# 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

# 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