Schedule, Notes/Handouts & Videos

Infinite-Dimensional Geometry December 07, 2013 - December 08, 2013

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Dec 07, 2013
Saturday

10:00 AM - 10:35 AM		The Intrinsic Flat Metric between Oriented Rectifiable Metric Spaces and Applications" Christina Sormani (CUNY, Graduate Center)
Location Evans Hall Video -- Abstract In the past 25 years great advances were made in the study of Riemannian manifolds and metric measure spaces endowed with a notion of lower bound on Ricci curvature by employing the Gromov-Hausdorff distance and various notions of metric measure convergence. However, in many settings one does not have such strong curvature controls including, for example, questions concerning manifolds with only lower bounds on scalar curvature or upper bounds on volume and diameter. In joint work with Stefan Wenger building upon work of Ambrosio-Kirchheim, we introduced the intrinsic flat distance between compact Riemannian manifolds and, more generally, oriented rectifiable metric spaces of finite Hausdorff measure. Wenger proved a compactness theorem requiring only a uniform upper bound on diameter and volume, and in joint work, we proved convergence with respect to this distance agrees with GH convergence on manifolds with nonnegative Ricci curvature. The relationship with smooth convergence away from singularities was explored in joint work with Sajjad Lakzian who then applied this to understanding Ricci flow through a rotationally symmetric neck pinch singularity. In joint work with Dan Lee, this notion has been applied to show asymptotically flat manifolds of nonnegative scalar curvature are close to Euclidean space if they have sufficiently small ADM mass. In current work in progress with Philippe LeFloch, the relationship between weak H^1 convergence of Riemannian metrics with uniform bounds on ADM mass and intrinsic flat convergence is being explored under rotational symmetry. Most recently, Raquel Perales has begun studying the intrinsic flat limits of Riemannian manifolds with boundary that have uniform control on boundary data without any assumptions on rotational symmetry. Supplements -- Show Detail
10:40 AM - 11:15 AM		A model for random Riemannian metrics Lior Silberman (University of British Columbia)
Location Evans Hall Video -- Abstract "I will describe a model for choosing a random deformation of a Riemannian metric, and some preliminary computations in this model. Supplements -- Show Detail
11:20 AM - 11:40 AM		Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
11:40 AM - 12:15 PM		Convenient Calculus and Differential Geometry in Infinite dimensions, with Applications to Diffeomorphism Groups and Shape Spaces Peter Michor (University of Vienna)
Location Evans Hall Video -- Abstract 1. A short introduction to convenient calculus in infinite dimensions. 2. Manifolds of mappings (with compact source) and diffeomorphism groups as convenient manifolds 3. A diagram of actions of diffeomorphism groups 4. Riemannian geometries of spaces of immersions, diffeomorphism groups, and shape spaces, their geodesic equations with well posedness results and vanishing geodesic distance. 5. Riemannian geometries on spaces of Riemannian metrics and pulling them back to diffeomorphism groups. 6. Robust Infinite Dimensional Riemannian manifolds, and Riemannian homogeneous spaces of diffeomorphism groups. We will discuss geodesic equations of many different metrics on these spaces and make contact to many well known equations (Cammssa-Holm, KdV, Hunter-Saxton, Euler for ideal fluids). Supplements -- Show Detail
12:20 PM - 02:20 PM		Lunch
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
02:20 PM - 02:55 PM		Random hyperbolic surfaces of large genus. Maryam Mirzakhani (Stanford University)
Location Evans Hall Video -- Abstract -- Supplements -- Show Detail
03:00 PM - 03:35 PM		The Cauchy problem for μHS equation: Weak solutions and integrability Feride Tiglay (Ohio State University)
Location -- Video -- Abstract μHS equation has a physical interpretation in the context of liquid crystals and is derived as an Euler-Arnold equation on the group of circle diffeomorphisms. Like its ''cousins'' in this framework, it has certain manifestations of integrability---a Lax pair, a bi-Hamiltonian structure and peakon solutions. I will briefly discuss existence of weak solutions and a way of constructing special functions to integrate the equation. Supplements -- Show Detail
03:40 PM - 04:00 PM		Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
04:00 PM - 04:35 PM		Diffeomorphism groups, infinite dimensional geometry and K-theory Guoliang YU (Texas A & M University)
Location Evans Hall Video -- Abstract I will discuss how geometry of infinite dimensional manifolds can be used to study K-theory for certain diffeomorphism groups. The main tool is a noncommutative space (C*-algebra) naturally associated to the infinite dimensional manifold. I will try to make this talk accessible to non-experts. This is joint work with Jianchao Wu. Supplements -- Show Detail
04:40 PM - 05:15 PM		Riemannian constructions in ideal hydrodynamics Gerard Misiolek (University of Notre Dame)
Location Evans Hall Video -- Abstract -- Supplements -- Show Detail
05:20 PM - 07:00 PM		Reception and Poster Sesssion
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail

Dec 08, 2013
Sunday

09:00 AM - 09:35 AM		On certain special solutions of incompressible Euler's equation. Vladimir Sverak (University of Minnesota Twin Cities)
Location Evans Hall Video -- Abstract Recently Tom Hou and Guo Luo suggested a new scenario for 3d incompressible Euler equation blow-up. The potential blow-up happens at the boundary. The example is related to classical secondary flows and motivated recent progress on the problem of the double exponential growth in 2d incompressible Euler. Joint work with A. Kiselev Supplements -- Show Detail
09:40 AM - 10:15 AM		Optimal Transport with Capacity Constraints Jonathan Korman (University of Toronto)
Location Evans Hall Video -- Abstract A variant of the classical optimal transportation problem is: among all joint measures with fixed marginals and which are dominated by a given density, find the optimal one. I will give some examples and discuss recent progress joint with Robert McCann Supplements -- Show Detail
10:20 AM - 10:40 AM		Break
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
10:40 AM - 11:15 AM		Gaussian free field, random measure and KPZ on R^4 Linan Chen (McGill University)
Location Evans Hall Video -- Abstract A highlight in the study of quantum physics was the work of Knizhnik, Polyakov and Zamolodchikov (1988), in which they proposed a relation (KPZ relation) between the scaling dimension of a statistical physics model in Euclidean geometry and its counterpart in the random geometry. More recently, Duplantier and Sheffield (2011) used the 2-dim Gaussian free field to construct the Liouville quantum gravity measure on a planar domain, and gave the first mathematically rigorous formulation and proof of the KPZ relation in that setting. Inspired by the work of Duplantier and Sheffield, we apply a similar approach to extend their results and techniques to higher even dimensions R^(2n) for n>=2. This talk mainly focuses on the case of R^4. I will briefly introduce the notion of Gaussian free field (GFF). In our work we adopt a specific 4-dim GFF to construct a random Borel measure on R^4 which formally has the density (with respect to the Lebesgue measure) being the exponential of an instance of the GFF. Further we establish a 4-dim KPZ relation corresponding to this random measure. This work is joint with Dmitry Jakobson (McGill University). Supplements -- Show Detail
11:20 AM - 11:55 AM		Metric measure spaces with Riemannian Ricci curvature bounded from below Giuseppe Savare (Università di Pavia)
Location Evans Hall Video -- Abstract The talk will present some recent results, in collaboration with Luigi Ambrosio and Nicola Gigli, concerning various characterizations and properties of Riemannian metric measure spaces with a lower bound of the Ricci curvature. They can be defined starting from the Lott-Sturm-Villani approach by optimal transport and entropy, and assuming that the natural Cheeger energy form is quadratic. We will quickly overview the link with the induced Sobolev spaces, the Heat flow and the gradient flows in spaces of probability measures, the stability with respect to Sturm-Gromov-Hausdorff convergence, the localization and tensorization properties. The connections with the equivalent Bakry-Emery approach by Dirichlet forms and Gamma-calculus will also be discussed Supplements -- Show Detail
12:00 PM - 02:00 PM		Lunch
Location SLMath: Atrium Video -- Abstract -- Supplements -- Show Detail
02:00 PM - 02:35 PM		Some analytic problems in two-dimensional conformal field theory Eric Schippers (University of Manitoba)
Location Evans Hall Video -- Abstract A central object of two-dimensional conformal field theory is the Friedan/Shenker/Vafa moduli space of Riemann surfaces with boundary parameterizations. Radnell and I showed that this moduli space can be identified with the (infinite dimensional) Teichmuller space of bordered surfaces up to a properly discontinuous fixed-point-free group action. In this talk I will give an overview of joint results with Radnell and Staubach, in which we use this correspondence to solve certain analytic problems in the construction of CFT. I will also discuss the relation of these problems to a refinement of Teichmuller space on which the Weil-Petersson metric converges. In particular, we constructed a Teichmuller space of genus g with n boundary curves which is a Hilbert manifold and has a convergent Weil-Petersson metric. This generalizes results of Takhtajan, Teo, Hui, Cui and others for the case of the disc. Supplements -- Show Detail
02:40 PM - 03:15 PM		Spectral geometry on the Riemann moduli space. Rafe Mazzeo (Stanford University)
Location Evans Hall Video -- Abstract I will discuss some problems and results concerning the definition and spectral properties of the Laplacian with respect to the Weil-Petersson metric on the Riemann moduli space of curves, as well as some recent work on the refined asymptotics of this metric near the singular divisors. Supplements -- Show Detail

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