MAT1309HS, Spring 2020
Class Location and Time: BA6180, Wednesday 12-2 and Friday 1-2. First lecture is on
Wednesday January 8.
Course Instructor: Yevgeny Liokumovich, ylio [at] math.toronto.edu, BA6189
Office hours: Wednesday
2-3, Friday 2-3.
main text will be Yu. D. Burago, V. A. Zalgaller, “Geometric
inequalities”, but we will also use many other sources.
About the course:
course will be about geometric constructions and we will draw lots of
pictures. We will try to develop intuition for how the sizes of various
geometric objects relate to each other. We will also encounter many
simple to state and surprisingly important questions, some of them
resolved only recently and many of them still open. We will try to
understand the reasons behind some technical definitions in Geometric
Measure Theory, but will not dwell on them for too long. The course is
mostly self-contained. Mathematical curiosity is required.
What we plan to talk about:
inequalities that relate the “sizes” of geometric objects: classical
isoperimetric inequality, Loomis-Whitney inequality, Besicovitch
inequality, coarea inequality.
- A brief tour of 3 approaches in
- Isoperimetric inequalities in higher
codimension: Ferder-Fleming Deformation Theorem and Wenger’s proof of
Gromov’s and Michael-Simon’s isoperimetric inequalities.
- A brief tour of minimal surfaces:
minimal cones, monotonicity formula, maximum principle.
- Systolic inequalities: theorems of
Lowener, Pu, Gromov and Guth.
- Some topology and geometry of the
space of flat cycles.
After that we will explore topics in
systolic geometry and/or Almgren-Pitts Min-Max theory guided by
(I will keep updating this section and
adding questions and notes as we go along).
- Loomis-Whitney inequality.
- Two versions of Besicovitch
inequality for topological cube.
- A rapid tour of Riemannian
- Consequences of Besicovitch
inequality: Papasoglu's isoperimetric inequality on the
- Consequences of Besicovitch
inequality: Loewner's and Pu's systolic inequalities. Brunn-Minkowski inequality (Burago-Zalgaller §8).
- Classical isoperimetric and
isodiametric (Bieberbach) inequalities, symmetrization (Burago-Zalgaller §9, §11.2).
- Outer measures. Equivalence between
n-dimensional Hausdorff and Lebesgues measures in R^n. (L. Simon,
"Lectures on Geometric measure theory", chapter 1).
- Rademacher theorem. Area and coarea
formulas. (Burago-Zalgaller §13, L. Simon, "Lectures on
Geometric measure theory", chapter 1).
- Proof of Sobolev inequality using
isoperimetric inequality and coarea formula. (Burago-Zalgaller §8.3)
- Waist inequality for the n-sphere. Expository
paper by Larry Guth.
- Federer-Fleming Deformation Theorem.
Discussion of the waist of Calabi-Croke sphere. Strange example of Katok.
- Discussion of coarea formula. Guth’s proof of systolic
inequality for tori.
READING WEEK: no
- Wenger’s proof of
Gromov-Michael-Simon’s isoperimetric inequality. Paper.
- Proof of Gromov’s systolic
by L. Guth.
- Birkhoff's Theorem on existence of a
closed geodesic on every Riemannian 2-sphere
- Lyusternik-Shnirelman theorem. Estimates for lengths of closed geodesics.
- Open Problems
IMPORTANT: if you did not reiceve an email from me this week (with Open problems file attached), please let me know!
Course is going online:
Please, submit your projects and/or record your presentations and send me a link to your video by April 3.
Email me to set up a skype meeting to discuss your questions about the course, homework and open problems.