The lecture series consists of three one-hour lectures and a departmental colloquium.
Speaker Information
Professor Felix Otto
Max Planck Institute for Mathematics in the Sciences, Leipzig
https://www.mis.mpg.de/people/felix-otto
Lecture series
TITLE: The thresholding scheme for mean curvature flow and De Giorgi's minimizing movements
ABSTRACT: The flow of interfaces by mean curvature was first formulated in the context of grain growth of polycrystalline materials. The computationally efficient and very intuitive thresholding scheme for mean curvature flow by Osher et.al. naturally extended to such a multi-phase situation. In this mini-course, we motivate this extension based on the fact that mean curvature flow can be regarded as a gradient flow on a suitable configuration space. This makes ideas of De Giorgi on the natural time discretization of such dynamics accessible, which we shall introduce. In fact, the thresholding scheme can be interpreted as such a minimizing movement scheme. If time permits, I'll explain how de Giorgi's ideas can be used to establish a convergence result.
- Lecture 1: 3pm-3:50pm, Wednesday, April 9, 2025; Thackeray Hall 427.
- Lecture 2: 4pm-4:50pm, Wednesday, April 9, 2025; Thackeray Hall 427.
- Lecture 3: 3pm-3:50pm, Thursday, April 10, 2025; Thackeray Hall 703.
Colloquium: 3:30pm, Friday, April 11, 2025; Thackeray Hall 704.
TITLE: A variational regularity theory for Optimal Transportation, and its application to matching
ABSTRACT: The optimal matching of two large point clouds is a special case of optimal transportation between measures, a ubiquitous variational problem. In statistics, it is natural to consider random points clouds that arise from sampling from a given distribution, like the uniform distribution. The setting in two space dimension is known to be critical, and its finer behavior has been predicted by Parisi et al. The predictions rely on the connection between the Monge-Ampere equation, which is the Euler-Lagrange equation for optimal transport, and its linearization, the Poisson equation from electrostatics. Ambrosio et al. established these predictions rigorously on a macroscopic level. A variational regularity theory, used as a large-scale regularity theory, allows to establish this connection down to the microscopic level. It mimics De Giorgi's approach to the regularity theory of minimal surfaces in the sense that a harmonic approximation result is at its center.
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Contact: Ming Chen & Dehua Wang