Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
Archived Summer Tutorials: 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 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 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 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. Places are filled on a first-come, first-served basis. 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. I will be out of town April 9 to 13th.) Although there are inevitably some late changes, please try and respond by April 30.


Peter Kronheimer

Stochastic Models in Biology by Ivana Bozic

Stochastic processes are ways of quantifying the dynamic behavior of sequences of random events. They are useful tools for analyzing the variability inherent in biological and medical processes. In this tutorial, we begin with a review of basic probability and introduce a variety of probabilistic techniques for mathematical modeling in biology and medicine, such as Markov chains in discrete and continuous time, branching processes and birth-death processes. We will show how different stochastic models can be used to study the same phenomenon using the example of tumor progression. We will pay special attention to applications that not only closely describe experimental data, but also shed a new light on the biological processes in question. In student projects, we could explore questions such as why are cells hexagonal or how to use stochastic techniques to determine the number of different types of cells in skin tissue. We could also look into the dynamics of escape from biomedical intervention, such as chemotherapy.

Prerequisites: Knowledge of elementary linear algebra, calculus, and differential equations at the level of Math 21a,b. Previous experience with probability will be a plus, but is not necessary.

Knot Theory (Tanya Kobylyatskaya and Ethan Street (,

What is a knot? Is it what keeps your shoes on your feet? Is it that scary mass of cables behind your desk you have to wade through every time you drop your pencil? To a mathematician, a knot is a precise mathematical object: a circle that has been mapped into 3-dimensional space in some (possibly complicated) way. Far from being simply a curiosity stemming from our real world experience, the study of mathematical knots has become absolutely fundamental to many parts of topology and other fields of math as well. This mini-course will be an introduction to the theory of knots and how it helps us think about topology. Beginning with a rigorous definition of knots and their notion of equivalence, we=92ll introduce some of the tools used to determine when two knots are mathematically different - the central problem of knot theory. These include the Alexander polynomial and the knot group.
The theme will be using knot diagrams (2-dimensional depictions of knots) to visualize knots in order to turn their geometry into algebraic information. After covering the basics we will have a significant amount of flexibility in choosing which more advanced topics to present. For example: applications to low-dimensional topology, =93surgery=94 on knots, more sophisticated invariants like Khovanov homology and the Jones polynomial, links (no pun intended) to algebraic geometry, slice knots and knot concordance, and braids. In addition, we could possibly cover some topics in higher dimensional knot theory such as knot "spinning". Whichever of these we don't do will be fair game for final papers. Along the way we will be sure to introduce many examples, and to draw plenty of pictures.

Prerequisites: The tutorial will be accessible to students with a basic understanding of point set topology, as may be gained in Math 131 or equivalent. Also preferred but not required is a background in algebra at the level of Math 122, say.

Additive Combinatorics by Le Anh Vinh ( )

Additive combinatorics is the theory of counting additive structures in sets. This subject has seen exciting developments in recent years, thanks to its connection with areas such as algebra, number theory, ergodic theory and graph theory. In this tutorial, we will give an elementary introduction to this fascinating field. More precisely, we aim to cover most of the following topics: the probabilistic method, sum-product estimates, the Erdos distance problem (over real and finite field spaces), graph theory method, and algebraic method.

Prerequisites: Familiarity with the Fourier transform.

Archive: Old Summer Tutorials, since 2001

Summer Tutorials: 2008 2007 2006 2005 2004 2003 2002 2001

Last update, 4/14/2010