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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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Convergence of Riemannian manifolds and Metric Measure Spaces
Christina Sormani (CUNY, Graduate Center)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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v1123
1.11 MB application/pdf
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10:30 AM - 11:00 AM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Higher order isoperimetric inequalities - an approach via the method of optimal transport
Sun-Yung Chang (Princeton University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- " Higher order isoperimetric inequalities --an approach via the method of
optimal transport."
One of the method to derive sharp isoperimetric inequality for domains
in the Euclidean is to apply the method of optimal transport; in this
talk, I will report some recent joint work with Yi Wang to extend the method
to prove some higher order isoperimetric inequalities with
integrands involving symmetric functions of the second fundamental form.
- Supplements
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v1124
170 KB application/pdf
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12:00 PM - 02:00 PM
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Lunch
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- Location
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- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Applications of optimal transportation and Wasserstein barycenters in computer vision, image and video processing.
Julie Delon (Télécom PARISTECH)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- The purpose of this talk is to present an overview of the applications of optimal transport in computer vision, image and video processing. The use of optimal transportation in these fields has been popularized twelve years ago by Rubner et al. for image retrieval and texture classification, with the introduction of the so-called Earth Mover's Distance (EMD). Nowadays, Monge-Kantorovich distances are used for applications as various as object recognition and image registration. The other interesting aspect of this theory lies in the transportation map itself and the possibility to define barycenters between multiple distributions. These notions permit many image and video modifications, such as contrast and color transfer, or texture mixing, to name just a few. However, optimal transport suffers from two important flaws for such applications. First, discrete optimal transport maps are generally irregular and tend to produce artifacts in images. The second drawback is that optimal transport generally leads to computationally expensive solutions, which make it impracticable in many real applications. We will see how these problems can be handled in practice, by introducing well chosen regularization and approximations
- Supplements
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v1125
615 KB application/pdf
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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03:30 PM - 04:30 PM
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Dynamics of kinematic aggregation equations
Andrea Bertozzi (University of California, Los Angeles)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- Aggregation equations are used to model nonlocal interactions in biology and related problems in which a population density moves with a velocity that depends nonlocally on the density through an aggregation interaction kernel.
The mathematical behavior of these equations has direct connection to problems in optimal transport and incompressible fluid dynamics. I will review recent results for these problems and the role of geometry in the dynamics of both smooth and singular solutions
- Supplements
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v1126
2.74 MB application/pdf
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04:30 PM - 05:30 PM
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Panel Discussion
Andrea Bertozzi (University of California, Los Angeles), Sun-Yung Chang (Princeton University), Marina Chugunova (Claremont Graduate University), Eleonora Cinti (Università di Bologna), Wilfrid Gangbo (University of California, Los Angeles), Maria Westdickenberg (RWTH Aachen)
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- Location
- SLMath: Commons Room
- Video
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- Abstract
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- Supplements
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06:30 PM - 08:30 PM
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Dinner at the Taste of the Himalayas
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