Talk

Yang-Mills heat flow on gauged holomorphic maps"

I consider the space of gauged holomorphic maps from a Riemann surface to
a compact Kähler manifold, and study the gradient flow of Yang-Mills
type functional, whose zeros are vortices. The flow exists for all time.
When the Riemann surface has boundary, the flow lines converge
to a vortex leading to a Hitchin-Kobayashi correspondence.

In this talk I will report on a joint project with H. Ruddat, D. Treumann and E. Zaslow, which constructs a 'skeleton' for  affine hypersurfaces in toric ambient space, and proves that
there exists a retraction of the hypersurface onto it. The skeleton, which is a half-dimensional CW complex, will be defined using toric data. Defining the retraction will involve techniques from log geometry. If time permits I will discuss applications to homological mirror symmetry for
degenerate Calabi-Yau hypersurfaces, with toric components.

A maximal degeneration of an algebraic variety is, in a
sense, a decomposition of the variety into primitive pieces. Log
geometry provides a framework to investigate such degenerations.
There is an underlying discrete geometry building the bridge to
tropical geometry. Corrections to the naive complex geometry can be
organized by a wall structure in tropical geometry. In favourable
situations one obtains canonical degenerations, and a basis of
canonical sections of an ample line bundle generalizing classical
theta functions.

We will study the asymptotics of the torsion in the cohomology
of certain arithmetic groups with coefficients in certain Z-modules
if the rank of the Z-modules tends to infinity. Our first main result is
that for cocompact groups the torsion grows exponentially where the
leading term is proportional to the volume of the underlying locally
symmetric space. Our method is based on studying the analytic torsion of
locally symmetric spaces with coefficients in flat vector bundles. If these space

Fix $n$. Let $U(sl_n)$ be the universal enveloping algebra of a Lie algebra $sl_n$. We say that an ideal $I$ of $U(sl_n)$ is integrable if it is an intersection of ideals of finite codimension in $U(sl_n)$. Such ideals allows (yet unknown, at least to me!) classification based on the representation theory of finite-dimensional modules of $sl_n$. First result which I will discuss is that, for big enough $N$, a radical of intersection of $I$ with $U(sl_n)$ is integrable for any two-sided ideal $I$ of $U(sl_N)$.