MATH 497A - Honors MASS Algebra

Groups and their connections to Geometry

Instructor: Anatole Katok, Raymond N. Shibley Professor of Mathematics

TA: Vaughn Climehaga

MWRF - 11:15-12:05pm

**Description**: In the introductory part of this course we will introduce principal classes of groups, both abstractly and as coming from various constructions in algebra, geometry and analysis, and develop basics of group theory. After that we will develop two principal themes:

- Groups related to geometric objects, and
- Geometric objects related to groups

Within the current mathematical landscape the course will provide introduction into several aspects of three major areas:

- Algebraic topology
- Theory of transformation groups
- Geometric group theory

**Readings**: There will be no single text for the course. The principal source will be lecture notes which will be developed and made available to students in real time. A variety of supplementary sources covering the background, various course topics, and directions for projects and future research, will be provided.

Math 497B - Honors MASS Analysis

Complex Analysis from a Fluid Dynamics Perspective

Instructor: Andrew Belmonte, Associate Professor of Mathematics

TA: Andong He

MWRF - 10:10-11:00am

**Description**: There is a deep connection between differentiable functions of complex variables and solutions to Laplace's equation. Due to this beautiful fact, there are many surprising connections between classical results in two-dimensional fluid flows and analytic functions in the complex plane. This course is an introduction to complex analysis from the perspective of these incompressible irrotational (ideal) fluid flows. We will additionally cover the opposite limit of very viscous fluids, which connects to a generalization of analytic functions known as polyanalytic functions. Topics will include: incompressible flows, vorticity, circulation, theorems of Kelvin and Helmholtz, streamfunctions, Euler's equation, Bernouilli's Theorems, analytic functions and the Cauchy-Riemann equations, conformal mapping, Kutta-Zhukovskii Theorem and lift on a swept wing, Milne-Thompson Circle Theorem, Blasius Theorem, d'Alembert's paradox, Cauchy's Theorem, Method of Residues.

**Readings**: T. Needham, Visual Complex Analysis (Oxford, 1999)

Math 497C - Honors MASS Geometry

Explorations in Convexity

Instructor: Sergei Tabachnikov, Director of MASS

TA: Pavlo Tsytsura

MWRF - 1:25-2:15pm

**Description**: The course is an introduction to the theory of convexity that plays an important role in analysis and geometry. Topics to be covered include Helly's theorem and its applications, geometry and combinatorics convex polyhedra, polar duality and its applications, lattice points in convex bodies.

**Readings**: A. Barvinok. A course in convexity (AMS, Providence, 2002)