Math 504-505-506
 Modern Algebra

Steve Mitchell

Aut/Win/Spr  2006-2007, Monday/Wednesday/Friday, 9:30

The course will be organized around three major themes:

The three themes will roughly align with the three quarters. All of the usual graduate algebra topics will be covered; the themes are intended to provide some focus.

Following are a rough description of part I and very brief descriptions of parts II and III; more detailed course outlines will appear at the beginning of each quarter.

I. Representation theory of finite groups.

Let G be any group, and let GlnF denote the group of invertible n x n matrices over a field F; for example F could be the real or complex numbers. A representation of G over F is just a group homomorphism  G → GLnF. Since matrices are very concrete, computable objects, studying the representations of G can greatly facilitate understanding G itself. Furthermore, representations arise naturally almost everywhere in mathematics: In differential geometry (where the group in question is usually a ``Lie group''), in Galois theory (G is a Galois group), in algebraic topology (every G imaginable), algebraic geometry, combinatorics, harmonic analysis, etc. etc. In this course we will focus on the purely algebraic representation theory of  finite groups.

In Part I, therefore, we'll begin by reviewing some basic group theory. I won't stand at the board and recite definitions you already know---that would be frightfully tedious for everyone---but rather we (meaning you) will study some interesting problems and examples. That way you can review and improve your understanding of the basics and have fun at the same time.

Then we'll move on to a deeper study of groups. For example, we'll study the elegant Sylow theorems. Since the importance of these theorems is not apparent at first sight, it may be worth mentioning that in my career as an algebraic topologist, I have used the Sylow theorems approximately 1,857 times.

Our ultimate goal in this section is to study in depth the representation theory of finite groups over C. The slogan here is ``Wedderburn meets character theory''. Wedderburn's theorem has to do with the structure of certain non-commutative rings, while character theory is to some extent a part of elementary linear algebra. When the two theories meet, beautiful theorems ensue. A long digression will be required, in which we develop some basic theory of rings, modules and linear algebra, and in the end are rewarded by proofs of dazzling simplicity. I assume that you are familiar with the basic definitions of ring theory, e.g. ideals, and with undergraduate linear algebra. I do not assume familiarity with modules.

II. Algebraic number theory.

The title ``Algebraic number theory'' is only a motivating theme. The actual contents of this section will include: In ``Commutative algebra I'' we'll study commutative rings R and their modules, but focusing mainly on ``one-dimensional'' rings such as Z, more generally principal ideal domains and even more generally Dedekind domains. All of these terms will be explained at the appointed time!

III. Algebraic geometry.

The core of this section will be ``Commutative algebra II'', in which we study ``higher-dimensional'' rings such as a polynomial ring in n variables over a field. This automatically brings us into contact with ``algebraic varieties'', the main objects of study in algebraic geometry.

Text: Abstract Algebra (third edition), by Dummit and Foote.

Prerequisites: A solid grasp of abstract algebra at the undergraduate level, as in our 402-3-4 course. I assume you are reasonably fluent in the basic language of groups, rings, fields, and vector spaces.