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


Peter Kronheimer

Quadratic Forms (Ana Caraiani,

Quadratic forms are some of the most beautiful objects in mathematics. In this tutorial, we will study them mostly from the point of view of number theory and use the opportunity to introduce some central ideas and principles in the field.
We will introduce the p-adic rationals and prove that quadratic forms = with rational coefficients satisfy the local-global principle. Then we will look at more instances in number theory where this principle arises (primes in arithmetic progressions, twin primes). For integral quadratic forms, we will prove the remarkable Fifteen Theorem which says that if a quadratic form with integral matrix represents all positive integers up to 15, then it represents all positive integers. We will also see how binary quadratic forms are related to ideal theory in quadratic fields via Gauss composition; this will prepare us to talk about recently discovered higher composition laws. We might even have some fun with Rubik's cubes!
In student projects, we could explore the mysterious connection of higher composition laws with the Dynkin diagrams of exceptional simple Lie groups or learn about the classification of simply-connected 4-manifolds. We may also investigate theta functions, which are constructed using quadratic forms.

Prerequisites: The only prerequisites are basic algebra at the level taught in Math 122 and very elementary number theory as covered in the first part of Math 124 (factorization, prime numbers, congruences). Having some familiarity with rings, fields and field extensions would be helpful, but not required.

Pointless Topology (Sam Isaacson,

Traditionally, we think of spaces as a collection of points with some added structure: namely, a collection of open sets satisfying a short list of axioms. But what if we throw out the points? The open sets of a topological space with the order imposed by inclusion forms a locale, or generalized space; and the closed-open sets of a space form a lattice under inclusion. Both the locale of open sets and the lattice of closed-open sets have attached to them algebraic operations given by union and intersection. In the 1930's, Marshall Stone inaugurated the study of generalized spaces by proving a famous representation theorems: compact Hausdorff totally disconnected spaces are precisely those spaces determined by their lattice of closed-open sets.
We'll explore two consequences of Stone's work. First, we'll embark on a systematic study of pointless topology. More general representation theorems let us regard locales as a substitute for spaces. The chief advantage of this perspective is that many results that required the axiom of choice in classical point-set topology have constructive analogues. Second, we'll study some of the consequences of Stone's work outside of the localic world. Stone's insight that algebraic objects (lattices, Boolean algebras, etc.) could act as substitutes for spaces is at the heart of harmonic analysis and algebraic geometry with the Gelfand-Naimark theorem and the notion of the spectrum of a commutative ring.

Prerequisites: The only prerequisite for this tutorial is a basic understanding of point-set topology -- Math 131 or an equivalent will be sufficient.

The Model Theory of Fields Katy Körner ( and Aaron Silberstein (

The properties of fields can be studied in a number of ways, some more daunting than others; we seek to show students the value and friendliness of a model theoretic approach. Model theory is the study of mathematical structures. In order to study model theory at all, students will be taken on a quick jog through the basics of logic and set theory. The content will vary depending on the background of the students -- those who have taken 141 or 143 can, if they choose, move quickly on to applications of model theory to their own areas of mathematical interest, while those with no such background will study notions of completeness and decidability, as well as exploring the ordinals and cardinals through proving the equivalence of well-ordering and the axiom of choice. This is valuable for its own sake, serving to give students some insight into the foundations on which the rest of mathematics is built.
Galois theory is the study of the symmetry of both fields and equations. First developed to determine the problem of how to solve a polynomial, it has since developed into a deep and fruitful way of understanding symmetries whether they arise in number theory, algebraic geometry, topology, or differential equations. Emil Artin, in the first half of the last century, made the discovery that the most fundamental questions in Galois theory should be expressed in terms of logic; since then, Galois theory has been a roiling synthesis of geometry, topology, arithmetic, and logic, blurring the lines between these fields and allowing each to provide insight into the other. After introducing Galois theory and some of its many applications (including solving polynomial equations!) we will delve into the model theory of fields and see how the model theory of a field is very closely linked to its Galois theory. Projects will then focus on aspects of logic and field theory, tailored to students' interests --- be they logic, number theory, topology, algebraic geometry, or group theory.

Prerequisites: Algebra at the level of Math 122.

Symplectic Geometry (Alexandar Subotic,

Symplectic manifolds are a special class of smooth manifolds that arises naturally in many problems in geometry. They are necessarily even-dimensional and in fact they are often important players in the big research area of four dimensional geometry. In some ways they are similar to complex manifolds (the manifolds of algebraic geometry) so they are also thought of as being in-between real and algebraic geometry. Finally they are an essential part of geometry related to modern and classical physics: mirror symmetry, string theory, and Hamiltonian mechanics.
This tutorial will provide a first introduction into symplectic geometry. We will start with introducing the symplectic form and the linear algebra associated to it. We will define symplectic manifolds, and will give many examples along with the fundamental Darboux's theorem on their local structure. We will introduce almost complex structures and explain the relationship between symplectic and complex geometry. We will explain hamiltonian mechanics and its relation to symplectic geometry, and time permitting we will introduce the notion of a moment map.

Prerequisites: Basic concepts of manifolds, differential forms and vector fields.

Archive: Old Summer Tutorials, since 2001

Summer Tutorials: 2007 2006 2005 2004 2003 2002 2001

Last update, 4/14/2008