Representation Theory of Finite Groups and Homological Algebra
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.
The first midterm will take place in class on Tuesday February 25th. It will consist of some subset of problems from Homework assignments 1, 2, and 3.
The second and final midterm will be in class on Thursday, April 3rd. It will consist of the following material. Some subset of problems from homework 4 and homework 5 adapted to the midterm setting. Also possibly problems regarding left and right derived functors. In particular, you should know the definitions, the basic properties, and how to prove them.
- The lectures will be on Tuesdays and Thursdays from 2:00 to 3:20 in room MATH ??
- 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.
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