Talk

A t-structure is a kind of truncation on a triangulated category that is analogous to the Postnikov approximation in homotopy theory. Using the thick subcategory theorem of Hopkins and Smith, work of Bousfield gives a classification of these truncations in the category of finite p-torsion topological spaces. After a gentle introduction, we will look at the analogous situation in the derived category of a commutative Noetherian ring D(R) and construct complete invariants of t-structures on a subcategory of D(R).

In this joint work with Stephan Baier, we prove a subconvexity bound for Godement-Jacquet $L$-functions associated with Maass forms for $SL(3,Z)$ The bound arrives from extending a method of M. Jutila (with new ingredients and innovations) on exponential sums with Fourier coefficients of holomorphic cusp forms for $SL(2,Z)$ to a $GL(3)$ setting.

I describe the construction of a moduli space for the connected subgroups of an algebraic group, and of a universal family. I give a quick illustration of the notion of universal families, trough Grassmann varieties, then discuss in turn the construction of a moduli space and of a universal family, balancing general statements and examples.

The stabilization hypothesis of Breen,Baez and Dolan states that k-fold monoidal n-category is "the same" as (k+1)-fold (and therefore $\infty$-fold) monoidal n-category if k is grater or equal to n+2. In the first half of my talk I will explain this hypothesis in an informal manner and I will relate it to the geometry of configuration spaces of k points in $R^n$. In somewhat more technical second half I will introduce n-braided operads and will give a sketch of a proof of a stabilization theorem for n-braided operads.

I will explain how the Bloch-Kato conjecture leads to the following conclusion: any large prime dividing a critical value of the L-function of a classical Hecke eigenform of level 1, should also divide a certain ratio of critical values for the standard L-function of a related genus 2 (and in general vector-valued) Hecke eigenform F. This can be proved in the scalar-valued case, and there is experimental evidence in the vector-valued case (where the relation between f and F is a congruence of Hecke eigenvalues conjectured by Harder).