Personal Profile of Prof. Christina Sormani Ph.D.

Biography

My research concerns the Gromov-Hausdorff, Metric Measure and Intrinsic Flat Convergence of Riemannian Manifolds and Metric Measure Spaces with applications to General Relativity and the Topology of Riemannian Manifolds. I was born in New York City, attended Hunter High School and then New York University. I completed my doctorate at Courant in 1996 with Jeff Cheeger (also taking courses with Peter Lax, Fanghua Lin, Louis Nirenberg, and Gang Tian). I had a first postdoc at Harvard with Shing-Tung Yau and a second postdoc at Johns Hopkins where I worked with William Minicozzi and Joel Spruck. I then returned to New York City taking an assistant professorship at CUNY's Lehman College where I am now a Full Professor. Since 2003 I have been a member of the doctoral faculty at the CUNY Graduate Center.

Talks

March 02, 2016 (09:00 AM PST - 11:00 AM PST)
Seminar Reading group on Convergence of Metric Spaces

January 14, 2016 - January 15, 2016

Workshop Connections for Women: Differential Geometry

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Jan 14, 2016
Thursday

03:30 PM - 04:30 PM

Sliced Filling Volumes and Intrinsic Flat Convergence
Christina Sormani (CUNY, Graduate Center)

December 07, 2013 - December 08, 2013

Workshop Infinite-Dimensional Geometry

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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

September 03, 2013 - September 04, 2013

Workshop Connections for Women: Mathematical General Relativity

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Sep 03, 2013
Tuesday

05:30 PM - 06:15 PM		Panel Discussion Andrea Bertozzi (University of California, Los Angeles), Manuela Campanelli (Rochester Institute of Technology), Carla Cederbaum (Tübingen University), Alexi Hoeft (Rutgers University), Christina Sormani (CUNY, Graduate Center), Iva Stavrov (Lewis and Clark College)
Location MSRI: Commons Room Video -- Abstract -- Supplements -- Show Detail

August 22, 2013 - August 23, 2013

Workshop Connections for Women on Optimal Transport: Geometry and Dynamics

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Aug 22, 2013
Thursday

09:30 AM - 10:30 AM

Convergence of Riemannian manifolds and Metric Measure Spaces
Christina Sormani (CUNY, Graduate Center)

Location

MSRI: Simons Auditorium

Video

Abstract

Recent advances in Geometric Analysis and in Optimal Transport have provided new insight into both fields. In Geometric Analysis we study the limits of sequences of Riemannian manifolds and produce limit spaces with a variety of structures. With lower bounds on Ricci curvature, Cheeger-Colding combined Gromov's Compactness Theorem and ideas of Fukaya to show that sequences of Riemannian manifolds with uniform lower Ricci curvature bounds have metric measure limits. They show these limits are metric measure spaces with a doubling measure that has many of the properties of a Riemannian manifold with a lower Ricci curvature bound. Sturm and Lott-Villani then generalized the notion of a lower Ricci curvature bound to metric measure spaces using notions from Optimal Transport. Sturm also defined a new notion of metric measure convergence of metric measure spaces based upon the Optimal Transport notion of a Wasserstein distance between probability measures. Other notions of convergence provide more structure on the limit spaces and do not require the lower Ricci curvature bound. One notion, with applications in General Relativity, is the intrinsic flat distance introduced in joint work with Stefan Wenger (building upon work of Ambrosio-Kirchheim) produces integer weighted countably $\mathcal{H}^m$ rectifiable limit spaces. A newer notion soon to be introduced in joint work with Guofang Wei (building upon work of Solorzano and on Sturm's metric measure convergence) preserves the structure of the tangent bundle. We provide a brief survey of these notions with examples. The details in the papers can be understood after the audience has attended the introductory workshop next week.

Supplements

v1123 1.11 MB application/pdf

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