Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
Archived Summer Tutorials: 2010 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 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 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 approximately , 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

Symplectic Geometry, by Aliakbar Daemi (

Symplectic geometry is the study of symplectic manifolds, i.e. smooth manifolds with a closed non-degenerate bilinear 2-form. First examples of these objects appeared in classical mechanics when Legendre tried to study motion of the planets in the solar system. Symplectic geometry also has been at the center of modern physics and plays a key role in areas such a string theory. From the math side, symplectic manifolds are very important objects in their own right: after something of a revolution in the second half of the 20th century, they have become a very active subject of research. They also have interesting interactions with other fields of mathematics such complex geometry, differential equations and Riemannian geometry. In this tutorial, we will start by explaining some examples from classical physics to motivate the definition of symplectic manifolds. In the first few sessions we will work on the simplest examples of symplectic manifolds, namely Euclidean spaces, which already have a surprisingly rich theory. After getting familiar with linear symplectic geometry, we will be ready to work on symplectic manifolds: we will talk about some elementary examples of symplectic manifolds, Darboux's theorem, basic properties of symplectomorphisms and important submanifolds of a symplectic manifold (e.g. Lagrangian submanifolds). A short discussion of contact geometry is possible afterward. Depending on the interests of the class, further topics may include symplectic group actions, generating functions, and non-squeezing theorem, to name a few.

Prerequisites: Some familiarity with smooth manifolds. Math 132 would be useful, but not required.

Coding Theory, by Nathan Kaplan, (

Coding theory is a rather young field, but it has become an extensive area of research among both pure and applied mathematicians. Suppose two parties want to exchange a message, a sequence of ones and zeros, over a noisy channel. That is, there is some probability p such that for each bit of the message the bit received does not match the bit sent. The sender and the receiver must build redundancy into their messages so that these errors can be detected and also corrected. Coding theory gives us a mathematical framework for studying this problem. In this course we will focus on algebraic questions, but we will talk about the more practical side of the subject as well. We will emphasize examples, discussing several well-known codes with amazing properties, and also discuss theoretical limitations of codes. We will also incorporate some of the incredible things that computer algebra systems can do in this subject. We will also study connections to several other areas including combinatorial designs, finite simple groups, sphere packings and lattices, and linear programming.

Prerequisites: Abstract linear algebra at the level of Math 23, 25 or 121.

Representation Theory, by Sam Raskin ( )

Representation theory, broadly speaking, is the collection of techniques combining two principles in mathematics: linearization and symmetry. Its province is therefore considerable, reaching across number theory, algebraic geometry, arithmetic geometry, homotopy theory, topology, mathematical physics and combinatorics, to name a few. Representation theory is the entry point for many applications of powerful algebraic techniques in other parts of mathematics and has become necessary working knowledge for many mathematicians. The subject readily presents difficult and deep problems, beautiful in their own right or in the presence of such applications. This course will be an introduction to the techniques of representation theory, with a focus on classical aspects of representations of finite groups. This subject continues to be an area of active research, besides being an accessible starting point to representation theory, group theory, and abstract algebra more broadly. The course will cover orthogonality relations between characters, Frobenius reciprocity, and examples, before branching out depending on the students' interest. Possible topics include (but are not limited to) Mackey's criterion, representations of finite groups of Lie type, Burnside's "pq" theorem, representations of the symmetric group, Wedderburn theory, and the Frobenius determinant theorem.

Prerequisites: Abstract algebra at the level of Math 122.

Archive: Old Summer Tutorials, since 2001

Summer Tutorials: 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001

Last update, 4/16/2011