Tim Austin: Rational group ring elements with kernels having irrational von Neumann dimension.
A 1972 question of Atiyah asks for examples of finitely-generated
groups G together with cocompact free proper G-manifolds whose L^2
Betti numbers are irrational. Building on earlier work that converts
this into a question about the von Neumann dimension of the kernel of
an element of the rational group ring of G, I will describe a recent
construction of a family of groups and group ring elements answering
this question.
Xiuxiong Chen: The space of Kaehler metrics.
In this talk, we will discuss geometric structure in the infinite dimensional
space of Kahler potentials.
In particular, we will discuss recent progress in Kaehler geometry
(existence and
uniqueness of extremal Kahler
metrics where Kahler Einstein is a special case). Moreover, we will
discuss some problems in Kahler geometry which might be useful to attack the existence problem
of extremal Kahler metrics via deformation methods.
Toby Colding:
Sharp Hölder continuity of tangent cones for spaces with a lower Ricci
curvature bound and applications.
I will discuss new estimates for manifolds and spaces with Ricci
curvature bounds and discuss various applications to both collapsed and
non-collapsed limits. This is ongoing joint work with Aaron Naber.
Larry Guth: Contraction of surface areas vs. topology of mappings.
The k-dilation of a mapping measures how much the mapping
stretches k-dimensional
areas. How does a bound on the k-dilation of a mapping restrict the
topology of the mapping?
We focus on the following simple-sounding problem: if F is a degree 1 map
from one n-dimensional ellipse to another, how small can the k-dilation of
F be? Work on this problem draws on tools from minimal surface theory
(minimax inequalities, isoperimetric inequalities), from
topology (cup powers, Hopf invariant, Steenrod squares), and from partial
differential relations
(h-principles including Smale's h-principle for immersions).
Jeremy Kahn: Essential Immersed Surfaces in Closed Hyperbolic Three-Manifolds.
Given any closed hyperbolic 3-manifold M and &epsilon > 0,
we find a closed hyperbolic surface S and an immersion f: S ---> M
such that f lifts to a 1+ &epsilon -quasi-isometry
from the universal cover of S to the universal cover of M.
In particular f induces an injection on the fundamental group of S;
thus there is an essential immersed surface in every closed hyperbolic 3-manifold.
I will explain why the mixing of the frame flow on M
implies the existence of a highly symmetric collection of pairs of pants,
which can then be assembled to form the desired surfaces S.
Gang Tian: Kahler-Ricci flow through finite-time singularities.
I will discuss singularity formation of Ricci flow in Kahler
geometry at finite time. We will show an approach of studying the
singularity through a new degenerate elliptic equation.
|