Description:

This course focuses on analysis on metric measure spaces and their applications to metric geometry. 

Analysis on metric measure spaces is a mathematical field to study metric spaces with no a priori smooth structure. One of the earliest motivations and applications of this field arose in Mostow’s celebrated rigidity work. Heinonen and Koskela later axiomatize several aspects of Euclidean quasiconformal geometry in the setting of metric measure spaces, which initiated the modern theory of analysis on metric measure spaces. 

Today, this is an active field with far-reaching applications to areas such as geometric measure theory, geometric function theory, geometric group theory, nonlinear PDEs, dynamics of rational maps, fractal geometry, and even theoretical computer science. 

We will mainly discuss the following content in this semester. 

Part 1. Basic framework in analysis on metric spaces. We will cover topics such as covering lemmas, fractal dimensions, modulus, quasiconformal mapping, quasisymmetry, Loewner spaces, Poincare inequality, conformal dimension, etc. 

Part 2. Application of techniques of analysis on metric measure spaces into geometric group theory. We will cover topics such as hyperbolic groups, the boundary at infinity, visual metric, Gromov Hausdorff convergence, weak tangents, Cannon conjecture, Kapovich-Kleiner conjecture, etc. 

If time permits, we may cover more subjects in related areas.

Prerequisite:

Real Analysis and Complex Analysis. Some knowledge of Topology is very helpful but not essential.

Reference:

1. L. V. Ahlfors, Lectures on Quasiconformal Mapping, American mathematical Society, Providence (2006).

2. J, Heinonen, Lectures on Analysis on Metric Spaces, Springer-Verlag, New York, (2001).

3. J, Heinonen and P, Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math., 181, (1998), 1--61.

4. D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, Grad. Stud. Math., 33, Amer. Math. Soc., Providence, RI, (2001).

5. I. Kapovich and N. Benakli, Boundaries of hyperbolic groups, Contemp. Math., 296, (2002), 39--93.

6. C. Bishop and Y. Peres, Fractals in Probability and Analysis, Cambridge Stud. Adv. Math., 162, Cambridge Univ. Press, (2017).

Target Audience: Undergraduate students, Graduate students

Teaching Language: English

Short Bio:

Wenbo Li is an assistant professor at YMSC, Tsinghua University. He obtained his Doctor of Philosophy in Mathematics in 2022 at the University of Toronto. His research is focused on Analysis on Metric Measure Spaces, Complex Analysis and Random Geometry. He is also interested in Complex Dynamics, Geometric Measure Theory, Geometric Function Theory, Geometric Group Theory, Fractal Geometry, Metric Geometry, etc

Registration: https://www.wjx.top/vm/QZCBxgY.aspx#