I will talk about the geometry of the moduli space $\mathcal M$ of holonomy $G_2$ metrics. In particular I will discuss the Hessian metric structure, the Yukawa coupling, and the sectional curvature of this moduli space. This is a combination of past work with Conan Leung and new work in progress with Christopher Lin.

The classical Watts' theorems identify functors which are tensor products or Hom functors by internal properties. We extend these theorems to homological algebra and algebraic topology. So, in the easiest case, we characterize all functors from the unbounded derived category $D(R)$ of a ring $R$ to $D(S)$ which are given by the derived tensor product with a complex of bimodules (recovering a result of Keller's in this case). We draw conclusions about Brown representability of homology and cohomology functors.

Note room change: MC107.

O-minimal structures are categories of sets and mapping having nice geometrical properties. To each o-minimal expansion of a real closed field, one can associate the set of germs at infinity of its unary functions, which form a Hardy field. Valuational properties of these Hardy fields give good information about the initial structure. After a lengthy introduction of all the previously named objets and motivated by a conjecture of L. van den Dries and a result of F.-V. and S. Kuhlmann, I will discuss whether an o-minimal expansions of the field of the reals is, in general, fully determined by its associated Hardy field. I will also relate this question to the Hilbert's 13th Problem.

This is a report on efforts to classify the endotrivial modules over the modular groups algebras of groups which are not $p$-groups. A classification of the endotrivial modules over $p$-groups was completed by the speaker and Th\'evenaz a few years ago, building on the work of many others, notably Dade and Alperin. The endotrivial modules form an important part of the Picard group of self equivalences of the stable category of modules over the group algebra. For groups which are not $p$-groups, the problem of determining the endotrivial modules often reduces to discovering when the Green correspondent of an endotrivial module is endotrivial. This investigation often involves a detailed study of the representation theory of the groups in question.

This talk will present an introduction to some continuing work being conducted with Eric Friedlander, Julia Pevtsova and Andrei Suslin. The work is concerned with some basic questions about sets of commuting nilpotent operators on vector spaces. The objects that we construct generalize the class of endotrivial modules that is important in the modular representation theory of finite groups. They can also be used to construct bundles on projective spaces and Grassmannians.

I will talk about the dynamical aspects of Linar Symplectic Widths, and then I will introduce nonlinear capacities.

The aim of the talk will be to look at non-abelian analogues of Kummer theory for function fields of Riemann surfaces and relate them to bundles. We will trace the ramifications of Weil's paper "Generalizations des fonctiones abeliennes".