Math 497A - Honors MASS Algebra

Function Field Arithmetic

Instructor: Mihran Papikian, Assistant Professor of Mathematics

TA: Evgeny Mayanskiy

MWRF - 1:25-2:15pm

**Description**: This course is a mixture of an abstract algebra course and a number theory course. Its aim is to explore the properties of F[T], the ring of polynomials over a finite field F, and to compare them to the properties of the ring of integers Z. From the point of view of Algebra, the rings Z and F[T] are quite similar. For example, both rings are principal ideal domains, both have the property that the residue class ring of any non-zero ideal is finite, both rings have infinitely many elements, and both rings have finitely many units. What we will see in the course is that the similarities of Z and F[T] extend beyond algebraic properties to the more subtle arithmetic properties, which are studied in Number Theory. For example, it is possible to define a zeta-function for F[T] having properties similar to the properties of the Riemann zeta-function. There are at least two different "zeta-functions" for F[T], and both will be discussed in the course. Especially intriguing is the zeta-function introduced by L. Carlitz in 1930s, which naturally leads to the analogue of Euler's formula for zeta(2), and to the theory of Drinfeld modules - an area on the frontier of current number theory.
The only prerequisite for this course is a background in linear algebra. A prior exposure to abstract algebra is certainly helpful, although all the necessary background material will be covered at the beginning of the course.

**Readings**:

- Lidl and Niederreiter,
Finite fields

- M. Rosen,
Number theory in function fields

- D. Goss,
Basic structures of function field arithmetic

Math 497B - Honors MASS Analysis

Differential equations from an algebraic perspective

Instructor: Nigel Higson, Evan Pugh Professor of Mathematics

TA: Tyrone Crisp

MWRF - 10:10-11:00am

**Description**: The aim of this course is apply concepts from algebra and algebraic geometry to the study of differential equations. The main object of study will be the so-called Weyl algebra of differential operators. This is a noncommutative algebra (since for example differentiating then multiplying by `x` is not the same as multiplying by `x` then differentiating.) However it is nearly commutative, and this makes it possible to analyze it using a variety of geometric techniques. The Weyl algebra originally arose in quantum theory, and we'll take a little time to discuss how.

**Readings**:

- W. Fulton, algebraic curves (for a bit of algebraic geometry)
- S. Coutinho, A primer of algebraic D-modules (for the Weyl algebra)
- Other readings covering background material and supplementary topics will be provided during the course.

Math 497C - Honors MASS Geometry

Dynamics, mechanics, and geometry

Instructor: Mark Levi, Professor of Mathematics

TA: Pavlo Tsytsura

MWRF - 11:15-12:05pm

**Description**: The course will introduce students to the following topics: Geometry theory of ordinary differential equations; Linear and nonlinear systems in the plane; deterministic chaos and fractals as they arise in simplest dynamical systems; Variational principle of mechanics and optical-mechanical analogy. Many of the ideas developed in the course will be illustrated by mechanical demonstrations and/or applied to explain some interesting mechanical phenomena and paradoxes.

**Readings**:

- V. Arnold, Ordinary Differential equations
- A. Andronov, A.A. Vitt and S.E. Khaikin, Theory of oscillators, Dover, 1966.
- V. Arnold, Mathematical methods of Classical mechanics, Springer Verlag.
- S. Strogatz, Nonlinear Dynamics and Chaos. Addison Wesley.