Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
Archived Summer Tutorials: 2011 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 or three times 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 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

Introduction to Ergodic Theory, by Hansheng Diao (

A dynamical system is a mathematical object intended to capture the idea of a "systems" in the real world that passes through different possible "states" as time passes. More precisely, a dynamical system consists of some set X of possible states and a transformation T from X to X specifying the movement of the system from one state to the next.

Ergodic theory is the qualitative study of the measure-preserving dynamical systems; especially, the study of the behavior of a dynamical system after running the system for a long time. The main results are summarized by various ergodic theorems. The ergodic theorems assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. In particular, when the dynamical system is ergodic, the time average is the same for almost all initial points. In other words, the system "forgets" the initial condition after a long time.

The concept of ergodicity plays a central role in the development of ergodic theory. Roughly speaking, the word "ergodic" means the transformation mixes the space very well. Similarly, there are notions of mixing and weakly mixing, corresponding to better mixed systems. Lots of applications of ergodic theory amounts to show the "well-mixed-ness" of the systems.

Ergodic theory has fruitful connections to many different aspects of mathematics. For example, ergodic theory has been adopted to study geodesic flow on Riemannian manifolds. Applications of ergodic theory also shows up in harmonic analysis, Lie theory, probability theory, differential geometry, and even number theory! Recent progresses include Elon Lindenstrauss's proof of Arithmetic Quantum Ergodicity Conjecture and Terrence Tao and Ben Green's proof of arbitrarily long arithmetic progressions of primes.

Reference: Any introductory textbook on Ergodic theory works. I recommend:

Other standard textbooks in ergodic theory are Some topics for projects:

Prerequisites: Real analysis at the level of Math 112 is strongly recommended. Knowledge of measure theory as in Math 114 will help, but is not required.

Knots and Primes, by Chao Li, ( and Charmaine Sia (


A knot is an embedding of a circle in 3-space. A prime number is a natural number with no positive divisors other than 1 and itself. They are the basic objects of study in knot theory and number theory respectively and have attracted the interest of many mathematicians over the course of history. In knot theory, mathematicians developed various knot invariants to determine when two knots are not equivalent; in number theory, mathematicians use the behavior of various primes to study arithmetic problems.
Surprisingly, these two seemingly unrelated concepts have a deep analogy discovered by Barry Mazur in the 1960s while studying the Alexander polynomial. This initiated the study of what is now known as arithmetic topology. Pursuing deeper connections between knot theory and number theory may raise new points of view and lead to interesting problems and progress in these fields. In this tutorial, we will explore some aspects of this beautiful analogy between knots and primes.

Tentative syllabus: We will introduce basic knot theory and number theory, providing many examples and focusing on the intuition and motivation behind the concepts introduced, with the aim of discussing analogies between linking numbers and quadratic reciprocity, fundamental groups and Galois groups, 3-manifolds and number rings, knots and primes, the Alexander polynomial and Iwasawa theory.

Reading material and Reference texts:

Possible final paper topics:

Prerequisites: Basic knowledge of point set topology on the level of Math 131 and rings and fields on the level of Math 123.

Archive: Old Summer Tutorials, since 2001

Summer Tutorials: 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001

Last update, 4/17/2012