Prerequisite: Consent of department. This course is required for all
incoming graduate students. It is a lab course meeting for an
average of two hours per week throughout the semester for
one-semester-hour's credit.
This course provides practical training in the teaching of mathematics at
the pre-calculus level, how to write mathematics for publication, and treats
other issues relating to mathematical exposition. Communicating Mathematics I and II are designed to provide training
in all aspects of communicating mathematics. Their overall goal is to teach
the students how to successfully teach, write, and talk about mathematics to
a wide variety of audiences. In particular, the students will receive
training in teaching both pre-calculus and calculus courses. They will
also receive training in issues that relate to the presentation of research
results by a professional mathematician. Classes tend to be structured to
maximize discussion of the relevant issues. In particular, each student
presentation is analyzed and evaluated by the class.
Algebra: An Approach via Module Theory; Adkins and Weintraub
This is the first semester of the first year graduate algebra sequence, and covers the material required for the Comprehensive Exam in Algebra. It will cover the basic notions of group, ring, module, and field theory.
Topics will include
GROUPS: finitely generated abelian groups, group actions, and the Sylow theorems;
RINGS and MODULES: Euclidean domains, principal ideal domains, unique factorization domains, polynomial rings, and modules over PIDs;
FIELDS: vector spaces, applications to Jordan canonical form, field extensions, and finite fields.
Prerequisite: Consent of department. This course is required for all
incoming graduate students. It is a lab course meeting for an
average of two hours per week throughout the semester for
one-semester-hour's credit.
Prerequisites: Basic Algebra and Topology; this is covered in Math 7210, 7510. Complex Analysis - familiarity with analytic functions; this is covered in Math 7350. It is helpful, but not required to have some exposure to Number Theory and Algebraic Geometry. There will be courses taught on these in Fall 2018 (Math 7230, 7240).
Text: No required text, but much of the material will be drawn from
A First Course in Modular Forms (Graduate Texts in Mathematics,
Vol. 228), Diamond & Shurman
A Classical Introduction to Modern Number Theory, Ireland &Rosen,
(Graduate Texts in Mathematics) (v. 84)
An Introduction to the Langlands Program, Bernstein& Gelbart (eds)
The focus is on L-functions. These are at the center of much research today. Themes: Euler products; analytic continuation and functional equations; special values; relations among different kinds of L-functions. Briefly there are two broad classes of L-functions: those that arise from automorphic forms and those that arise from geometry.
There are conjectural relations between the L-functions of the first kind with those of the second kind. We start with the classical Dirichlet L-functions and move on to the L-functions defined by modular forms. We give an exposition of Tate's thesis which gives a unified account of the L-functions that arise from abelian characters of Galois groups. We also give an introduction to L-functions arising from algebraic varieties, for instance algebraic curves. We then describe relations, proven and conjectural among these. This course is more like a seminar. The topics and level will depend on the students. Each student will have a project to do, with a written report and a lecture exposition.
Prerequisites: Math 7410 and 7490 (Matroid Theory) or permission of the department.
Text: Matroid Applications edited by Neil White (Chapter 6: The Tutte Polynomial and its Applications by Thomas Brylawski and James Oxley)
The theory of numerical invariants for matroids is one of many aspects of matroid theory having its origins within graph theory. Most of the fundamental ideas in matroid invariant theory were developed from graphs by Veblen, Birkhoff, Whitney, and Tutte when considering colorings and flows in graphs. This course will introduce the Tutte polynomial for matroids and will consider its applications in graph theory, coding theory, percolation theory, electrical network theory, and statistical mechanics.