Abstract Algebra & Number Theory
Research Topics: Field theory, error-correcting codes, cryptography, primality testing, eliptic curves, mathematics of the Rubik’s cube, classifying symmetries of patterns, Galois theory, applications of group theory in combinatorics, and other topics.

Program I topics are introduced through five motivating problems such as limitations of straight-edge and compass constructions, classification of patterns in two dimensions, error-correcting codes, cryptography, and the analysis of symmetry in structures. 

The mathematics central to solving these problems comes from the areas of abstract algebra and number theory. Abstract algebra originated in the early part of the 19th century through the study of polynomial equations. This branch of mathematics lies at the core of many areas of modern mathematical research. Number theory concerns properties of the integers, and has its origins in ancient mathematics. Number theory remains a very active field of study with interesting open problems and important applications in computer science.

Algebraic Topology
Research Topics: Moduli spaces, homotopy groups, cohomology, configuration spaces, Lie groups, Lie algebras, knot theory, hyperbolic geometry, and other topics.

Program II centers on algebraic topology, a major area of current mathematics research.

Topology is the study of the properties of shapes that remain unaffected by deformations. For example, a sphere made out of rubber can be deformed into the shape of a cube. While it may appear that a sphere and a cube don't have that much in common, it turns out that they are topologically equivalent and in a way that can be made precise mathematically. This course will explore different ways of analyzing topological properties of shapes using algebraic concepts, such as the notion of group.