L2

An important problem in algebra is the study of algebraic objects

defined over fields and how they behave under field extensions,

for example the Brauer group of a field, Galois cohomology groups

over fields, Milnor K-theory of a field, or the Witt ring of bilinear

forms over

a field. Of particular interest is the determination

of the kernel of the restriction map when passing to a field extension.

We will give an overview over some known results concerning the

kernel of the restriction map from the Witt ring of a field to the

Witt ring of an extension field. Over fields of characteristic

not two, general results are rather sparse. In characteristic two,

we have a much more complete picture. In this talk, I will

explain the full solution to this problem for extensions that are

given by function fields of hypersurfaces over fields of

characteristic two. An important tool is the study of the

behaviour of differential forms over fields of positive

characteristic under field extensions. The result for

Witt rings in characteristic two then follows by applying earlier

results by Kato, Aravire-Baeza, and Laghribi. This is joint

work with Andrew Dolphin.

Let L be a positive definite lattice. There are only finitely many positive definite lattices

L' which are isomorphic to L modulo N for every N > 0: in fact, there is a formula for the number of such lattices, called the Siegel mass formula. In this talk, I'll review the Siegel mass formula and how it can be deduced from a conjecture of Weil on volumes of adelic points of algebraic groups. This conjecture was proven for number fields by Kottwitz, building on earlier work of Langlands and Lai. I will conclude by sketching joint work (in progress) with Dennis Gaitsgory, which uses topological ideas to attack Weil's conjecture in the case of function fields.

Since the work of Feigenbaum and Coullet-Tresser on universality in the period doubling bifurcation, it is been understood that crucial features of unimodal (one-dimensional) dynamics depend on the behavior of a renormalization (and infinite dimensional) dynamical system. While the initial analysis of renormalization was mostly focused on the proof of existence of hyperbolic fixed points, Sullivan was the first to address more global aspects, starting a program to prove that the renormalization operator has a uniformly hyperbolic (hence chaotic) attractor. Key to this program is the proof of exponential convergence of renormalization along suitable ``deformation classes'' of the complexified dynamical system. Subsequent works of McMullen and Lyubich have addressed many important cases, mostly by showing that some fine geometric characteristics of the complex dynamics imply exponential convergence.

We will describe recent work (joint with Lyubich) which moves the focus to the abstract analysis of holomorphic iteration in deformation spaces. It shows that exponential convergence does follow from rougher aspects of the complex dynamics (corresponding to precompactness features of the renormalization dynamics), which enables us to conclude exponential convergence in all cases.

The topic of this talk is the representation theory of Hopf-Galois extensions. We consider the following questions.

Let H be a Hopf algebra, and A, B right H-comodule algebras. Assume that A and B are faithfully flat H-Galois extensions.

1. If A and B are Morita equivalent, does it follow that the subalgebras A^coH and B^coH of H-coinvariant elements are also Morita equivalent?

2. Conversely, if A^coH and B^coH are Morita equivalent, when does it follow that A and B are Morita equivalent?

As an application, we investigate H-Morita autoequivalences of the H-Galois extension A, introduce the concept of H-Picard group, and we establish an exact sequence linking the H-Picard group of A and

the Picard group of A^coH.(joint work with Stefaan Caenepeel)

We shall explain how to give substance to an old dream of Tits, to invent exotic new zeta functions, and discover the skeleton of algebraic varieties (toric manifolds and tropical geometry).

An emerging set of methods enables an experimental dialogue with biological systems composed of many interacting cell types---in particular, with neural circuits in the brain. These methods are sometimes called “optogenetic” because they employ light-responsive proteins (“opto-“) encoded in DNA (“-genetic”). Optogenetic devices can be introduced into tissues or whole organisms by genetic manipulation and be expressed in anatomically or functionally defined groups of cells. Two kinds of devices perform complementary functions: light-driven actuators control electrochemical signals; light-emitting sensors report them. Actuators pose questions by delivering targeted perturbations; sensors (and other measurements) signal answers. These catechisms are beginning to yield previously unattainable insight into the organization of neural circuits, the regulation of their collective dynamics, and the causal relationships between cellular activity patterns and behavior.

We shall report on the use of algebraic geometry for the calculation of Feynman amplitudes (work of Bloch, Brown, Esnault and Kreimer). Or how to combine Grothendieck's motives with high energy physics in an unexpected way, radically distinct from string theory.