Math 497A - Honors MASS Algebra

Finite Fields and their Applications

Instructor: Gary Mullen, Professor of Mathematics

MWRF - 10:10-11:00am

This course will consist of an introduction to the theory of finite fields. We will also discuss some of the many practical applications of finite fields. These applications will include algebraic coding theory for the error-free transmission of information, and cryptology for the secure transmission of information. Finite fields are also of great use in the construction of various kinds of combinatorial designs.

Typical Reading: lecture notes provided by instructor

Math 497C - Honors MASS Geometry

Aspects of symmetry: from representations to Quantum Field Theory

Instructor: Adrian Ocneanu, Professor of Mathematics

MWRF - 11:15-12:05pm

We shall describe the structure of group, illustrated by the finite rotation groups in 3 dimensions. We shall construct concretely the representations of these groups and their tensor product decompositions. With the coefficients obtained this way we shall construct topological quantum field theories and obtain knot and manifold invariants. The structure and representations of the 3 dimensional rotation group SO(3) and of its cover SU(2) will also be discussed. We shall describe these groups in several concrete ways, several of which use the departmental sculpture.

Prerequisites: We start from scratch, but a working knowledge of linear algebra is recommended.

Math 497B - Honors MASS Analysis

Mathematical theory of waves

Instructor: Alberto Bressan, Professor of Mathematics

MWRF - 1:25-2:15 pm

The first part of the course will be concerned with linear waves: sound waves and vibrations in elastic strings. We shall derive a partial differential equation describing wave motion, and study various properties of solutions; in particular, the superposition principle and the speed of propagation of disturbances. The second part of the course will be devoted to non-linear wave motion. In this case, the wave profiles can change in time, eventually leading to the formation of shocks. Although shock waves are described by discontinuous functions, they can still be described in terms of a P.D.E, taking the form of a conservation law. Various models and applications to gas dynamics and to traffic flow will be discussed.

Basic reference: Roger Knobel, An Introduction to the Mathematical Theory of Waves. AMS 2000

Prerequisites: basic linear algebra, calculus in several variables.

Math 497D - MASS Interdisciplinary seminar

Instructor: Sergei Tabachnikov

T - 10:10-12:05pm

This seminar is designed to focus on selected interdisciplinary topics in algebra, geometry and analysis to coordinate core courses and to prepare students to MASS Colloquium. Seminar sessions may include presentations from student research projects.