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Tutorial Topics, Fall and Spring

(2003-2004), (For Summer Tutorials 2003, look here )

This is a Pamphlet from the Undergraduate page of the Harvard mathematics department
Archived old tutorial abstracts: Tuturials 2000-2001 Tuturials 2001-2002 Tuturials 2002-2003 Tuturials 2003-2004

Fall Tutorials:

This is an archived page.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

1. Geometric Topology (fall) taught by Curt McMullen

Prerequisites: algebra (Math 122) and topology (Math 131), some complex analysis (Math 113) would be useful as well.
Description: How is a sphere different from a torus? A square-knot from a granny knot? In this tutorial we will discuss topology in dimensions one, two and three, with an emphasis on problems and examples. Topics may include: the fundamental group and covering spaces; free groups and graphs; topology and geometry of surfaces; knot theory and polynomials; 3-dimensional spaces; and hyperbolic geometry. References: Adams, The Knot Book Rolfsen, Knot Theory Stillwell, Classical Topology and Combinatorial Group Theory

2. Morse Theory (fall) taught by Erick Matsen and Ciprian Manolescu

Prerequisite: a basic knowledge of manifolds (such as in Math 134 or Math 135).
Description: Morse theory is an extremely simple tool which has revolutionized fields of mathematics several times over. Morse himself developed the theory and applied it to mathematical physics. Later, Bott took these ideas and used them to prove his celebrated periodicity theorem. Then Smale used it to prove the h-cobordism theorem, which implies the generalized Poincare conjecture in dimensions five and above. More recently Andreas Floer applied the ideas in the symplectic setting to prove the Arnol'd conjecture, and in the process invented an important new homology theory. This tutorial, however, will have the goal of introducing the basic ideas and proving Bott's periodicity theorem. This theorem is concerned with the topology of matrix groups, and demonstrates a very beautiful and surprising fact about their higher homotopy groups. In a larger sense, though, another goal for the course will be to play with manifolds and the tools that we use to understand them. We will get to talk about Lie groups and differential geometry (two extremely important parts of mathematics) in very elementary, hands-on ways. It should be a lot of fun!

Spring Tutorials:

Two tutorials will be offered this spring:
  • Computational Group Theory taught by Nick Rogers, nfrogers@math
  • The Incredible Edible Sphere taught by Matt Bainbridge, matt@math
  • Class Field Theory taught by Grigor Grigorov, grigorov@math
If you are interested in taking a tutorial, then you should come to the tutorial Introductory/organizational meetings:
  • On Tuesday, February 3rd, at 5:30 PM , The tutorial introductory meeting - the first spring meeting of the Math Club at Mather House in Dining Rooms A&B.
  • On Wednesday, February 4th, at 2:00PM in room 507 The tutorial organizational meeting.

3. Computational Group Theory (spring) taught by Nick Rogers

Prerequisite: group theory (such as in Math 122).

During the last few decades, there has been a great deal of work done and progress made in computational aspects of finite groups. Classifying all of the groups of a given order, expressing an element of a group as a word in the generators, and determining the subgroup generated by a set of group elements are all problems that become computationally intensive once the group in question is sufficiently large and complex. One of the most difficult and celebrated theorems of the 20th century, the classification of finite simple groups, required for its proof a great deal of computation to eliminate the myriad possibilities.
list of sporadic groups
Choosing an element at random from a large group is an important problem with a great deal of computational flavor. The idea of this tutorial is to introduce students to the basic techniques of computational group theory, and give them a chance to explore recent progress in this field through their research projects. The only prerequisite for this tutorial is an elementary understanding of group theory, as presented in Math 122. Students with specialized knowledge in combinatorics, representation theory, probability and stochastic processes, or theoretical computer science will be encouraged to explore problems of current interest that take advantage of this additional background. If you're fascinated by finite groups, and want the chance to learn more about them in a very hands-on way, then this is the tutorial for you!

4. The Incredible Edible Sphere (spring) taught by Matt Bainbridge

Prerequisites: topology (Math 131) and experience with smooth manifolds (such as in Math 134 or Math 135).

Source: Geometry Center
The sphere is one of the most basic objects in mathematics. You're probably familiar with spheres from your everyday life, and you may even have on sitting on your neck. What's neat about spheres is that some of the simplest questions that you can ask about them can lead to extremely surprising and deep mathematics. In this tutorial, we will study the mathematics of spheres, concentrating on their topology. One question that we will study is "Can you turn the sphere inside out?" To make this possible, you have to assume that the sphere is made of a material which can pass through itself. To make this question nontrivial, we'll impose a technical requirement which amounts to saying that you can't "crease" the sphere, We'll see that under these conditions, you can indeed turn the sphere inside-out. If you don't believe me, you can see a movie of a sphere turning inside-out.
Further topics will depend on the interests of the students. Here are a few topics I would like to cover: Milnor's exotic spheres -- these are 7-manifolds which are homeomorphic but not diffeomorphic to the 7-sphere; homotopy groups of spheres; Hopf's theorem -- this classifies maps from an n-manifold to the n sphere up to homotopy; the hairy ball theorem; the Banach-Tarski paradox; and horned spheres -- these are wild embeddings of the two sphere into the three sphere. Along the way, we'll learn some useful techniques in topology. To take this class, you should have taken a basic topology class such as 131, and you should be familiar with multi-variable calculus.

5. Class Field Theory (spring) taught by Grigor Grigorov

Prerequisite: group theory (such as in Math 122). Students who are taking the tutorial should have some basic knowledge of general topology and some familiarity with integration and measure theory in the classical context. Good knowledge of algebra at the level of Math 122 and 123 is required. Also the people in the tutorial should either know some algebraic number theory or should be concurrently taking Math 129.

First Meeting Tuesday Feb 10 at 5pm, Room 507


Class field theory is one of the fundamentals of modern number theory. It classifies the abelian extensions of a given number field in terms of the arithmetic of this field. It is quite easy to formulate the fundamental theorems of the theory but the proofs are hard. In this tutorial we are going to follow Weil's approach which is based on integration on locally compact groups and the study of central simple algebras. The goal of the tutorial would be to go in detail through the book \cite{Weil}. We will focus on understanding the proofs of the main theorems in class field theory and this will require active participation from the students. One of the reasons for choosing Weil's book is that it is hard to read on your own and hopefully that will encourage the students to work together. In addition to the main reference the book by Ramakrishnan and Valenza is a useful reference for the theory of integration on locally compact groups. Another source which might be helpful are Milne's notes.

  • Weil, Andre, Basic number theory, Springer-Verlag, 1995
  • Ramakrishnan, Dinakar and Valenza, Robert J., Fourier analysis on number fields, Graduate Texts in Mathematics, Springer-Verlag, 1999
  • Milne, J. S., Class Field Theory

For more information regarding tutorials, please contact Andy Engelward.

Last update: 8/8/2003 Back to the Math department homepage Back to the undergraduate information page