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## Spring Tutorial 2017

# Representations and Cohomology of Groups

**Description:**Group theory has been fundamental in modern mathematics and mathematicians study the structure of groups from different viewpoints. Among them, group cohomology and group representations play an important roles. This tutorial will begin with the basics of the cohomology of groups, representations of groups and its applications to group theory. We will interpret materials from the viewpoint of both algebra and topology. We may move deeper to Atiyah's representation of groups and equivariant cohomology, and Quillen's algebraic K- theory. Topics may include, but are not limited to:

- Classification of k[G]-modules,
- Homological algebra: Ext and Tor,
- Basic equivairiant cohomolgy,
- Basic algebraic K theory,

**Prerequisites:**Math 122 and Math 131.

**Contact:**Meng Guo (mengguo@math.harvard.edu)

# Category Theory

**Description:**

Category theory is a branch of abstract algebra which has found applications in many areas of pure and applied mathematics, computer science, logic, and even linguistics. In some subfields of algebraic topology, algebraic geometry, and representation theory, it is completely indispensable as the language in which theorems are formulated. The organizing principle is that the relations, or "morphisms," between mathematical objects should always be studied alongside the objects themselves. This perspective applies equally well to continuous maps between topological spaces, homomorphisms of algebraic structures, and implications between logical propositions. In this tutorial we will learn the basic concepts and results of category theory, encountering many examples along the way. We will also discover some tensions with standard set theory, which have lead to renewed interest in alternative foundations of mathematics like type theory.

**Prerequisites:**Category theory has no true prerequisites other than some experience reading and writing proofs. I will select examples from algebra, differential and algebraic geometry, and topology, and so recommend that you be familiar with at least one of these subjects. Some knowledge of algebraic structures like groups, rings, and modules would be especially helpful.

**Contact:**Justin Campbell (Campbell@math.harvard.edu)

## Fall Tutorial 2016

# Arithmetic of elliptic curves

**Description:**This tutorial is an introduction to the geometry and arithmetic of elliptic curves. These curves have played a central role in number theory and algebraic geometry in the past century, and encode some of the biggest mysteries in the field, for example, the BSD conjectures. A major goal of the tutorial is to study the rational points on elliptic curves, and prove the Mordell-Weil theorem, which states that for an elliptic curve defined over a number field, the set of rational points forms a finitely generated Abelian group. Then we will (hopefully have time to) explore more properties of these curves, for example, the theory of complex multiplication (and its link with class field theory), as well as modular forms (and how they enter the proof of Fermat's last theorem).

**Prerequisites:**Students should be familiar with Galois theory. Some familiarity with basic algebraic number theory, p-adic numbers and some previous experience with algebraic curves (or its analytic equivalence -- the theory of Riemann surfaces) would be useful, but not required.

**Contact:**Zijian Yao (zyao@math.harvard.edu)