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 termtime tutorials, with the
tutorial leader lecturing in the first few meetings and students
lecturing later on. Unlike the termtime 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 Harvardrelated 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 5page 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 kronheim@math.harvard.edu. When you sign
up, please list at least one other choice in case your preferred
tutorial is either oversubscribed or undersubscribed. Places are
filled on a firstcome, firstserved 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.
Yours,
Peter Kronheimer
Stochastic Models in Biology by Ivana Bozic ibozic@math.harvard.edu
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 birthdeath 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 (tkobyly@fas.harvard.edu,estreet@math.harvard.edu)
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 3dimensional
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 minicourse 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 (2dimensional 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 lowdimensional
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 (vinh@math.harvard.edu) )
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, sumproduct 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 
