Let me express my heartfelt thanks to the organizers of the Shanghai 2017 NCG-event

as well as to each participant. Your presence and talks were the most wonderful gift showing so many lively facets of mathematics and physics displaying the vitality of NCG! The whole three weeks were a pleasure due to the wonderful hospitality of our hosts, Xiaoman Chen, Guoliang Yu and Yi-Jun Yao. It is great that the many talks of the three workshops have been recorded in video which will be put online after being edited into appropriate format and upon approval of each speaker. For now the list of abstracts (which will be updated when I get the missing abstracts) is available here, with a few last minute nuances which I will mention:

Farzad Fathizadeh could not come because of obvious worries on getting back in US after a trip abroad in the DT times. So his talk on the second day of the first workshop was delivered by Masoud Khalkhali. On the second week, Dennis Sullivan gave a wonderful improvised talk (not the same as in the abstract) on the subject of "Sobolev manifolds" i.e. manifolds in which the natural regularity of functions is to have a first derivative in L^p where the critical case is when p=d is the dimension of the manifold. Dennis was as he used to be in IHES, making key comments in every talk on the second week. His generosity with his time and devotion to "understanding" showed at their best. There were many many interactions during these three weeks and it will take time to digest them, but modernity does have some advantages and the recording of the talks should help a lot! For sure there was a general feeling of ongoing excitement to see how the field of NCG has developed and branched so diversely after 40 years, and each of the words connected to this special event went straight to my heart.

ps: Here is the "histoire courte" with Jacques Dixmier, so remarkably well elaborated by Anne Papillault and Jean-François Dars, which was shown on the large screen at the banquet on 01/04!

# Noncommutative Geometry

## Sunday, April 16, 2017

## Monday, April 10, 2017

### DAVID

**It is with immense emotion and sorrow that we learnt the sudden death of David Goss.**

**He was our dear friend, a joyful supporter of our field and a constant source of inspiration through his great work, of remarkable originality and depth, on function field arithmetics.**

**We are profoundly saddened by this tragic loss.**

**David will remain in our heart, we shall miss him dearly.**

## Tuesday, February 7, 2017

### Connes 70

I am happy to report that to celebrate Alain Connes' 70th birthday, 3 conferences on noncommutative geometry and its interactions with different fields are planned to take place in Shanghai, China. Students, postdocs, young faculty and all those interested in the subject are encouraged to participate. Please check the Conference webpage for more details.

## Tuesday, January 3, 2017

### Gamma functions and nonarchimedean analysis

Happy New Year!

I view blog writing as a great opportunity to reach out to members of the mathematics community and especially the younger members; so in this sense blog writing is, for me, very similar to writing for Math Reviews. I have enjoyed doing both for many years (and many many years for MR!). Recently I wrote a review for MR on the paper ``Twisted characteristic p zeta functions'' written

by Bruno Angles, Tuan Ngo Dac and Floric Ribeiro Tavares (``MR Number: MR3515815''). I am attaching the review here with the permission of Math Reviews. You can find it, in preprint form, here with the original (with live hyperlinks to papers) on the MR site.

The paper being reviewed makes some demands of the reader. But the devoted reader will be rewarded with an early view of a beautiful new world. Those readers familiar with Drinfeld modules know that they exist in incredible profusion: One starts with a smooth projective, geometrically connected curve

is concentrated on this particular

Of course, the theory of the zeta function is intimately connected with the theory of the Gamma function and so one should also expect analogs of Gamma functions to appear in the characteristic p theory with the correct one being given decades ago by Greg Anderson and Dinesh Thakur in the polynomial case. Their function appears firstly as an element of the Tate algebra of functions in

higher genus as opposed to the affine line). This is the beautiful new world I mentioned above…..

I view blog writing as a great opportunity to reach out to members of the mathematics community and especially the younger members; so in this sense blog writing is, for me, very similar to writing for Math Reviews. I have enjoyed doing both for many years (and many many years for MR!). Recently I wrote a review for MR on the paper ``Twisted characteristic p zeta functions'' written

by Bruno Angles, Tuan Ngo Dac and Floric Ribeiro Tavares (``MR Number: MR3515815''). I am attaching the review here with the permission of Math Reviews. You can find it, in preprint form, here with the original (with live hyperlinks to papers) on the MR site.

The paper being reviewed makes some demands of the reader. But the devoted reader will be rewarded with an early view of a beautiful new world. Those readers familiar with Drinfeld modules know that they exist in incredible profusion: One starts with a smooth projective, geometrically connected curve

*X*over the finite field*F*_q with q elements. Then one chooses a fixed closed point \infty of*X*and defines the algebra*A*to be the Dedekind domain of functions regular away from \infty; so*A*plays the role of the integers*Z i*n the Drinfeld theory. One instance of such an*A*is, of course, the ring*F*_q[\theta] which is, like*Z*, Euclidean, and indeed most of the work done so faris concentrated on this particular

*A*as it is both easy to work with and very similar to classical arithmetic. However, ultimately, the theory should work for general*A*just as the theory of Drinfeld modules (and generalizations) does. As general*A*is very far from factorial, one can imagine that many interesting issues arise (and the paper being reviewed discusses them from an axiomatic viewpoint).Of course, the theory of the zeta function is intimately connected with the theory of the Gamma function and so one should also expect analogs of Gamma functions to appear in the characteristic p theory with the correct one being given decades ago by Greg Anderson and Dinesh Thakur in the polynomial case. Their function appears firstly as an element of the Tate algebra of functions in

*t*converging on the closed unit disc. One fascinating aspect of the paper being reviewed is that this Tate algebra is replaced by Tate algebras created out of the general rings*A*(and so lie inside curves ofhigher genus as opposed to the affine line). This is the beautiful new world I mentioned above…..

## Sunday, July 24, 2016

### A motivic product formula

The 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 mountains of terminology there is a fantastic similarity between these two beautiful papers and I have created a blog to bring this to the attention of the community. Please see:

https://drive.google.com/open?id=0BwCbLZazAtweamZYckpaTy15cFU

https://drive.google.com/open?id=0BwCbLZazAtweamZYckpaTy15cFU

## Tuesday, June 14, 2016

### What 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 expectation that the zeroes are also real. In characteristic p the best one can hope is also that the zeroes will be as rational as the coefficients (though this statement needs to be modified to take care of standard factorizations as well as the great generality of Drinfeld's base rings A).
For those interested, a two page pdf can be found at the following link: https://drive.google.com/open?id=0BwCbLZazAtweTmNIa1ZSc0h2UEE

## Saturday, March 26, 2016

### An 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) coefficients of a number. I recently discovered a congruence using these permutations and the product of two binomial coefficients that I decided to blog about. The proof is an

I put all of this into a three page pdf which, if you are interested, you can find here:

https://drive.google.com/file/d/0BwCbLZazAtweN293bkxwYUZEYVk/view?usp=sharing

one seems to be unable to avoid the group obtained by permuting p-adic (or q-adic) coefficients of a number. I recently discovered a congruence using these permutations and the product of two binomial coefficients that I decided to blog about. The proof is an

*indirect*consequence of Lucas and is perhaps more interesting than the result itself. One is then led to look for something related with the Carlitz polynomials, which are the function field analog of the binomial coefficients.I put all of this into a three page pdf which, if you are interested, you can find here:

https://drive.google.com/file/d/0BwCbLZazAtweN293bkxwYUZEYVk/view?usp=sharing

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