Talk

In G-equivariant (Bredon) cohomology the integer grading is often extended to a grading over the real representations of G. We consider the general question of what other indices we could grade cohomology on. In a precise sense we will show that the universal group to grade on is the Picard group of the G-equivariant stable category. When G is a finite cyclic group we identify this group and show that it is generated by representation spheres. We then calculate the Pic(G) graded cohomology of a point which turns out to already be an interesting calculation.

We will study a system of Galois representations associated to algebraic varieties,
with values in an algebraic group. In particular we will focus on questions of rationality
and l-independence.

Several recent works make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold M and the symplectic geometry of the cotangent bundle of M is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle.

Given a compact exact Lagrangian submanifold of the cotangent bundle of M, we can add a variable and associate with it a conic Lagrangian submanifold of the cotangent bundle of MxR, say L. We will see that it is possible to build a sheaf on MxR with microsupport L, in a canonical way. We recover from this construction an earlier result of Abouzaid, which says that the projection to M induces a homotopy equivalence between L and M.

In my talk, I will discuss the differentiability of Fourier Series of
the form

F_k(\tau)=\sum_{n=1}^{\infty}\sigma_{k-1}(n) n^{-k-1}e^{2\pi i n \tau} for k even.

These series are related to Eisenstein Series. Using modular (and
quasi-modular) properties of Eisenstein Series, we can find functional
equations for F_k, from which we can draw some conclusions on
differentiability of F_k. This approach was introduced by Itatsu in
1981 in a paper on Differentiability of Riemann's Function.

In the main part of my talk I will focus on the case when k=2. The
imaginary
part and the real part of F_2 exhibit different behaviour while
considering the
differentiability at both rational and irrational points. We find
that the differentiability of the imaginary part of F_2 at an irrational
point x depends
on the properties of the continued fraction expansion of x.

Then I will talk about the general case k even.