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This is an archived page. See the current page
 Archived old Fall-Spring tutorial abstracts: Archived old Summer tutorial abstracts: Tuturials 2000-2001 Tuturials 2001-2002 Tuturials 2002-2003 Tuturials 2003-2004 Tuturials 2004-2005 Tuturials 2005-2006 Summer 2001 Summer 2002 Summer 2003 Summer 2004 Summer 2005 Summer 2006

### Fall 2006 Tutorials

 This is an archived page.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

### Enumerative Geometry

Description: Enumerative geometry has been an active and attractive research subject in math for a long time. Many enumerative problems originally from classical geometry can be solved now by ad hoc techniques in modern algebraic geometry. The purpose of this tutorial is to give an introduction to this fascinating subject. We will mostly focus on curves in low dimensional projective spaces. On one hand, we do not need too complicated theory to deal with them. On the other hand, those examples can still provide us some clue about how to play with enumerative problems. We hope that after this class, students will be familiar with some basic techniques in enumerative geometry and be able to pursue more advanced knowledge in this field. Some possible topics are:
1. Geometry of plane curves, dual curves, nodes, cusps, flexes and bi-tangents.
2. Degeneration method, counting the number of lines incident with four lines, conics incident to eight lines in P3 etc.
3. An introduction to Grassmannians and Schubert calculus.
4. Counting plane rational curves, a simple proof of Kontsevich's formula (or to understand the Fields Medalists work in 30 minutes!).
5. Counting lines on low degree surfaces and threefolds. 6. An introduction to tropical curves, with applications to counting low degree and genus plane curves.
A lot more topics are available and up to student interests. If you really appreciate the beauty of math, this tutorial is the right one for you!

Prerequisites: Familiarity with some ideas and techniques in Math 137 or equivalent would be helpful.

### Morse Theory

Description: Initiated by Morse in 1930s, Morse theory has been proved to be an extremely powerful tool in studying the geometry of smooth manifolds. In geography, one can understand the landforms by looking at a contour map. In the same manner, the fundamental lemma of Morse enables us to see the "shape" of a space by understanding the level sets of reasonably good functions (Morse functions) on it.

Given a Morse function, one can get the cell-decomposition of the manifold. Applying this cell-decomposition method, Smale proved the generalized Poincare' conjecture. With this applied to algebraic geometry we will prove that a hypersurface in complex projective n-space (for n bigger than 2) is simply-connected. A more detailed argument leads to a proof of the Lefschetz hypersurface theorem.

Instead of a smooth manifold itself and a Morse function on it one can consider its path space and the energy functional defined on it. This leads to an analogue of Morse theory, in which geodesics happen to be exactly the critical points of the energy functional. It gives a lot of information about the geometry of the original manifolds. As an example we will deal with the existence of closed geodesics . Along this line, Bott gives an elegant proof of his periodicity theorem for classical groups.

This tutorial serves as a motivating starting point for more advanced literature such as Bott's and Milnor's lecture notes on Morse theory, Smale's work on the generalized Poincare' conjecture, and Witten's revised version of Morse theory. If you are interested in understanding the geometry of manifolds, this tutorial is for you!

Prerequisites: Math 135 or familiarity with ideas in differential geometry.

### Geometry and Physics

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.

### Elliptic functions

Description: The theory of elliptic functions was a great unifying theme of 19th century mathematics. It germinated the development of complex function theory and algebraic geometry. At the same time, elliptic functions serve as motivating examples of these theories. The story can be traced back to the time of Euler. Motivated by results of Fagano on the arc length of lemniscate, Euler was led to the study of elliptic integrals. These are integrals of functions that involve rationally the square root of a polynomial of degree less than or equal to 4. If the degree is less than or equal to two, then these can be expressed in terms of inverse trigonometric functions. Therefore the interest lies in the case where the degree is 3 or 4. It was already realized early in the study that elliptic integrals define only multi-valued functions, i.e. values of these integrals depend on the path of integration on the complex plane, the ambiguity being called, in modern terms, a period. Gauss, and later Abel, Jacobi, made the discovery that, for each elliptic integral, there are actually two fundamental periods , which are linearly independent over the real numbers, such that any others can be written as integral linear combinations of these two.

In analogue with the trigonometric case, Gauss, Abel and Jacobi obtained what is now called elliptic functions, by considering the inverse functions of elliptic integrals. The key point for the inversion is that elliptic functions are single-valued analytic functions on the complex plane: they are periodic with respect to the lattice spanned by the two fundamental periods. In other words, elliptic functions are functions on compact surfaces of genus one, a view which is generalized as the theory of Riemann surfaces. In this course, well begin the study with the function-theoretic point of view, construct important examples of elliptic functions, and then explain the relation with the algebra-geometric point of view. If time permits, well study some of the arithmetic applications.

Prerequisites: Algebra on the level of Math 122, and complex analysis on the level of Math 113.