MAT1309HS, Spring 2020

Geometric Inequalities

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.

Textbook: Our main text will be Yu. D. Burago, V. A. Zalgaller, “Geometric inequalities”, but we will also use many other sources.

About the course:

The 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:

1. Basic inequalities that relate the “sizes” of geometric objects: classical isoperimetric inequality, Loomis-Whitney inequality, Besicovitch inequality, coarea inequality.

2. A brief tour of 3 approaches in measure theory.

2. Isoperimetric inequalities in higher codimension: Ferder-Fleming Deformation Theorem and Wenger’s proof of Gromov’s and Michael-Simon’s isoperimetric inequalities.

4. A brief tour of minimal surfaces: minimal cones, monotonicity formula, maximum principle.

5. Systolic inequalities: theorems of Lowener, Pu, Gromov and Guth.

6. 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 students’ interest.

Lectures

(I will keep updating this section and adding questions and notes as we go along).

1. Loomis-Whitney inequality.      Original paper.      Questions.

2. Two versions of Besicovitch inequality for topological cube.

2. A rapid tour of Riemannian geometry.   Problem set 1

2. Consequences of Besicovitch inequality: Papasoglu's isoperimetric inequality on the two-sphere.  Questions

2. Consequences of Besicovitch inequality: Loewner's and Pu's systolic inequalities. Brunn-Minkowski inequality (Burago-Zalgaller §8).

3. Classical isoperimetric and isodiametric (Bieberbach) inequalities, symmetrization (Burago-Zalgaller §9, §11.2).

3. Outer measures. Equivalence between n-dimensional Hausdorff and Lebesgues measures in R^n. (L. Simon, "Lectures on Geometric measure theory", chapter 1).
   

4. Rademacher theorem. Area and coarea formulas. (Burago-Zalgaller §13, L. Simon, "Lectures on Geometric measure theory", chapter 1).

5. Proof of Sobolev inequality using isoperimetric inequality and coarea formula. (Burago-Zalgaller §8.3)

6. Waist inequality for the n-sphere. Expository paper by Larry Guth.
   

6. Federer-Fleming Deformation Theorem. Discussion of the waist of Calabi-Croke sphere. Strange example of Katok.

6. Discussion of coarea formula. Guth’s proof of systolic inequality for tori.

7. Wenger’s proof of Gromov-Michael-Simon’s isoperimetric inequality. Paper.
   

9. Proof of Gromov’s systolic inequality. Gromov's paper. Notes by L. Guth.

9. Birkhoff's Theorem on existence of a closed geodesic on every Riemannian 2-sphere

9. Lyusternik-Shnirelman theorem. Estimates for lengths of closed geodesics.
   

9.  Open Problems