Geometric Analysis and Topology Seminar

Fall 2014

The seminar's usual time is Friday at 11:00am in 517 Warren Weaver Hall (Directions). Special times and dates are marked in red. Click on the title of a talk for the abstract (if available).

Upcoming seminars:

Feb 6
Raanan Schul
(Stony Brook)
Rectifiable measures 517 WWH
Feb 13
Hisham Sati
(University of Pittsburgh)
Cohomological structures in geometry and topology inspired by mathematical physics 517 WWH
Feb 23
Mark Sapir
Asymptotic cones of groups 517 WWH
Apr 3
Spyros Alexakis
The impossibility of periodic motion in general relativity 517 WWH
Apr 10
Xiuxiong Chen
(Stony Brook)
On a new continuity path for canonical metrics in Kaehler geometry 517 WWH
Apr 24
Bing Wang
(UW Madison)
The Kähler Ricci flow on Fano manifolds 517 WWH
May 8
Dan Knopf
(UT Austin)
Universal behaviors in geometric heat flows 517 WWH

Organizers: Sylvain Cappell, Jeff Cheeger, Robert Haslhofer, Bruce Kleiner, and Robert Young.

Previous semesters:


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 1-dimensional 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 K-theory, and then generalizing to the newly constructed twisted elliptic cohomology, and twisted Morava K-theory and E-theory. Then I will discuss recent work on desirable differential geometric extensions of these theories, otherwise constructed by homotopy-theoretic means, building on Cheeger-Simons differential characters and Deligne-Belinson 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 Gromov-Hausdorff 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 slow-motion approximation, the derivation of the equations of motion to first post-Newtonian 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 time--the problem of eternal return. For asymptotically simple space-times, 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 Cheeger-Colding-Tian theory for non-collapsed Einstein manifolds, we develop the compactness of the moduli of non-collapsed Kähler Calabi-Yau 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 Hamilton-Tian conjecture and the partial-C0-conjecture 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 diffusion-reaction structure that makes them prone to finite-time 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.

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