Harvard Mathematics Logo Summer Tutorials 2005
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Witt Hasse Alexander Wirtinger
Serre Grothendieck Aharohov Bohm

Welcome Message


What follows describes the 2005 version of our summer tutorial program. The goal of the program is to offer you some interesting mathematics if you have an ulterior motive for being in the Boston area during July and the beginning of August. In particular, the tutorials 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 roughly at the beginning of 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. Here, I should say that these tutorials have no official Harvard status, so you won't 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 student participant a stipend of $700 and you can hand in your final paper from the tutorial for you junior 5-page paper requirement.

We are offering four tutorials this summer; the topics and leaders are as follows:

  • Bilinear forms in algebra, topology and arithmetic; led by John Francis
  • Knot theory; led by Andrew Lobb
  • Spectral sequences; led by Mike Hill
  • How to talk to physicists: Groups, symmetry and topology; led by Daniel Larson.

Below, you will find a brief description of each tutorial topic. Moreover, the tutorial leaders will describe their tutorials at the April 5 Math Table. You can sign up for a tutorial by emailing your desire to me at chtaubes@math. When you do sign up, please list at least one other choice in case your desired tutorial is either over subscribed or under subscribed. What follows are some brief descriptions of the tutorial topics. If you have further questions about any given topic, contact the tutorial leader via the email address given below. If you have questions about the administration of the tutorials, ask me.

                          -- Cliff Taubes

Tutorial: Bilinear forms in algebra, topology and arithmetic

Witt Hasse Symmetric bilinear forms are simple to define, but have an elegant structure theory. Moreover, they have applications all over mathematics, from manifolds to number fields to finite simple groups to motivic homotopy. They date from Gauss work on quadratic reciprocity, and were further studied by the likes of Eisenstein, Minkowski, Witt, Hasse. They are a beautiful subject within mathematics. In this tutorial we will develop their structure theory over integral domains and then see the variety of their occurrences and uses. No one will be lost due to lack of background; no one will be over-qualified, even those who have taken Math 250 and 260; and no one will be bored. Topics will include: Grothendieck-Witt rings of finite fields and number fields; type I and type II inner product spaces over Z; lattices. Final projects can involve subjects as diverse as the cohomology rings of manifolds, class fields, lattices, coding theory, and algebraic K-theory.

Prerequisites:: Basic of algebra and linear algebra as can be found in Math 121 and 122.

For further information: Contact John Francis, jnkf@math.mit.edu.

Tutorial: Knot theory

Alexander Wirtinger A knot is the same thing to a mathematician as it is to anyone else - an embedding of the circle into 3-space. Surely knots are such simple objects that they've been studied to death and there's nothing else interesting to figure out? No! In fact the more we seem to learn about the theory of knots, the deeper and richer and more mysterious that theory becomes.

This tutorial will first provide an introduction to classical knot theory including notions such as the group of a knot, the Wirtinger presentation, the Alexander module and polynomial (these are all based on analyzing the fundamental group of the complement of the knot in the 3-sphere). Then we'll look at newer invariants of knots such as the Jones and HOMFLY polynomials and their realization as quantum invariants. These newer invariants differ in flavor from the classical ones by virtue of the fact that they are not defined from intrinsic geometric properties of the knot or its complement. Rather, they are defined directly from a picture of a knot using data that is coded in the over and under crossings that the picture presents. We'll aim to look at very current areas of study, Vassiliev invariants and Khovanov homology, at the end of the course. Khovanov homology, in particular, has a very beautiful "algebra-free" interpretation where everything is defined in terms of pictures of surfaces with boundary. (I intend either to explain to you or to get one of you to explain to your classmates!)

Prerequisites: Knot theory has the advantage that it can be studied without a lot of prerequisites, but I would prefer students to have had some topology (at the 131 level, say) and a little algebra (122/123).

For further information: Contact Andrew Lobb, (lobb@math)

Tutorial: Spectral sequences

Serre Grothendieck Spectral sequences are one of the most powerful tools at the modern mathematicians' disposal, arising ubiquitously across the discipline. In this tutorial, we will discuss the spectral sequences from an algebraic perspective, focusing on how one actually uses them and how to interpret the results. People who take this will fearlessly utter statements like "an elementary spectral sequence argument shows иии."

We will begin by reviewing the algebraic underpinnings of spectral sequences, focusing on the concepts of a chain complex of vector spaces or algebras and moving to the topics of modules and differential graded algebras. With this background in place, we will then move to the meat of the course: How spectral sequences arise and how to understand them. The discussion here starts by introducing filtrations of an algebra and the resulting spectral sequences. These sorts of spectral sequences are perhaps the most ubiquitous of all; and so some time will be spent on how to understand their properties. We will then move on to study exact couples and use them to motivate the more difficult concept of convergence. After discussing convergence, we will, time permitting, study additional topics such as product structures and the behavior of spectral sequences in the presence of additional constraints.

Students will have an opportunity in the final weeks of the tutorial to present lectures on various topics that relate to spectral sequences. Students can lecture on important computational applications, such as to the homology of the loopspace of a sphere. There can be lectures on more advanced algebraic structures, such as comultiplication and its interaction with spectral sequences. There can also be lectures on certain specific spectral sequences, such as those of Serre and Grothendieck.

Prerequisites: Since the focus will be algebraic, the only prerequisite is a familiarity with basic algebra as is taught in Math 122. No specialized knowledge from particular fields will be required, as we will focus on the general form of spectral sequences rather than the nitty-gritty of particular examples.

For further information. Contact Michael Hill, (michael.a.hill@gmail.com)

Tutorial: How to talk to a physicist: Groups, symmetry, and topology

Aharonov Bohm It is often said that mathematics is the language of physics. Though this may be true, it sometimes seems that mathematicians and physicists speak very different dialects. In this tutorial we will bridge this gap by investigating how various algebraic and topological notions are used by physicists to understand the world around us. In particular, the powerful concept of symmetry will be explored in the context of quantum mechanics and quantum field theory. By the end of the tutorial you should be able to explain to both physicists and mathematicians the answers to the following questions:
  • What are the differences between classical and quantum mechanical symmetries? Answering this question will require discussion of Lie groups and will help uncover the mysterious quantum mechanical property of the electron called "spin".
  • What is the difference between a "global" and a "local" symmetry? The answer to this question leads us to consider gauge theories, especially electromagnetism and some of its interesting geometrical properties like the Aharanov-Bohm effect and magnetic monopoles.
  • What does topology have to do with the vacuum? To answer this we will need to bring homotopy theory to bear on non-Abelian gauge theories.
Prerequisites: Some group theory, topology, or quantum mechanics will help, but I will strive to make the tutorial completely self-contained.

For further information: Contact Daniel Larson (dtlarson@fas) and look at the website.

Archive: Old Summer Tutorials, since 2001

Summer Tutorials: 2004 2003 2002 2001

Last update, 3/22/2005