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. The tutorials will run for six weeks, meeting twice or three times per week in the
evenings (so as not to interfere with day time jobs). The tutorial will start early in July or late in June,
and run to mid August. The precise starting dates and meeting times will be arranged for the convenience
of the participants once the tutorial roster is 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
approximately , and you can hand in your final paper from the tutorial for your junior paper
requirement for the Math Concentration.
The topics and leaders of the tutorials this summer are:
Hyperbolic 3-manifolds and Kleinian groups
Yongquan Zhang
Abstract: Hyperbolic geometry in dimension three connects many topics in mathematics, including three-dimensional topology, Kleinian groups, Teichumuller theory, circle packings and many more. This tutorial focuses on the connection between three dimensional hyperbolic geometry and conformal dynamics on the extended complex plane, via Kleinian groups. The conformal action of Kleinian groups on the extended complex plane provides a visual guide to our study, and we will look at (and try to make!) many fascinating pictures throughout the tutorial. In the end, we will try to venture into the more theoretic side, and depending on the interest and background of the participants, we can talk about structure of cusps, geometric finiteness, arithmetic hyperbolic manifolds, reflection groups and many more.
Category Theory
Morgan Opie
Since Eilenberg and Mac Lane introduced category theory in the 1940s, the subject has developed rapidly and become a fundamental part of modern mathematics. Category theory provides a framework to study mathematical structures and constructions, and makes precise analogies between different areas. This tutorial will introduce the tools of category theory, focusing on examples tailored to participants' backgrounds. Fundamental concepts such as categories, functors, natural transformations, representability, limits and colimits, and adjunctions will be covered in detail. We'll also venture into more advanced topics, which, depending on the interests of participants, might include monoidal categories, abelian categories and derived categories, model categories, or infinity categories. In my Math Table talk, I'll give a sense of what category theory is all about and explain why a categorical perspective is useful with a few illustrative examples.
Elliptic Curves and Drinfeld Modules
Robert Cass
Abstract: Elliptic curves are central objects of study in number theory. Elliptic curves can be defined algebraically as the set of solutions to a polynomial equation, but this definition hides one of their fundamental properties: the set of points on an elliptic curve forms a group. In the first half of this tutorial we will establish the basic properties of elliptic curves and explore their connection with Galois theory over the rational numbers. One can also consider elliptic curves over fields of positive characteristic. However, another type of object called a Drinfeld module often contains more interesting arithmetic information. In the second half of this tutorial we will develop the theory of Drinfeld modules while emphasizing their analogies with elliptic curves.