Geometric Analysis and Topology Seminar

Fall 2016


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
1314 WWH
Bruce Kleiner
(NYU)
Uniqueness of weak solutions to Ricci flow, and Perelman's convergence conjecture
Oct 5,  11am
1314 WWH
Sylvester Eriksson-Bique
(NYU)
Poincare inequalities via quantitative connectivity, and differentiability in metric measure spaces
Oct 26,  11am
1314 WWH
Guy David
(NYU)
Differentiability and rectifiability on metric planes
Nov 2,  11am
1314 WWH
Jeff Cheeger
(NYU)
Bounded Ricci curvature and the codimension 4 conjecture
Nov 4,  11am
517 WWH
Yair Minsky
(Yale)
Fibrations, subsurface projections and veering triangulations
Nov 10, 10am
1314 WWH
Assaf Naor
(Princeton)
A new vertical-versus-horizontal isoperimetric inequality on the Heisenberg group,
with applications to metric geometry and approximation algorithms
Nov 16,  11am
1314 WWH
Andrea Schioppa
(ETH)
The structure of currents in R^n and in metric spaces


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

Abstracts:

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 weights.

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.
A new vertical-versus-horizontal isoperimetric inequality on the Heisenberg group, with applications to metric geometry and approximation algorithms, Assaf Naor. In this talk we will show that for every measurable subset of the Heisenberg group of dimension at least 5, an appropriately defined notion of its ``vertical perimeter" is at most a constant multiple of its horizontal (Heisenberg) perimeter. We will explain how this new isoperimetric-type inequality solves open questions in analysis (an endpoint estimate for a certain singular integral on W^{1,1}), metric geometry (sharp nonembeddability into L_1) and approximation algorithms (asymptotic evaluation of the performance of the Goemans-Linial algorithm for the Sparsest Cut problem). Joint work with Robert Young.
The structure of currents in R^n and in metric spaces, Andrea Schioppa.  I will talk about recent progress in understanding the structure of normal currents in Euclidean spaces and of Ambrosio-Kirchheim metric currents in general metric spaces. In particular, I will focus on the solution of the following question of Giovanni Alberti: Can a k-dimensional normal current be represented as an integral of k-rectifiable currents?


Previous semesters:

Please email comments and corrections to bkleiner@cims.nyu.edu.