Geometric measure theory, fall 2016

Last modified by iholopai@helsinki_fi on 2024/03/27 10:50

Scope: 10 cr

Type: Advanced studies

Teaching: Weeks 36-42 and 44-50, Tuesday 12-14 and Thursday 10-12 in room C124. Note: No lectures (nor home work classes) during weeks 43 (25-26.10) and 44 (1-3.11).

Between 9.11 - 14.12 we will have extra classes/home work sessions on Wednesdays 12-14 in room B322.

Topics:
From the book Geometric measure theory: A beginner's guide by F. Morgan:

"Geometric measure theory could be described as differential geometry, generalized through measure theory to deal with maps and surfaces that are not necessary smooth, and applied to the calculus of variations."

The quotation above describes very well the goal of the course. This course gives an introduction to the theory of varifolds and currents that are kind of generalized surfaces. They have been used in many geometric variational problems, in particular, in connections with higher dimensional minimal surfaces.

Some topics that might be discussed:

Review of measure theory

measures and outer measures
metric outer measures
regularity of measures, Radon measures
Hausdorff measure and dimension
Riesz representation theorem

Lipschitz mappings and rectifiable sets

extension of Lipschitz functions
Rademacher's theorem
area and co-area formulae
rectifiable sets
approximate tangent space
densities
the structure theorem

Varifolds

general varifolds
rectifiable k-varifolds
first variation formula
monotonicity formula and its consequences
regularity theorem

Currents

k-vectors and k-covectors
differential forms
currents (definition and basic notions)
spaces of currents
slicing
deformation theorem
isoperimetric inequality
rectifiability and compactness theorems

Mass (area) minimizing currents

existence
properties of mass minimizing currents
regularity of mass minimizing currents

Prerequisites:

Good knowledge on measure and integration theory (courses like Measure and integral and Real analysis I). The course Real analysis II would be helpful but is not necessary.

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News

No lectures (nor home work classes) during weeks 43 (25-26.10) and 44 (1-3.11).

Teaching schedule

Weeks 36-42 and 44-50, Tuesday 12-14 and Thursday 10-12 in room C124. Note: No lectures (nor home work classes) during weeks 43 (25-26.10) and 44 (1-3.11).

Between 9.11 - 14.12 we will have extra classes/home work sessions on Wednesdays 12-14 in room B322.

Exams

The course can be passed by an exam or preferably by giving presentations in the home work classes and writing essays.

How to pass the course by giving presentations? The plan is that a student gives two presentations (1-2 hours each) on a chosen topic and writes an essay on the topic (and, of course, follows the presentations of others).

Course material

Books

L. Evans and R. Gariepy: Measure theory ans fine properties of functions, CRC Press, 1992.

H. Federer: Geometric Measure Theory, Springer, 1969.

F. Lin and X. Yang: Geometric Measure Theory: An Introduction, International Press, 2002.

P. Mattila: Geometry of sets and measures in Euclidean spaces, Cambridge University Press,1995.

F. Morgan: Geometric Measure Theory, A Beginner's Guide, Academic Press, 1987. An e-book available for students of UH.

L. Simon: Lectures on Geometric Measure Theory, Australian National University, 1983. An updated version.

Lecture notes

P. Mattila: Currents and varifolds. Hand-written lecture notes, fall 2011.

I. Holopainen: Geometric measure theory (first 88 pages + appendix). Proof of the Deformation theorem, discussions on regularity results (stationary varifolds, mass minimizing currents) will be added later.

Registration

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Exercises

Exercises 1 Solutions 1
Exercises 2 Solutions 2
28.9.2016, Presentation on the Riesz representaton theorem by Ville Marttila (see Appendix in the lecture notes)
5.10.2016, Presentation on weak convergence of measures and on the compactness of Radon measures by Janne Siipola (see Appendix in the lecture notes)
12.10.2016, Presentation on Rademacher's theorem by Akseli Haarala
19.10.2016, Presentation on the area formula by Krishnan Narayanan
8.11.2016 (Tuesday), Presentation on the co-area formula by Otte Heinävaara
9.11.2016, Presentation on the Whitney extension by Valter Lillberg
9.11.2016, Presentation on submanifolds and mean curvature by Ville Karlsson
16.11.2016, Presentation on approximate tangent spaces and on rectifiable sets (part I) by Ville Marttila
16.11.2016, Presentation on approximate tangent spaces and on rectifiable sets (part II) by Janne Siipola
23.11.2016, Presentation on the monotonicity formula and isoperimetric inequality by Akseli Haarala
23.11.2016, Presentation on the rectifiability theorem and tangent cones (part I) by Krishan Narayanan
30.11.2016, Presentation on the rectifiability theorem and tangent cones (part II) by Otte Heinävaara
30.11.2016, Presentation on the regularity theory (part I) by Valter Lillberg
7.12.2016, Presentation on the regularity theory (part II) by Ville Karlsson

Exercise classes

Group	Day	Time	Room	Instructor
1.	Wednesday	14-16	C129	Ilkka Holopainen

Course feedback

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