We will discuss families of Pell's equation in polynomials
with one complex parameter. In particular the relation between
the generic equation and its specializations. Our emphasis will
be on families with a triple zero. Then additive extensions enter
the picture.
I will present a basic overview of finiteness conditions, group cohomology, and related quasi-isometry invariance results. In particular, I will show that if a group satisfies certain finiteness conditions, group cohomology with group ring coefficients encodes some structure of the `homology at infinity' of a group. This is seen for hyperbolic groups in the work of Bestvina-Mess, which relates the group cohomology to the Čech cohomology of the boundary.
I will talk about a remarkable theorem by Paulin, which says
that if a one-ended hyperbolic group has infinite outer automorphism
group, then it splits over a two-ended subgroup. In particular, this
gives a condition which ensures a hyperbolic group doesn't have property
(T).
In the last few years, it has become clear that there are striking connections between supersymmetry and geometric representation theory. In this talk, I will discuss boundary conditions in three dimensional gauge theories with N = 4 supersymmetry. I will then outline a physical understanding of a remarkable conjecture in representation theory known as `symplectic duality.
In this talk I will give several perspectives on the role of
quasi-abelian categories in analytic geometry. In particular, I will
explain why a certain completion of the category of Banach spaces is a
convenient setting for studying sheaves of topological vector spaces on
complex manifolds. Time permitting, I will also argue why this category
may be a good candidate for a functor of points approach to (derived)
analytic geometry.
I will collect some results about the study of topological and algebraic invariants of this moduli space by using non-abelian Hodge theory. Some keywords are: Higgs bundles, Mixed Hodge structures.