Talk

In this talk I will present a topological approach to Euler characteristics of categories in terms of finiteness obstructions. This approach is compatible with almost anything one would want, for example products, coproducts, covering maps, isofibrations, and homotopy colimits. Classical constructions are special cases, for example, under appropriate hypotheses the functorial $L^2$-Euler characteristic of the proper orbit category for a group $G$ is the equivariant Euler characteristic of the classifying space for proper $G$-actions.

We show that for a prime $p$ the smallest $a$ with $a^{p-1}$ that is not congruent to 1 modulo ${p^2}$ does not exceed $(\log p)^{463/252 + o(1)}$ which improves the previous bound $O((\log p)^2)$ obtained by H. W. Lenstra in 1979. We also show that for almost all primes $p$ the bound can be improved to $(\log p)^{5/3 + o(1)}$.

In my talk I will present several topics related to Algebraic K-theory of number fields. I will discuss the connection of divisible elements in even K-groups to the conjectures of Kummer-Vandiver, Iwasawa, Coates-Sinnott and Quillen-Lichtenbaum. I will show the construction of the Stickelberger's splitting map to the boundary map in the Quillen localization sequence and its relation with Coates-Sinnott conjecture and the divisible elements.

In this talk we are going to construct a pairing on the dual Selmer group over the $GL_2$-extension $Q(E[p^{\infty}])$ of an elliptic curve without complex multiplication and with good ordinary reduction at $p$ whenever the dual Selmer satisfies certain--conjectured--torsion properties. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of the $p$-adic $L$-function.

The theory of buildings has been developed by J. Tits in order to study semi-simple algebraic groups from a combinatorial point of view. He classified several classes of buildings and showed that for several other classes there is no hope for a classification. However, the question about a possible classification of compact hyperbolic buildings of higher rank is still open. The only known examples are buildings arising from Kac-Moody groups. In my talk I will explain a joint result with A. Devillers and H. Van Maldeghem which provides a combinatorial characterization of these buildings.