronski, Vandermonde and Moore) in one simple formula which I will present below and then pose a related question.
I David Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com1tag:blogger.com,1999:blog-6912603287930240451.post-56711905525298146172013-01-28T01:19:00.000+00:002013-01-28T01:19:53.055+00:00Informal video series on the Carlitz moduleDear All: My student, Rudy Perkins, and his fellow graduate student, Tim All, are creating an informal video lecture series on the Carlitz module. If you are interested, please check out http://rudyperkins.wordpress.com/ . DavidDavid Gosshttps://plus.google.com/105910031825143642379noreply@blogger.com0