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Geometric Measure Theory

by Katharine Chartrand — last modified 2007-08-13 05:38

Geometric measure theory (GMT) and significant chunks of areas like harmonic analysis and PDE, permit the development of deep insights into the nature of sets, measures, and functions. These insights are critical to the development of methods capable of dealing with the enormous data challenges that now confront us. Examples of relevant work include Jones' Beta, the recent work of Allard, Esedoglu, and Vixie on TV functionals, established results on the Mumford-Shah functional, and various new works focused at analysis of shapes and their evolution and dynamics. This class will be a rather intensive introduction to geometric measure theory and its applications with 2 hour lecture sessions combining lecture, problems, and questions and answers. Lecturers are Kevin R. Vixie, Simon P. Morgan and Peter F. Schultz. 7/9-8/15.

More details are available here.

The first 5 classes will take place July 9-13, 12am-2pm in the Coyote Conference Room, Room 163, Building 410:

Mon 7/9

Introduction and Panorama: The zoology and physiology of sets, functions and measures in Euclidean space.

Tue 7/10

Functions of Bounded variation: a path into GMT I

Wed 7/11

Functions of Bounded variation: a path into GMT II

Thu 7/12

Applications I: Rudin-Osher-Fatemi (ROF), Mumford-Shah, L1TV, TV denoising and reconstruction of images. Question motivating the next 3 lectures: L1TV provides decompositions in codimension 1. What about higher co-dimension? Five minutes with currents, minimal surfaces and the isoperimetric inequality.

Fri 7/13

Isoperimetric inequality: integration on manifolds and Hausdorff measure

The last 11 classes will take place on Mondays and Thursdays, 3pm-5pm in the Coyote Conference Room, Room 163, Building 410 except for July 19th, August 2nd and August 16th. Those classes will take place in the nearby Roadrunner Conference Room in SM 410, room 115.:

Mon 7/16

Currents 1

Thu 7/19

Currents 2 Roadrunner Conference Room

Mon 7/23

Applications II: part a) L1TV and the (new) flat norm with scale and part b) Geomeasures. This last application suggests the usefulness of densities and measure theoretic generalizations of various notions: how much information do these measures really contain? Besicovitch, Marstrand, and Priess as motivation for studying, Densities, tangent cones and rectifiable sets.

Thu 7/26

Densities, Tangent Cones and Rectifiable Sets I

Mon 7/30

Densities, Tangent Cones and Rectifiable Sets II

Thu 8/2

Regularity for Rectifiable Sets and Sets of Finite Perimeter: Approximate tangent spaces and structure theory for Cacciopoli Sets.Roadrunner conference room

Mon 8/6

Applications III: What practical use can be made of regularity results? What insights do they add? ROF, L1TV, Mumford Shah and Exact solutions. Motivation for the next 3 lectures: Classic questions of regularity: minimal surfaces spanning arbitrary boundaries. What can we know about them?

Thu 8/9

Minimal Surfaces: variational picture, result for graphs, and calibrations. But what about existence, uniqueness, and regularity in general?

Mon 8/13

Slicing, Closure, Deformation, and Compactness I

Thu 8/16

Slicing, Closure, Deformation, and Compactness II Roadrunner Conference Room

Mon 8/19

Big Picture Again. More questions and little tastes of the many subjects we didn't cover: Brunn-Minkowski, Curvature Measures, Varifolds, and Stochastic Connections