Geometric Analysis and Topology Seminar
|PLEASE NOTE CHANGE OF REGULAR TIME AND LOCATION:The seminar's usual time is Wednesday at 11:00am in 1314 Warren Weaver Hall (Directions). Special times and dates are marked in red. Click on the title of a talk for the abstract (if available).|
|Sept 21,  11am |
| Bruce Kleiner
|Uniqueness of weak solutions to Ricci flow, and Perelman's convergence conjecture|
|Oct 5,  11am |
| Sylvester Eriksson-Bique
|Poincare inequalities via quantitative connectivity, and differentiability in metric measure spaces|
|Oct 26,  11am |
| Guy David
|Differentiability and rectifiability on metric planes|
|Nov 2,  11am |
| Jeff Cheeger
|Bounded Ricci curvature and the codimension 4 conjecture|
|Nov 4,  11am |
| Yair Minsky
|Fibrations, subsurface projections and veering triangulations|
|Nov 16,  11am |
| Andrea Schioppa
|Three questions of Giovanni Alberti on the structure of normal currents and Lipschitz maps|
|Organizers: Sylvain Cappell, Jeff Cheeger, Bruce Kleiner, and Robert Young.|
|Uniqueness of weak solutions to Ricci flow, and Perelman's convergence conjecture, Bruce Kleiner. In his proof of Thurston's geometrization conjecture, Perelman proved the existence of a Ricci flow with surgery starting from any given compact smooth Riemannian 3-manifold. In the same papers, he conjectured that when the surgery parameters are sent to zero, the flow with surgery converges to a limiting "flow through singularities", yielding a canonical generalized Ricci flow. The lecture will briefly cover some background on uniqueness questions for weak solutions to geometric evolution equations (Ricci flow, mean curvature flow and harmonic map heat flow), and then discuss recent joint work of Richard Bamler and myself, giving a proof of Perelman's convergence conjecture.|
Poincare inequalities via quantitative connectivity, and differentiability in metric measure spaces, Sylvester Eriksson-Bique.
The talk will start by discussing some background of Poincare
inequalities on metric measure spaces. We will then introduce a new
condition that is equivalent to admitting a Poincare inequality. This
metric condition is flexible enough to have a number of applications, and
we immediately obtain new classes of spaces admitting Poincare
inequalities. We also observe a connection to classical analysis on
Euclidean space through Orlicz-Poincare inequalities and Muckenhoupt
In the second half of the talk we introduce an asymptotic version of our condition, and explain how it guarantees a new type of rectifiability result in terms of spaces admitting Poincare inequalities. This result is applied to resolve, in a particular context, a conjecture of Cheeger and Kleiner on so called differentiability spaces. The talk should be understandable without particular knowledge of the applications.
|Differentiability and rectifiability on metric planes, Guy David. Since the work of Cheeger, many non-smooth metric measure spaces are now known to support a differentiable structure for Lipschitz functions. The talk will discuss this structure on metric measure spaces with quantitative topological control: specifically, spaces whose blowups are topological planes. We show that any differentiable structure on such a space is at most 2-dimensional, and furthermore that if it is 2-dimensional the space is 2-rectifiable. This is partial progress on a question of Kleiner and Schioppa, and is joint work with Bruce Kleiner.|
|Fibrations, subsurface projections and veering triangulations, Yair Minsky. When a hyperbolic 3-manifold fibers over the circle, its geometric features can be read from the fine structure of its monodromy map, specifically the "subsurface projections" of the stable and unstable foliations to the arc complexes of subsurfaces of the fiber. While this correspondence is useful when the topological type of the fiber is fixed, it is not well-understood in general. A good laboratory for studying this is a single 3-manifold that fibers in infinitely many different ways, as organized by Thurston's norm on homology. In this setting there are canonical triangulations due to Agol, which can be studied very explicitly via a construction of Gueritaud. We explore how the subsurface projections of monodromies for all the fibers can be seen in the structure of this triangulation, and how this leads to a nice combinatorial picture with estimates that do not depend on complexity of the fibers. Joint work with Sam Taylor.|
Three questions of Giovanni Alberti on the structure of normal currents and Lipschitz maps, Andrea Schioppa.
I will explain how techniques originating in Analysis on Metric Spaces can be applied to solve three questions of Alberti in Euclidean Geometric Measure Theory:
Q1. Can a k-dimensional normal current be represented as an integral of k-rectifiable currents?
Q2. If mu is a finite Radon measure and g:R^N ---> R^N is bounded Borel measurable function, can you solve grad f = g on a set of large measure?
Q3. If mu is a singular Radon measure can you construct a Lipschitz f:R^N ---> R with a prescribed bad non-differentiability behavior on a set of large measure?
The solution of Q2 and Q3 is joint work with Andrea Marchese (University of Zurich).