Registration is now open for this master class. For more details please check their website here.

# Category Archives: Blog

# Noncommutative Geometry and Index Theory for Group Actions and Singular Spaces

This weeklong workshop will be an opportunity for young researchers in noncommutative geometry to share their ideas with their peers and with some of the leaders of the field. Apart from lectures by young researchers there will be mini-courses by Pierre Albin, Nigel Higson and Shmuel Weinberger. Topics will include the analysis and geometry of hypoelliptic operators, group actions on aspherical manifolds, index theory on singular spaces, and secondary invariants.

Some funding is available for participants, and there are still opportunities for contributed talks. For further information, see the link below.

# Maryam Mirzakhani (May 3, 1977 – July 15, 2017)

Some twenty five years ago when I was told by one of her teachers that she is truly brilliant and exceptional I could have not imagined that I will see this terribly sad day. A shining star is turned off far too soon. Heartfelt condolences to her husband, family, and mathematics community worldwide.

“Two eyes are gone

Thousands of minds withered”

# Great Thanks!

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

and here is a link to the photos Jean-François took on April first !

# 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.**

# Connes 70

# 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 *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 far

is 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 of

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

# 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

# What is a functional equation?

Like all number theorists I am fascinated (to say the least) with the functional equation of

*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).

# Summer School “Coarse Index Theory”, September 26-30, 2016, Freiburg

**Topic: ** Index theory is a prime example of fruitful interaction between analysis, geometry, topology and operator algebras.

The index is associated to a global differential operator and is computed from the set of solutions of the associated differential equation. It turns out, however, that the index has remarkable stability properties and can often be computed a priori without solving the differential equation. This uses index theorems and the underlying topology. On the other hand, the most interesting operators are tied to the geometry and the geometry determines the set of solutions. The most powerful implementations of this idea that the relevant operators lie in operator algebras which are specific to the situation at hand. The indices are then naturally defined as elements in K-theory groups of these operator algebras.

It turns out that a particularly useful setup uses the ideas of “coarse geometry”. The basic idea is to study (non-compact) metric spaces; but considering only their large scale features. A lot of this can be captured in appropriately associated C*-algebras; the coarse C*-algebras of the space (often called Roe algebra). This tool also applies to compact spaces, by passing first to their universal covering.

The corresponding manifestation of index theory in this context is “coarse index theory” or “large scale index theory” and has many interesting properties and applications.

The summer school will explain the relevant general background in index theory, operator algebras; and then focus on large scale geometry and index theory and its numerous applications.

**Program: **There will be lecture series by

- John Roe: Coarse geometry and index theory
- Thomas Schick: (Secondary) Coarse index and applications
- Rudolf Zeidler: K-theory of C*-algebras

along with daily exercise and discussion sessions in the afternoon.

**Schedule: Preliminary schedule**

**Funding: **As a general rule, you are supposed to arrive with your own funding but there are also some limited funds available.

**Contact**: Please send an informal email to enroll until **September 1, 2016, latest, **to Mrs. Ursula Wöske, **coarse16@math.uni-freiburg.de**

**Organizer: **Nadine Große

**Speakers: **John Roe, Thomas Schick, Rudolf Zeidler

**Poster: Poster Summer School 2016**