Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
Archived Summer Tutorials: 2009 2008 2007 2006 2005 2004 2003 2002 2001

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 presenting 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 Harvard College student participant a stipend of , and you can hand in your final paper from the tutorial for you junior 5-page paper requirement for the Math Concentration.
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 only by emailing me at When you sign up, please list at least one other choice, if possible, in case your preferred tutorial is either over-subscribed or under-subscribed. Places are filled on a first-come, first-served basis, but with priority being given to math concentrators. In the past, some tutorials have filled up quickly. If you have further questions about any given topic, contact the tutorial leader via email. Please contact me if you have questions about the administration of the tutorials.


Peter Kronheimer

Teichmüller Theory and Thurston's Classification of Mapping Classes, by Ronen Mukamel (

Let S be a surface of genus g. A mapping class on S is a homotopy class of homeomorphism from S to itself. How can we measure how complicated a mapping class is? To give an answer this question, which at first seems purely topological, Thurston took a detour through the geometry of Teichmüller space--the classifying space for complex structures on S. In this class we will study Teichmüller space with a view towards understanding Thurston's theorem. Throughout we will emphasize example, and in particular the case of genus 1.

Prerequisites: Math 113 and 131

Linear Algebraic Groups, by Ryan Reich, (

The study of groups in mathematics needs no motivation, since it is applied virtually everywhere. First courses focus on finite groups and their actions, but any general description tends to have as many exceptions as it has classifications. We will learn about a particular kind of topological group, belonging to algebraic geometry, whose examples can be viewed as matrix groups. On the one hand, the topological structure makes it possible to study the local structure of the group operation, embodied by its Lie algebra; on the other, the linear structure allows it to act naturally (be "represented") on vector spaces and thus be studied by the elementary methods of linear algebra. As a simple example, we describe finite group representations, then turn to representations of algebraic groups By combining the two we introduce Dynkin diagrams and show how these simple combinatorial objects describe a large class of algebraic groups.

Prerequisites: Math 122 (Abstract algebra: groups and vector spaces). Exposure to topics from Math 123 (rings and modules) or 137 (algebraic geometry, in particular affine algebraic varieties) will be helpful, but I will fill in for them as required by the students' backgrounds.

p-adic Arithmetic and p-adic Analysis, by Jack Thorne ( )

The real numbers can be constructed by filling in the holes in (i.e. taking the completion of) the set of rational numbers, with respect to the usual notion of distance. Close to the beginning of the last century, Hensel noticed that there is another notion of distance, distinct from the usual one, coming from modular arithmetic. Taking the completion one thus obtains what are known as the p-adic rational numbers. In this tutorial we'll explore the arithmetic of p-adic numbers, their Galois theory, and their relationship with arithmetic over finite fields. The connection with analysis will be a central theme. For example, it's not difficult to give a proof of the Chinese remainder theorem using only epsilons and deltas. Not so obvious is that any polynomial with p-adic coefficients is soluble in radicals. Students who take this tutorial will have a good understanding of the Galois theory of the p-adics and some basic notions from p-adic analysis, such as the p-adic logarithm and Newton polygons. These will be necessary for the climax of the course: given a polynomial equation in several variables, with coefficients in a finite field, one can ask about the number of solutions to the equation as the base field varies. This sequence of solution counts can be put into the form of a generating function, called the zeta function of the equation. An amazing fact is the zeta function of such an equation is always a rational function. In particular, the number of solutions over any extension of the base field is determined by the first few solution counts! This was proved by Dwork in 1960 as a stunning application of p-adic analysis, and it is his proof we will give.

Prerequisites: We will assume familiarity with basic notions from real analysis, and with finite fields and Galois theory. Math 123 would be sufficient.

Archive: Old Summer Tutorials, since 2001

Summer Tutorials: 2009 2008 2007 2006 2005 2004 2003 2002 2001

Last update, 4/14/2009