Representation Theory of Finite Groups and Homological Algebra
Link to Canvas Page
This course is Math 423/502 and consists of two parts:
- Representation Theory of Finite Groups. A representation of a finite group is an embedding of the group into a matrix group. Representations arise naturally, for example, when studying the set of symmetries of a geometric or combinatorial object. Representations can be uniquely decomposed into irreducible representations; we will classify the irreducible representations of a group. Using the theory of characters we will learn how to effectively decompose an arbitrary representation into its irreducible constituents. We will apply the theory of characters to solve a nice problem in the topology of surfaces. The key idea is the concept of a 1+1 dimensional Topological Quantum Field Theory and its relationship to Frobenius algebras.
- Homological algebra. Homology and cohomology arise in a variety of subjects across pure mathematics and they are essential in algebraic topology, algebra, and algebraic geometry. We will study homological algebraic in the setting of modules over a commutative ring which is broad enough to encompass most applications. In this context, we will see how a complex and its cohomology naturally arises when studying a module via it generators, the relations among the generators, the relations among the relations, and so on. We will discuss basic ideas in homological algebra including derived functors. We will possibly discuss spectral sequences and/or the derived category.
Grades will be based on two midterm (in class) exams, one on representation theory, one on homological algebra. Dates of the midterms to be announced. Weekly homework will be assigned but not graded — however, the problems on the exams will be a subset of the homework problems.
Midterm 1:
The first midterm will take place in class on Tuesday, February 25th. I have reserved the room for an extra hour. You will be given 120 minutes. It will consist of some subset of problems from Homework assignments 0, 1, and 2.
NEW GRADING PLAN FOR SECOND HALF OF COURSE
Instead of having a final, I will be collecting homework for grades. This will apply to TQFT homework 1, Homological algebra homework 0, and homological algebra 1. I think that we will not get as far as homological algebra homework 2, so that might be extra credit. Homework should be sent to me by email if you have LaTeXed it or have a high quality scan of it. You can also put it under my door in the math department.
TQFT and Homological Algebra Notes
Here is a scan of my lecture notes for the material on the TQFT associated to ZCG, the center of the group algebra. It has some topological details that I will not go over in class and that you are not formally responsible for, but may be interested in, especially if you have seen the fundamental group before.
Here is a scan of my lecture notes for Homological Algebra. A lot of this material is drawn from Appendix A in Eisenbud’s book on Commutative Algebra.
basic info:
- The lectures will be on Tuesdays and Thursdays from 11:00 am to 12:20 pm ~in room MATHX 1102~ online using Collaborate Ultra through the course Canvas page.
- I have added a Piazza discussion page for this course. It is accessible from the Canvas page.
- Basic references for the course are Serre’s book “Linear Representations of Finite Groups”, Fulton and Harris’s “Representation Theory” (Part I only), Weibel’s “An Introduction to Homological Algebra”, and Appendix 3 (“Homological Algebra”) of Eisenbud’s “Commutative Algebra” book. For a basic refence on TQFTs I suggest “Frobenius Algebras and 2D Topological Quantum Field Theories” by Joachim Kock. Also this expository paper by Greg Moore might be helpful. The paper Derived Categories for the Working Mathematician is a helpful introduction to the derived category.
Homework assignments
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