Rectifiable measures, Raanan Schul.
When does a (locally) finite Borel measure $\mu$ on $R^n$ give full measure
to a countable family of Lipschitz images of $R$? We discuss the question,
interesting examples and known results. We will distinguish between two
special cases: $\mu$ is absolutely continuous with respect to the
1dimensional Hausdorff measure $H^1$, vs. $\mu$ is singular with respect
to $H^1$. We will only assume first year graduate measure theory.

Cohomological structures in geometry and topology inspired by
mathematical physics, Hisham Sati.
Cohomology comes in various flavors and plays important roles in
many areas of mathematics. Recent interactions with mathematical physics
have led to several interesting cohomological constructions in geometry and
topology, including categorified notions of bundles associated to higher
connected covers of Lie groups and twisted versions of generalized
cohomology theories. I will describe mathematical results in this area,
starting with twisted de Rham cohomology and twisted Ktheory, and then
generalizing to the newly constructed twisted elliptic cohomology, and
twisted Morava Ktheory and Etheory. Then I will discuss recent work on
desirable differential geometric extensions of these theories, otherwise
constructed by homotopytheoretic means, building on CheegerSimons
differential characters and DeligneBelinson cohomolog.

Asymptotic cones of groups, Mark Sapir.
Asymptotic cones of groups are the major tools in studying "large scale" properties of finitely generated groups such as growth, isoperimetric functions, "representation varieties" of one group in another group, etc. In this talk I will survey the main applications of asymptotic cones. If the group grows polynomially (say, it is Abelian) asymptotic cones are GromovHausdorff limits of rescaled Cayley graphs of the group (in a way, similar to obtaining Brownian motion from random walks). In general one needs ultraproducts to define asymptotic cones. In turn, ultraproducts connect geometry of groups with model theory which lead to very powerful result. For example, recent results of Hrushovsky imply that if one asymptotic cone of a group is locally compact, then the group is virtually nilpotent.

The impossibility of periodic motion in general relativity, Spyros Alexakis.
The problem of motion of bodies in general relativity dates back to the
early days of the theory. Initially considered in the slowmotion
approximation, the derivation of the equations of motion to first
postNewtonian order is due to Einstein, Infeld and Hoffman, with much more
precise approximations obtained since. A natural question considered in this
connection is whether there exist solutions which are periodic in timethe
problem of eternal return. For asymptotically simple spacetimes, we show
that this is not possible, at least near infinity. We relate this to the
problem of reconstructing a solution of the Einstein equations from
knowledge of the radiation it has emitted towards infinity. Joint work with
V. Schlue, and (partly) A. Shao.

On a new continuity path for canonical metrics in Kaehler geometry, Xiuxiong Chen.
Calabi’s original program is about the existence of constant scalar
curvature Kähler metrics. In this talk, I propose a “new” continuity path
to attack the existence problem which links cscK metric equation to certain
second order elliptic PDE. This will be a preliminary report on the
interesting questions such as openness, relation to stability of underlying
line bundle etc.

The Kähler Ricci flow on Fano manifolds, Bing Wang.
As a generalization of CheegerColdingTian theory for
noncollapsed Einstein manifolds, we develop the compactness of the moduli
of noncollapsed Kähler CalabiYau spaces with mild singularities. Based on
this compactness, we set up a structure theory for polarized Kähler Ricci
flows with proper geometric bounds. As applications, we prove the
HamiltonTian conjecture and the partialC0conjecture of Tian. This is a
joint work with X.X. Chen.

Universal behaviors in geometric heat flows, Dan Knopf.
Geometric heat flows like Ricci flow and mean curvature flow have
proven to be remarkably successful tools to investigate the geometry and
topology of manifolds. As is well known, these nonlinear PDE have a
diffusionreaction structure that makes them prone to finitetime
singularities. But these singularities are aids, not obstacles, to these
flows’ applications, because regions of high curvature tend to be very
special. In this talk, we will survey these phenomena and present evidence
in favor of the conjectures that (1) solutions asymptotically acquire extra
symmetries as they become singular, and (2) generic solutions may be
constrained to a small catalog of universal asymptotic profiles.
