Math 191 Fall 2019
Section 1 - Topics in Geometry and Topology (Chen)
My topic course breaks into two topics:
On moduli space of algebraic curves and
mapping class groups.
The moduli space of curves is a central topic in low dimensional topology, algebraic geometry and complex analysis. It is both the moduli space of algebraic curves, hyperbolic surfaces and Riemann surfaces. It is also virtually the classifying space of surface bundles. Thus, the study of the moduli spaces always has many different angles. In this course, we will present some of Harer’s result on the comparison of moduli space with arithmetic groups. For example, we will prove high connectivity of curve complex, Harer stability, computation of Euler characteristic of moduli space. We will introduce Fenchel-Nielsen coordinate and a magic formula of Wolpert and draw connection with Mirzakhani’s work on counting simple closed curves on hyperbolic surfaces.
On the structure of classical diffeomorphism groups.
The second half of this course is on transformation groups of manifolds; i.e., groups of diffeomorphism of homeomorphism of manifolds. Those groups share some common properties with Lie groups. We will present basic properties of those groups: for example, simplicity, perfectness, automatic continuity (recent work of Hurtado, Mann and others), and some version of classification of closed subgroups of transformation groups (some even more recent work of Mann and myself). We will follow two books/papers: John L. Harer: The cohomology of moduli space of curves & Augustin Banyaga: The structure of classical diffeomorphism groups.
Section 2 - Geometrical Paradoxes (Kechris)
This course will provide an introduction to the striking paradoxes that challenge our geometrical intuition. One of the most famous ones is the Banach-Tarski Paradox (1924): A pea can be decomposed into finitely many pieces which can be rearranged in space to form a ball the size of the sun. A popular account of these paradoxes can be found in the book: The Pea and the Sun, A Mathematical Paradox, by Leonard M. Wapner, A.K. Peters, 2005.
Topics to be discussed include: geometrical transformations, especially rigid motions; free groups; amenable groups; equidecomposability and invariant measures; Tarski's Theorem; the role of the Axiom of Choice; old and new paradoxes, including the Banach-Tarski Paradox; Laczkovich’s solution of the Tarski Circle-Squaring Problem; the Dougherty-Foreman solution of the Marczewski Problem (equidecomposition with Baire measurable pieces); recent work of Grabowski-Máthé-Pikhurko and Marks-Unger on measurable and explicit equidecompositions; and the continuous movement version of the Banach-Tarski Paradox (due to Trevor Wilson, a Caltech undergraduate, solving the de Groot Problem).
The treatment of the material will be as elementary as possible. The course should be accessible to students who are familiar with basic algebra (Math 5 or equivalent).