We will plan talks and speakers for the semester and decide if this is the best time to meet

*Note the new room.

We find and describe the irreducible components of the space of rational curves on moduli spaces M of rank 2, stable vector bundles, with fixed determinant of odd degree, on curves C of genus g\geq2. We prove that the maximally rationally connected quotient of such a component is either the Jacobian of the curve C, or a direct sum of two copies of the Jacobian. We show that moduli spaces of rational curves on M are in one-to-one correspondence with moduli of rank 2 vector bundles on the surface P^1\times C.

Lie algebra cohomology will be defined and compared to corresponding de Rham cohomology on smooth Lie groups.

This will be an introductory talk in the semester-long seminar being organized by Brian Martensen. I will describe some examples and indicate various directions of current research interest.

Abstract: I will give an elementary introduction to Fabers conjectures on the moduli space of curves -- assuming no prior knowledge of algebraic geometry, and then give an analogous conjecture for the locus of hyperlliptic curves. The analog can be stated in very concrete and elementary terms; if you know about Poincare duality for real manifolds you know more than enough to understand the question. None the less I think it is quite non-trivial and surprising, and a solution would involve a nice mix of geometry and combinatorics.

This will be a quick meeting to set up the talks for this semester's SophEx series. SophEx is a seminar intended for first- and second-year students, all of whom should plan to attend!

These talks are intended to serve as an introduction to the models of the interest rate term structure. I hope to discuss models of both the short rate and the forward rate, as well as the issues of pricing and hedging of contingent claims under such models.

We will try to set the schedule of speakers for the Fall 2002 Semester.

This is a continuation of last week's talk.

This seminar is intended for graduate students. Questions related to the Navier-Stokes and the Euler equations will be the main topic. We will study some old and more recent papers in this area. Interested faculty are welcome too.

I'll explain what inverse limit spaces are, and then show that (nearly) all spaces of tilings can be modeled as inverse limits of branched manifolds.

We will get organized for the coming semester. R. de la Llave will report on a conference that took place Sept 4-7. The regular meeting time will be Wed. at 4:00pm in RLM 9.166

These talks are intended to serve as an introduction to the models of the interest rate term structure. I hope to discuss models of both the short rate and the forward rate, as well as the issues of pricing and hedging of contingent claims under such models.

This seminar will begin on September 24.

We will describe a (still open) problem of Arveson dating to 1969: Does every unital non-self-adjoint operator algebra have an irreducible boundary representation? We will detail recent progress of Muhly and Solel (1998) and Dritschel and McCullough (2001) towards an affirmative answer, as well as evidence of Hopenwasser (1973) suggesting a negative answer.

We will classify the semisimple Lie algebras over C by way of root systems and Dynkin diagrams. This will basically be my option talk from last May, with some more detail and more examples.

I'll give a construction for AF C*-algebras from substitution systems and shifts of finite type and explain what information is contained in their K-theory.

I'll examine the possible dimensions the commutator space of a particular n by n matrix can have. I will use the Jordan form and partitions of n to prove results about these dimensions, and use them to compute the dimensions for small values of n. I will move on to a few quirky little facts about the ground field that you may find surprising...

Gaussian affine and quadratic term structure models are generalised by replacing Gaussian processes with Levy processes. In the affine term structyre models (ATSM) the prices of contingent claims of European type are computed explicitly by using the Fourrier transform method. For many parametrized families of Levy processes, the bond prices can be calculated explicitly, and the pricing of contingent claims of European type requires one additional integration.

This series of three lectures aims to introduce the audience to a bunch of related problems in hyperkahler (quaternionic, N=4 supersymmetric) geometry. In the first lecture I start with listing several famous and by now classical results from Kahler (complex, N=2 supersymmetric) geometry. I quickly go through the definition of Kahler manifolds, the Kahler quotient construction, Hodge theory and the Hard Lefschetz theorem and index theory on them. I finish with three exciting applications. The first is Stanley's solution of McMullen's conjecture about the classification of face vectors of simple convex polytopes (as an application of the Hard Lefschetz theorem). The other two are 2+1 dimensional Topological Quantum Field Theories. One leads to the Casson invariant of three manifolds (as an application of intersection theory), the other (as an application of an index theorem) leads to the Jones polynomial and its generalization the Reshetikhin-Turaev-Witten invariant. These invariants are cooked out from the cohomology of a particular Káhler manifold: the moduli space of stable bundles on a complex curve. This first talk of the lecture series serves to explain the Kahler package that I am intending to carry through in the hyperkahler setting.

Eugene Tevelev is a recently arrived lecturer is is a candidaye for an assistant professor position.

This is a continuation of last week's talk on classifying semisimple Lie algebras over C. We will finish the root space decomposition and start on root systems. Even if you missed the first talk, you should be able to follow this one, knowing what a Lie algebra is.

We will discuss several software packages (K. Brakke surface evolver, level set methods) and demonstate various models.

We do global calculations on moduli spaces of pointed curves to compute the graded Betti numbers of general sets of points on any nonhyperelliptic canonically embedded curve (thus proving the Minimal Resolution Conjecture for canonical curves). These results have surprising connections with linear series on moduli spaces of stable vector bundles on curves and geometric divisors on moduli spaces of pointed curves. We also give an positive answer to a conjecture of Lazarsfeld that all exterior powers of the normal bundle of a curve inside its Jacobian have a theta divisor which is identified with a difference variety in the Jacobian. These vector bundles are of great interest because because Green's Conjecture on syzygies of canonical curves can be reduced to computing their number of sections. This is joint work with M. Mustata and M. Popa.

We study a model of informed trading in which trades arrive sequentially, uninformed trades arrive as a Poisson process. We characterize an equilibrium in which the single informed trader plays a mixed strategy - a point process with stochastic intensity. In this equilibrium, informed and uninformed trades arrive probabilistically, as they are assumed to do in Glosten-Milgrom models. We study a sequence of such markets in which uninformed trades become amaller and arrive more frequently, approximating a Brownian motion. We show that if the equilibria converge then their limit is the equilibrium of a Kyle-type continuous auctions model.

Completely opposite to the first talk, this second one will be a thorough introduction to hyperkahler manifolds with lots of examples. After some general ideas and definitions about hyperkahler manifolds I will go through the simplest hyperkahler manifolds, namely 4-dimensional gravitational instantons. In particular I will explain the Gibbons-Hawking ansatz, which gives the multi-Eguchi-Hanson metrics and multi-Taub-NUT metrics, they are all 4-dimensional hyperkahler manifolds. The next topic of the talk will be a more general method of constructing hyperkahler manifolds: the hyperkahler quotient construction. As an example I will detail the toric case, that is the construction of toric hyperkahler varieties; the simplest of this family, the Calabi metric on the cotangent bundle to CP^n will be treated explicitely. Finally, if time permits, I may talk about the moduli space of Higgs bundles, which also has a natural hyperkahler metric.

I really will finish talking about the root space decomposition this time. I'll introduce root systems and mention the important facts we need. This will set up the classification for semisimple Lie algebras via Dynkin diagrams next week.

The Rohlin property for automorphisms of C*-algebras can be viewed as a topological analog to the classical Rohlin lemma in ergodic theory. I'll explain how it is useful and why this property holds for a class of automorphisms arising from shifts of finite type.

In applications of elliptic curves to cryptography, one often needs to construct elliptic curves with known number of points over a given prime field , where is a prime. In the context of primality proving, Atkin and Morain suggested the use of the theory of complex multiplication to construct such curves. One of the steps in this method is the calculation of a certain Hilbert class polynomial $H_D(X)$ modulo for a certain fundamental discriminant . The usual way of doing this is to compute $H_D(X)$ over the integers and then reduce modulo . We present a modified version of the Chinese remainder theorem to compute $H_D(X)$ modulo directly from the knowledge of $H_D(X)$ modulo enough small primes. This is joint work with K. Lauter and R. Venkatesan.

We will present different definitions of solutions to PDEs and we will motivate the need for a new concept of solution, viscosity solutions. This concept is distinct from the previous definitions in its structure and we shall show that it is the appropriate one to solve even seemingly easy problems.