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Mathematics 99r, Tutorial Topics, Fall and Spring

(2005-2006), (For Summer Tutorials 2006, look here )

This is a Pamphlet from the Undergraduate page of the Harvard mathematics department
This is an archived page. See the current page
Archived old Fall-Spring tutorial abstracts: Tuturials 2000-2001 Tuturials 2001-2002 Tuturials 2002-2003 Tuturials 2003-2004 Tuturials 2004-2005
Archived old Summer tutorial abstracts: Summer 2001 Summer 2002 Summer 2003 Summer 2004 Summer 2005

This is an archived page. The currently offered tutorials are here

Fall Tutorials:

Binary quadratic forms

Description: The theory of binary quadratic forms was initiated by Fermat and culminated with Gauss Disquisitiones Arithmeticae in 1801. The main question addressed in Disquisitiones is: Given a binary, quadratic form, which prime numbers can be represented by it? To be more explicit, a binary quadratic form is a function of real variables X and Y having the form Q(X,Y) a X2+b XY+cY2 with a, b and c integers. An integer, m, is said to be represented by Q when the equation Q(X, Y) = m has solutions where X and Y are both integers.

The solution to this question in Disquisitiones hints at many subsequent, fundamental developments in algebraic number theory. This said, the tutorial will study Gauss theory of quadratic forms while emphasizing the relations to modern concepts in algebraic number theory. Beginning with quadratic reciprocity and the reduction theory of quadratic forms will lead to a rudimentary answer to the main question in Disquisitiones. To advance further, the tutorial will develop Gauss theory of the class group and the genus theory, both from the point of view of quadratic forms and quadratic fields. Following Dirichelet, analytic methods will then be applied, and these will lead to the theory of L-functions and the class number formula. If time permits, Gauss class number problem will also be studied. Number theory has been developed to a degree that hides most traces of the classical sources. The latter, though central, are hidden in their modern incarnations. If you believe that this makes for a missing link in number theory education, then this is the tutorial for you.

Prerequisites: Algebra on the level of Math 122 and 123, and complex analysis on the level of Math 131.

For more information, contact Chung Pang Mok ( and Kai-Wen Lan (

Holomorphic vector bundles

Description: Just as the study of integers spurred the development of beautiful mathematics in number theory, vector bundles have been a major driving force in geometry and topology for a long time. Vector bundles can be viewed as parameterized families of vector spaces. For example, the simplest, but non-trivial vector bundle is called the Mobius bundle; its parameter space is the circle, and it is the set of points (t,(x,y)) where t is the angle around the circle and (x,y) in R2 is orthogonal to the vector (cos(t/2),sin(t/2)). (This Mobius bundle is not the stupid product bundle, S1 x R, in disguise.) In general, vector bundles serve as bridges between linear phenomena and non-linear phenomena. In particular, they are central to diverse and very topical areas of mathematics and theoretical physics. Vector bundles can be studied in topological contexts, in geometric contexts and in the context of complex numbers and holomorphic functions. Vector bundles in the latter context (holomorphic vector bundles) have the most structure and elegance. Their simplest parameter spaces are Riemann surfaces, but even here, their mathematics is incredibly rich and leads to the heart of a great deal of current mathematical research. This said, the goal of the tutorial is to first introduce vector bundles to the uninitiated, and then proceed to uncover some of the beautiful results about holomorphic vector bundles. The aim is to first understand one dimensional, complex vector bundles on Riemann surfaces, and then study higher dimensional vector bundles on the sphere and on tori. These cases are almost completely understood and are a rich source of beautiful ideas and techniques. (Meanwhile, there are a slew of profound and not yet answered questions about holomorphic bundles on surfaces with genus large than 1.) This tutorial can serve as a good step into the subject of holomorphic vector bundles; thus a first step into a hot area of current research in mathematics.

Prerequisites: Some complex function theory as can be found in Math 113, an introduction to algebra as in Math 122, and a bit of algebraic topology as taught in Math 131.

For more information, contact Sabin Cautis (

Spring Tutorials:

Height functions in Diophantine geometry

Description: The search for integer or rational number solutions to polynomial equations has intrigued mathematicians for at least two and half thousand years. Such equations are called Diophantine equations. The name invokes Diophantus, a Greek mathematician who wrote about such equations. The simplest examples of Diophantine equations are linear equations such as aX + bY = c where a, b and c are integers. Perhaps the most famous is the Fermat equation Xn + Yn= Zn; here n is a positive integer. There is an algorithm for finding integer solutions to the former. As for the latter, a proof that the n larger than 2 case has no non-trivial solutions requires some of the most sophisticated mathematics yet developed. So where does geometry come in, and what is Diophantine Geometry? The idea is to view the solutions to a Diophantine equation as defining a surface or related geometric object. Diophantine Geometry considers the interplay between the number theory and the geometry of the corresponding geometric object. The goal of this course is to provide an introduction to some of the foundational concepts in Diophantine geometry. In particular, a height function plays a central role in the subject. A height function tells us about the arithmetic complexity of a rational solution to the given equation, and it also tells us about the complexity of the geometry of the corresponding surface near the solution point. Height functions lie at the heart of several important theorems in Diophantine geometry. We will discuss classical applications of height functions as well as some new results and open problems. This tutorial should be a great lead in for a senior topic!

Prerequisites: The prerequisites are fairly modest. I will assume that you have had a first course in abstract algebra, such as 122-123, and also some complex function theory as can be found in Math 113.

For more information, contact Sonal Jain (

Sphere packing

Description: How densely can you pack non-overlapping spheres of the same size in Euclidean space? This problem in geometry has attracted the attention of mathematicians and other sorts of scientists for centuries.It is an extremely hard problem in general, but there are many ingenious methods that are used to obtain upper and lower bounds for the maximal possible density. Likewise, there are ingenious methods that go towards understanding how the maximal density changes with the Euclidean space dimension. Lower bounds arise by exhibiting a particular packing, and some of the best packings arise from some truly elegant parts of modern mathematics. This tutorial will focus on the theory of sphere packing and related problems such as:
  1. The kissing problem:How many congruent non-overlapping spheres can be packed so that all touch a given sphere?
  2. The lattice packing problem: What is the tightest packing if the spheres are constrained so that their centers sit at points in a regular lattice? This problem has some deep connections with the quadratic forms in their number theoretic guise.
  3. The covering problem: What is the most efficient covering of Euclidean space by congruent overlapping spheres. Thus, if every point is to be in at least one sphere, what covering gives the minimum fraction of space wasted by overlapping?
  4. Spherical codes: This is a sphere packing on a sphere! Give a non-zero angle and determine the maximum number of points on a sphere with pairwise separation no less than the given angle.
The tutorial will introduce techniques for tackling these problems and look at some concrete answers for small dimensional Euclidean spaces and spheres.

Prerequisites: None, save for an interest in geometry and some facility with analysis.

For more information, contact Abhinav Kumar (

Geometry and Physics

Description: Mathematics and physics have always been inextricably linked subjects, and mathematical methods have for a long time been essential to the physical understanding of the world. However, in recent years the deep insights offered by methods of physics to the understanding of classically mathematical questions has taken mathematicians somewhat by surprise. The interaction that ensued has been spectacular and has in particular led to some great work in geometry, from unexpected manifold invariants to ways of counting rational curves. This tutorial is designed for students who are interested in learning some basics of the machinery that is used in modern physics, from a mathematicians point of view, so that they could acquire a better understanding of these important developments. We will start by reviewing some dynamics, in context of field theories, and then introduce the notion of a quantum field theory (QFT) which will be central to the tutorial. There are good references on QFT for a mathematical audience and we will use a selection of these, since most of them cover the topics most interesting for our purposes. The wealth of topics is still great, and a lot of choices will depend on student interests.

Prerequisites: advanced undergraduate preparation would be preferred.

For more information contact Aleksandar Subotic (

Last update: 2/2/2006 Back to the Math department homepage Back to the undergraduate information page