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== Introduction ==
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.
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== The Lectures ==
In this 16 lecture course we will cover many topics that have connection both the the theoretical or pure aspects of GMT as
well as the emerging applications of these tools and insights
to various data analysis challenges.
These "lectures" will each be 2 hours in length due to the
fact that both problem breaks and Q&A breaks will be
integrated into the lectures. While this lengthens the time
at each lecture, the students will come away with a much
firmer grip on the concepts and will be able to apply the
ideas with little extra work.
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== Instructors and Speakers ==
The main instructors are [http://ddma.lanl.gov/team/Members/vixie Kevin R. Vixie], [http://ddma.lanl.gov/team/Members/morgan Simon P. Morgan] and
[http://ddma.lanl.gov/team/Members/pschultz Pete Schultz]. There will be additional lectures from
'''guest lecturers''' (not included in the list below). Additionally, we
expect problem sessions (also not shown below) in which '''students''' present ideas
and solutions to problems as well as new ideas of their
own.
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== References ==
1. '''Frank Morgan's ''Geometric Measure Theory: A beginners guide'', 2000'''
2. '''Evans and Gariepy's ''Measure theory and Fine Properties of Functions'', 1999'''
3. Krantz and Parks' ''Geometric Integration Theory'', preprint 2006
3. Mattila's ''Geometry of Sets and Measures in Euclidean Spaces'' 1995
4. Leon Simon's ''Lectures on Geometric Measure Theory'', 1983
5. Lin and Yang's ''Geometric Measure theory'' 2002
6. Federer's ''Geometric Measure Theory'', 1969
As hinted at above, if you are only going to buy two texts to begin with, get Morgan and Evans and Gariepy. Krantz and Parks is free. If you are serious about the subject, get all 7. (And buy the final published version of Krantz and Parks when it is published.)
See also the expanded annotated bibliography page found
[https://ddma.lanl.gov/Members/vixie/.personal/Bib.GMT.Sum.2007/ here].
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== Syllabus ==
'''Lecture 1''': Introduction and Panorama: The zoology and physiology of sets, functions and measures in Euclidean space.
'''Lecture 2''': Functions of Bounded variation: a path into GMT I
'''Lecture 3''': Functions of Bounded variation: a path into GMT II
'''Lecture 4''': 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.
'''Lecture 5''': Isoperimetric inequality: integration on manifolds and Hausdorff measure
'''Lecture 6''': Currents I
'''Lecture 7''': Currents II
'''Lecture 8''': 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.
'''Lecture 9''': Densities, Tangent Cones and Rectifiable Sets I
'''Lecture 10''': Densities, Tangent Cones and Rectifiable Sets II
'''Lecture 11''': Regularity for Rectifiable Sets and Sets of Finite Perimeter: Approximate tangent spaces and structure theory for Cacciopoli Sets.
'''Lecture 12''': 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?
'''Lecture 13''': Minimal Surfaces: variational picture, result for graphs, and calibrations. But what about existence, uniqueness, and regularity in general?
'''Lecture 14''': Slicing, Closure, Deformation, and Compactness I
'''Lecture 15''': Slicing, Closure, Deformation, and Compactness II
'''Lecture 16''': Big Picture Again. More questions and little tastes of the many subjects we didn't cover: Brunn-Minkowski, Curvature Measures, Varifolds, and Stochastic Connections
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== Additional Information ==
