Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132

This is an archived document. You can find the current tutorials here.

Archived Summer Tutorials 2005 2004 2003 2002 2001
Andreas Floer Emmy Noether Felix Klein Gotthold Eisenstein Henry Cartan Marston Morse Samuel Eilenberg Saunders MacLane

Welcome Message

The summer tutorial program offers some interesting mathematics to those of you who will be in the Boston area during July and August. Each tutorial will run for six weeks, meeting twice per week in the evenings (so as not to interfere with day time jobs). The tutorials will start early in July and run to mid August. The precise starting dates and meeting times will be arranged for the convenience of the participants once the tutorial rosters are set.

The format will be much like that of the term-time tutorials, with the tutorial leader lecturing in the first few meetings and students lecturing later on. Unlike the term-time tutorials, the summer tutorials have no official Harvard status: you will not receive either Harvard or concentration credit for them. Moreover, enrollment in the tutorial does not qualify you for any Harvard-related perks (such as a place to live). However, the Math Department will pay each student participant a stipend of $700, and you can hand in your final paper from the tutorial for you junior 5-page paper requirement.

The topics and leaders of the four tutorials this summer are:

A description of each topic is appended below. You can sign up for a tutorial by emailing me at When you sign up, please list at least one other choice in case your preferred tutorial is either over-subscribed or under-subscribed. If you have further questions about any given topic, contact the tutorial leader via the email. Please contact me if you have questions about the administration of the tutorials.


Peter Kronheimer

Category theory (Thomas Barnet-Lamb,

Analogies between different theories are a key driver of progress across most of modern mathematics. Many vital new concepts are identified by looking for concepts analogous to well-understood concepts elsewhere. Category theory is a fundamental tool for making these analogies. It is the systematic study of a special kind of picture called a commutative diagram: it turns out (rather miraculously) that many concepts can be defined using these pictures, and (even more miraculously) that once you define something in terms of such a picture, you can use it to translate the definition to other areas of mathematics and this has an almost magic ability to pick out the "right" analogous object.

The language of categories has thus become a fundamental part of the language of modern mathematics, and is now vital in much of modern topology and number theory. Alas, however, people tend to assume that students will "pick up" category theory as they go along. This is a shame, because although the category theoretic way of looking at things is beautifully slick and clean once you "get it", it unfortunately can seem a little bewildering at first. (The situation is made worse by a propensity of category theory textbooks to teach the theory in a very dry and abstract way, with little reference to its applications to the rest of mathematics.)

This course is meant to fix that problem. The course will provide a leisurely introduction to the category theoretic way of thinking about mathematics. We'll cover a number of the "buzzwords" of the theory: categories, functors, natural transformations, the Yoneda embedding, representability, limits, colimits, and adjunctions. At every point, we'll be focusing primarily on doing enough examples and looking at enough pictures so that students "get" what's going on, so that when they come across the category theoretic way of doing things elsewhere in mathematics, they'll feel at home.

Prerequisites: Math 122 will be essential. Math 131 would be very useful. (Since we'll be working with a lot of examples, you'll need to be familiar with those areas of math from which the examples will be drawn!)

Group cohomology (John Francis,

In the 1920s, Emmy Noether realized that the Betti numbers of spaces a la Riemann and Poincare were associated to a richer structure: cohomology. Other invariants of spaces, such as curvature, had natural interpretations in cohomology, and this had enormous conceptual and computational impact. Twenty years later, Samuel Eilenberg and Saunders MacLane were working on separate problems in topology and algebra, and they realized a deep connection between these topological invariants and invariants of groups. They uncovered a new cohomology theory for groups that likewise encoded classical algebra in a new and richer structure. The following development of homological algebra and category theory by Cartan/Eilenberg, MacLane, and Grothendieck changed the face of mathematics, and the field of group cohomology reached across topology, algebraic number theory, Lie groups, combinatorics, and invariant theory.

This tutorial will begin with the basics of the cohomology of finite groups and its applications to group theory. For instance, to build up the classification of simple groups to consider all groups, one has to figure out all the ways that simple groups can be "twisted" together to make new groups. We will study this problem as interpreted by Eilenberg-MacLane in group cohomology. Related topics will include homological algebra, categories, the ring of symmetric functions, invariant theory, and applications to group cohomology. We will focus on understanding concrete examples, such as symmetric groups, finite groups of Lie type, and possibly Galois groups. Final projects may include diverse subjects in algebra, topology, and number theory; for instance the cohomology of the sporadic groups, Hochschild cohomology of algebras, group actions on trees and graphs, classifying spaces of groups, and Galois cohomology.

Prerequisites: group theory and algebra at the level of 122 or another 100-level algebra class.

Modular forms (Mike Hill,

This summer we will embark on an exciting quest to understand the now ubiquitous topics of elliptic curves and modular forms. Elliptic curves and modular forms already have a very long and rich mathematical history, starting out life in the world of number theory and moving of late into algebraic topology. Their relative omnipresence alone sets them up as worthy of study, and over the course of the tutorial, we will learn to appreciate them in all their glory.

The goal of this tutorial is to build an understanding of the approaches to modular forms and elliptic curves. We will start by looking at elliptic curves over the complex numbers, as here we can build solid geometric intuition. From elliptic curves over C, we will move to modular forms over the complex numbers. Here there is a nice classification which is interesting to work out. Building on our work over C, we will branch out to look at elliptic curves over other rings. The concepts of modular forms also carry over, and we will look at the various ways they can be defined. Towards the end of the tutorial, we will shift attention slightly to classifications of elliptic curves, looking at how one can represent them as points on a curve.

Prerequisites: Though this topic has many "high brow" applications, I wish to present everything at a very basic level. The only prerequisites needed will be a good understanding of algebra at the level of 122 and 123. Some knowledge of algebraic geometry would be helpful but not required, as most of the concepts presented will be done in much more elementary language.

Morse Theory (Andrew Lobb,

Morse theory is the skeleton key of geometry, finding applications to many diverse problems including the famous Bott periodicity theorem and the h-cobordism theorem. In recent years, Floer's symplectic analogue of Morse theory has given rise to many new topological invariants. In this tutorial will we be discovering how we can make use of Morse theory in the study of the topology of three- and four-dimensional manifolds. Four-dimensional manifolds in particular are interesting to mathematicians because it is in this dimension that the notion of diffeomorphic manifolds is first distinct from the notion of homeomorphic manifolds. We shall see examples of manifolds that have the same topology but different smooth structures.

We will start by looking at classical applications of Morse theory to differential geometry (such as the study of geodesics), before using Morse theory to establish a beautiful pictorial language with which we can describe all low-dimensional manifolds. After acquiring a facility with this language we shall use it to answer questions that we would otherwise find intractable.

Prerequisites: This course can be taken without a lot of prerequisites, but some topology is desirable, preferably at the level of Math 131.

Archive: Old Summer Tutorials, since 2001

Summer Tutorials: 2005 2004 2003 2002 2001

Last update, 4/10/2006
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