Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132

Archived old Fall-Spring tutorial abstracts: 2000-2001 2001-2002 2002-2003 2003-2004 2004-2005 2005-2006 2006-2007 2007-2008 2008-2009 2009-2010 2010-2011 2011-2012 2011-2012
Archived old Summer tutorial abstracts: 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013: none

Spring 2014 Tutorials

Arithmetic of Quadratic Forms

Description: The theory of quadratic forms is one of the great achievements of classical number theory. It is an entry point for the theory of arithmetic groups, one of the most active areas of research in modern mathematics. Moreover, the theory of quadratic forms is applied throughout much of mathematics, including number theory, representation theory, geometric topology and algebraic geometry.
The course will initially focus on the representation of rational numbers by quadratic forms. We will discuss the Hasse principle, which says that this question is governed by congruence (alias: local) constraints. Possible further topics include the theory of integral representations; spin groups, classical reduction theory, and Gauss's composition law.
Prerequisites: The basic language of abstract algebra and linear algebra will be important. In particular, students should be comfortable with kernels, quotients, morphisms, and linear algebra over general fields. Some comfort with tensor algebra and with Galois theory will be helpful, but is not necessary. Students may also benefit from elementary experience with the p-adic numbers, but these will be developed in a self-contained way during the course.
Experience with elementary number theory at the level of Math 124 will be helpful. In particular, some experience with rational and integral Diophantine equations, Fermat's theorem on the representation of integers as sums of two squares, finite fields, and quadratic reciprocity will be helpful.
Contact: Sam Raskin (

Dynamics Of Rational Billiards

Description: In the absence of friction and other external forces, the motion of a billiard ball around a rectangular table, enjoying elastic collisions with its boundary, behaves in a well-understood way. Its trajectory follows either a periodic motion or distributes uniformly around the table. This old and beautiful result can be traced back to the time of Weyl and Kronecker. Because of this dichotomy in the behavior of the ball in a rectangular billiard table we say that its dynamics are optimal. One of the goals of this tutorial is to explore which polygonal billiard tables have optimal dynamics. The study of the dynamics and geometry of billiards in polygons is a large and still very active area of research. An open problem, striking for its simplicity, is to determine whether there exists an(obtuse) triangular table without any periodic trajectories. Besides their intrinsic mathematical beauty, the study of billiards has been mostly motivated by physical problems such as modeling of Boltzmann gases and electron transport in metals.
In this tutorial we will concentrate on polygonal billiard tables whose angles are rational multiples of # which we will call rational polygons. The significance of this condition is that there is only a finite set of directions for the trajectory of a billiard ball, as it reflects off the sides of the polygon. Via an unfolding procedure understanding the billiard path is reduced to studying the straight lines(geodesics) in a compact surface endowed with a flat metric with a finite number of singular points. This approach will be our entry point to the subject. One of the gems that we will prove is a theorem of W. Veech, saying that all regular polygons have optimal dynamics. The key idea in the proof of this theorem is the concept of renormalization, a powerful method in dynamical systems that has its origins in physics and one can think of as a time-acceleration machine. We will put this method in a more geometric context by studying a natural action of the group SL(2;R) on the space of unfolded billiard tables. The theory of rational billiards has connections with many other topics in algebraic geometry, number theory and topology of surfaces. If time permits we can cover some of these topics depending on students' background and interests. Our goal throughout the tutorial will be to explain the important ideas using basic examples and plenty of pictures, starting with the classical case of square billiard tables. There will be many suggestions for final projects, some of which will be computational.
Prerequisites: You should be familiar with calculus (in one and several variables) and the basics of group theory and metric spaces. It is not necessary but it would be useful if you have taken a course on the topology or geometry of surfaces, and basic complex analysis. If you would like to take the tutorial, but you are not sure whether your mathematical background is adequate, don't hesitate to send me an e-mail to discuss this together. Texts. Introduction to Rational Billiards, John Smillie A very nice introduction to what we will be discussing in this tutorial. It consists of three lectures given by John Smillie at the MSRI institute in 2007. You can find them in .pdf format here.
The conformal geometry of Billiards, Laura DeMarco A great survey article. You can download it from her homepage
Geometry and Billiards, Serge Tabachnikov A whole book devoted to billiards, emphasizing their connections to geometry. It should be accessible and easy to read, it explains nicely some of the topics we will cover. (e.g. the unfolding procedure). Chapters 1,2 and 7 would be the most relevant. You can download it from here.
Flat surfaces, Anton Zorich A nice article covering the topics of this tutorial and much more. Although it is paced at a more advanced level, I hope that it would still be useful browsing through it and trying to read those parts that you feel most comfortable with. You can download it from here.
Contact: Stergios Antonakoudis (

Fall 2013 Tutorial

Structure, Randomness and Pseudorandomness

Description: The celebrated theorem of Szemeredi states that every subset of natural numbers of positive upper density necessarily contains arbitrarily long arithmetic progressions. And it is not only a nice combinatorial result that was an open problem for several decades, but more importantly exposes to us the following fascinating phenomenon. There are several proofs of the Szemeredi's theorem making use of different tool sets such as graph theory, ergodic theory or harmonic analysis, but each of them deeply relies on the idea that a subset of natural numbers of positive upper density can either be incredibly structured or essentially random (or a mixture of these two regimes). So on one hand, when a subset is structured, this regularity leaves it no choice but to contain arbitrarily long arithmetic progressions. And on the hand, if it is essentially random, then somewhere in it we should be able to find a long arithmetic progression, but for a different reason. Of course the main question that remains is what should it mean for an object to be "essentially structured" or "essentially random"? There are several ways to formalize the answer for the subsets of natural numbers and yet more answers and questions for other mathematical objects. This interplay of randomness and structure will be a recurring theme of the course and we will try to take a look at several examples in which it emerges. The story of arithmetic progressions is already sufficiently deep to be a topic of several courses (culminating with a famous proof by Green and Tao that primes contain arbitrarily long arithmetic progressions), so we will devote to it a substantial part of the course. Other topics will include probabilistic methods in combinatorics, Ramsey Theory, chaos in dynamical systems, applications of quasirandomness and so on depending on time and interests of the audience.

Prerequisites: the course will require a basic knowledge of calculus, some mathematical maturity and affinity to combinatorial problems. There will be no textbook, but references and other sources of information will be provided on the way (and will be mostly available online). Students will be expected to write a short expository paper and give a presentation on the topic of their choice (within the scope of the course).

Contact: Konstantin Matveev (