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.

This is the third of three talks.

Finally we are going to classify simple (and thus semisimple) Lie algebras via Dynkin diagrams. Classifying the Dynkin diagrams is fun and easy, with lots of pictures. Even people who have no idea what a Lie algebra is will love this talk!

We review mathematical issues related to the theory of Large Eddy Simulation of turbulent flows. We show that most LES models have mathematical counterparts that have been introduced to resolve questions such as uniqueness, existence of a maximum principle, convergence to entropy solutions, and convergence in graph norms. As a conclusion to this review, we suggest mathematical criteria for selecting LES models.

I plan on introducing Kazhdan's Property (T) for groups and its generalization, Property (\tau). I will state and prove some basic facts, outline an interesting result related to Riemannian manifolds and expander graphs, and finish with some open questions.

A Moore graph of type (k,d) is a connected regular graph of degree k, diameter d and girth (2d+1). We will discuss the special case when Moore graphs are of type (k,2). Apparently such graphs exist only for k=1,2,3,7 and maybe 57.

About a century ago, Marshall suggested that a firm should invest as long as the present value of the expected marginal profit is non-negative. This rule ignores the option value of investment opportunities which affects the irreversible investment under uncertainty. A Marshallian-like form for the investment rule presented in the paper uses the infimum process of the price instead of the price process itself. The result is obtained for for a wide family of Levy processes and for random walks; the form of the result is novel even for the Gaussian case. The crucial tool is the Wiener-Hopf factorization method. We also derive a formula for the expected level of the capital stock in terms of the infimum and supremum price processes.

We'll finish classifying Dynkin diagrams, which will finish the classification of our Lie algebras. This will be the last one. Really.

Hypertoric varieties are hyperkahler (ie quaternionic) analogues of toric varieties. Like toric varieties, hypertoric varieties are combinatorially defined, and many of their elegant geometric properties can be read off from simple combinatorial diagrams. In the first hour of this talk we will discuss the different combinatorics that go into the kahler and hyperkahler settings, and relate them by considering certain circle actions on hypertoric varieties. The second hour of this talk will be on quiver varieties, another class of combinatorially defined algebraic varieties with deep connections to many different fields, such as representation theory, singularity theory, and physics. Like toric varieties, quiver varieties come in Kahler and hyperkahler flavors. I will give a bunch of examples, and discuss some questions about quiver varieties that are analogous to the questions that were asked in the first half of the talk, but much more difficult to answer.

As the title says, every positive integer can be expressed as the sum of four perfect squares, for instance, 5=4+1+0+0, 21=16+4+1+0 or 127=121+4+1+1. This is known as Lagrange's Theorem. I would also like to say a few introductory words about the more general problem of expressing a number as a sum of k th powers (Waring's problem).

I will present some recent results on the existence of the solution of the utility-maximization problem with random endowment in incomplete semimartingale markets. The convex duality method used gives rise to interesting process-theoretical results and gives a novel characterization of the positive supermartingales in terms of finitely-additive measures. (Candidate for Assistant Professor position.)

The speaker is an Assistant Professor candidate

This is the first of three talks.

In this talk, we describe a method for deriving and justifying a hierarchy of "reduced equations" for the dynamics of a scalar hyperbolic equation on thin elastic media, i.e. starting with a PDE defined on a three-dimensional domain, we will show that its solutions can be approximated by the solutions of equations defined on a two-dimensional domain, and, furthermore, there is a sequence of approximating equations, each of which afford a successively better approximation. The approach is based on ideas from Hamiltonian mechanics, and is related to Nekhoroshev theory. The work is joint with C. Eugene Wayne of Boston University.

No seminar this week. Next week, Lorenzo is speaking.

A rather standard introduction to Coxeter groups and graphs. We will cover definitions, presentations, Coxeter Graphs, and, briefly, Dynkin diagrams and their classification. If we have time we will finish with a look at H_3, and H_4: the symmetry groups of the icosohedron and 120-cell respectively.

Credit derivatives are financial products tied to changes in credit quality of underlying credit names. Thus, adequate valuation and hedging of credit derivatives relies on adequate models of changes of credit quality of underlying credit names. That is, it relies on adequate models of credit migrations (defaults in particular). In case of so called basket credit derivatives the issue becomes adequate modeling of (possibly dependent) credit migrations of several underlying credit names. This is also the basic issue regarding management of credit portfolios. In this talk I shall present a survey of approaches to mathematical modeling of dependent defaults and credit migrations. This is not an easy issue, as modeling has to account for scarcity of data needed for model estimation and calibration. I will focus on models based on the theory of copulas, and models based on the theory Markov chains and diffusion processes. I will present recent results obtained by several researchers in this area.

Are you tired of being lost in the first 5 minutes of seminar because you don't know what a moduli space is? The next few weeks of Jr. Geometry will work to change that. Starting from the the very basics, we will talk about coarse and fine moduli spaces and consider lots of examples of each. If time permits we will use this opportunity to introduce the notion of a stack.

these lectures will place in context some old and some recent results whose proof utilize time-frequency analysis (wavelets + Gabor), as well as to discuss a few conjectures.

Let $\phi(z)$ be a polynomial of degree at least 2 with coefficients in a number field . Iterating gives rise to a dynamical system and a corresponding canonical height function $\hhat_\phi$, as defined by Call and Silverman. We will define the notion of "transfinite diameter" for a subset of a metric space and prove a simple product formula relating the transfinite diameters of the filled Julia sets of over various completions of . We will then apply this formula to give a generalization of Bilu's equidistribution theorem for sequences of points whose canonical heights tend to zero. This is joint work with Liang-Chung Hsia.

This will be a basic, spooky introduction to quadratic reciprocity (in honor of Halloween).

LIBOR market model is the benchmark model for interest-rate derivatives. It has been a challenge to extend the standard LIBOR market model so as to cope with the volatility smiles and skews that are pronounced in the swaption markets. In this paper we extend the standard LIBOR market model, which takes forward rate or swap rate as state variables, by adopting stochastic volatility.

This is a prelim option talk

Eugene Tevelev is a candidate for an assistant professor position here.

More examples of moduli spaces for more moduli space satisfaction.

In time reversal acoustics, a wave field from a source is recorded by an array of tranceducers, time reversed and sent back into the medium. The back propagated wave focuses approximately on the source. I will talk about the time reversal phenomenon and its applications to imaging. In particular I will explain super-resolution and self-averaging for time reversal in random medium. Then I will discuss how to use eigenvectors and eigenvalues of the response matrix of an active array to image both the location and the size of a target.

We discuss rationales for kinetic descriptions of traffic dynamics and present a class of new models of Vlasov-Fokker-Planck type. These models incorporate (nonlocal and time-delayed) braking and acceleration terms which are consistent with realistic time scales. Correlation assumptions are made such that braking and acceleration terms depend only on macroscopic densities and the relative speeds with respect to the average speed; the braking term includes lane change probabilities, and reasonable assumptions on the dependencies of these probabilities on the traffic situation lead to multivalued fundamental diagrams, consistent with traffic observations. This part is based on joint work with Axel Klar and Thorsten Materne. In the spatially homogeneous case the equations reduce to a class of nonlinear drift-diffusion equations in velocity space; the drift and diffusion coefficients depend on the macroscopic (i.e., average) speed. In analogy to more conventional drift-diffusion equations we investigate the existence and behaviour of (traffic) entropy functionals. This analysis implies that there will either be traffic concentrations, or the traffic pattern will converge to one (of possibly several) equilibria given by the fundamental diagram. This is joint work with Jean Dolbeault, Paris.

This series of lectures places in context some old and some recent results whose proof utilize time-frequency analysis (wavelets + Gabor), and discusses a few conjectures.

Kiran Kedlaya is a candidate for an assistant professor position. Abstract: The Weil Conjectures make a variety of predictions about the numbers of points on algebraic varieties over finite fields. These conjectures were first established using Grothendieck's etale (l-adic) cohomology. We outline an alternate approach that has recently succeeded in reestablishing the conjectures, and which can be used for explicitly computing zeta functions of varieties. This approach uses Berthelot's rigid cohomology, based on Dwork's p-adic analytic proof of part of the Weil Conjectures.

The talk will be an introduction to how fractals arise and how the computer images of fractals are created. I will discuss fractals that arise when Newton's method is applied to complex variables, as well as the "connections" between Julia sets and the Mandelbrot set. If there is time, I will draw some fractals and prove some theorems used to determine the Mandelbrot.

We describe a method to obtain bounds on the hedging parameters of discretely exercisable options using Monte-Carlo simulation. The method is based on a combination of the duality formulation of the optimal stopping problem for pricing discretely exercisable options and Malliavin calculus. For a given computer budget and exercise strategy we provide an algorithm that achieves the tightest bounds and present numerical results. Our method can handle arbitrary payoff functions, general diffusion processes, and is independent of dimension.

Abstract: The monodromy of a surface bundle has translation distance, in the curve complex of the fibre, bounded by twice the genus of any strongly irreducible Heegaard splitting. It follows that, in bundles with high translation distance monodromy, the standard splitting is the unique minimal genus splitting. This is joint work with Saul Schleimer.

Catherine O'Neil is a candidate for an assistant professor position. Abstract: In trying to solve diophantine equations, a familiar method for proving the nonexistence of solutions is to show none exists modulo some integer. Conversely, if modulo every integer a solution exists, one might expect a solution to exist in the integers. This is called the Hasse Principle, and as we will see it sometimes holds and, even more interestingly, it sometimes doesn't quite hold.

Catherine O'Neil is a candidate for an assistant professor position. Abstract: There are various reasons that there is no fine moduli space for genus one curves with fixed Jacobian. A classical compromise is to create a coarse moduli scheme. I will explain why we instead choose another compromise, that of a sampling space. This is work in progress.

We got yer moduli spaces right here.

Introduced by Whitney in 1935, matroids are a combinatorial abstraction of linear independence. Beginning in the 70's the study of real and complex hyperplane arrangements brought matroids to geometry and topology. We will examine these applications and more recent ones involving quotients of spheres by elementary abelian p-groups, and arrangements of homotopy spheres.

This talk will introduce Lie groups, one of the cornerstones of modern geometry, and the exponential map.

Several systematic deviations from Bayesian updating have been discussed in the psychology literature and some of these have been incorporated into modeling exercises in behavioral finance. This paper presents a unifying axiomatic framework for a broad range of updating behavior including some of these disparate specifications. The key is to study preference on (contingent) menus of acts. Given suitable axioms, the functional form for utility features two measures, where one is the 'usual prior' p and the other determines how updating deviates from Bayesian updating of p. A dynamic version of the Anscombe-Aumann model is obtained as the special case where the two measures coincide.

Traveling waves in reaction-diffusion equations have been very successful in modeling various phenomena in physics, biology and other sciences. Recently there has been a lot of mathematical works on the possible effect of a fluid flow or wind on front propagation. A flow may siginificanlty enhance the front speed as well as extinguish a flame. We will review our results that exhibit and quantify these phenomena as well as discuss some progress in the analysis of the reactive Boussinesq system.

After briefly describing the geometric meaning of Oseledec's multiplicative ergodic theorem as it applies to volume-preserving flows, we will present a proof of the theorem appearing in a paper of David Ruelle's that is due to M. S. Raghunathan. The proof contains both analytic and geometric ideas; being a seminar on geometry, we will highlight the geometric ideas and suppress (that is, state, but not prove) some of the analytic detail.

It is well known that strange attractors occur in the H\'enon family of maps. Benedicks and Carleson proved that such attractors, called H\'enon-like, coincide with the closure $\overline{W^u(O)}$ of the unstable manifold of some hyperbolic periodic orbit $O=\{p_1,\dots,p_n\}$ of saddle type. On the other hand, numerical studies show the occurrence of attractors with a quasi-periodic component in the dynamics, called quasi-periodic H\'enon-like. As models for such attractors, we study families of diffeomorphisms of the cylinder $\mR^2 \times \mS^1$. Our main result shows the occurrence of H\'enon-like strange attractors when is a skew-product obtained by driving a H\'enon-like family $H$ in $\mR^2$ by the Arnold family of circle maps. The dynamics of the Arnold family on $\mS^1$ is assumed to be of Morse-Smale type. The attractors we obtain are H\'enon-like and occur on a set of parameter values of positive measure. Furthermore, occurrence is persistent under sufficiently small $C^3$-perturbations of the H\'enon-like family $H$. We also consider the case when the family has an invariant circle . Under suitable hypotheses, we show that the closure $\overline{W^u(S)}$ attracts an open set of initial states.

I will discuss main requirements for quantum computation. Elementary quantum gates, swap operation, creation of entanglement, teleportation will be described. A perturbation theory for scalable solid-state quantum computation with many qubits will be discussed. The ideas of error correction will be explained. Finally, I will describe the unwanted effect of decoherence for quantum bits.

CANCELLED

The level set method for capturing moving fronts was introduced in 1987 by Osher and Sethian. It has proven to be phenomenally successful as a numerical device. For example, typing in "Level Set Methods" on Google's search engine gives roughly 3300 responses. Applications range from capturing multiphase fluid dynamical flows, to special effects in Hollywood to visualization, image processing, control, epitaxial growth, computer vision and many more. In this talk we shall give an overview of the numerical and analytical technology and a few new applications.

This is the speaker's candidacy talk. Abstract: This talk concerns recent work on evaluations of L-series related to Zagier's Kronecker limit formula for real quadratic fields. More specifically, I will define three L-series related to a certain universal function appearing in Zagier's Kronecker limit formula, and show that each of these series has a meromorphic continuation to the complex plane. I will then derive a limit formula which determines the values of these L-series at the negative even integers.

We sketch, in as much detail as possible in 50 minutes, a proof of Oseledec's multiplicative ergodic theorem. We follow the presentation in a paper of David Ruelle's of a version of the proof that is due to M. S. Raghunathan. The proof contains both analytic and geometric ideas; being a seminar on geometry, we will highlight the geometric ideas and suppress some of the analytic detail. The presenter's notes for this talk, which can be found at http://www.ma.utexas.edu/users/kelliher/Geometry/Geometry.html contain all the details of the proof.

A mathematical analysis of the ``time reversal mirror" is given. the emphasis is put on phenomena which are described by the genuine acoustic equation with Dirichlet or ``impedance" boundary conditions. An ideal situation is first considered then relation between the question of the local decay of energy and the accuracy of the method are exploited. At variance with other approaches no randomness is introduced and only deterministic phenomena are considered. However instead randomness deterministic ergodicity is analyzed and it positive effects are explained with the (now classical) theorems on Quantum Chaos.

We define the reciprocal Mahler's measure to be Mahler's measure restricted to the subspace of reciprocal polynomials with complex coefficients of fixed degree. We view the reciprocal Mahler's measure as a symmetric, homogeneous distance function in the sense of the geometry of numbers. The distribution function associated with the monic reciprocal Mahler's measure encodes statistical information about the range of values of Mahler's measure takes on the reciprocal subspace. We take the Mellin transform of this distribution function to encode this statistical information into a function analytic in a half-plane. This analytic function extends to a rational function with simple poles at small integers and a zero of high multiplicity at zero. From this rational function we explicitly compute the distribution function, as well as the volume of the star body (non-convex unit ball) associated with the reciprocal Mahler's measure. In particular the distribution function is a reciprocal Laurent polynomial, and the volume of the reciprocal star body is a power of pi times a rational number.

After proving his revolutionary hyperbolization theorem for Haken 3-manifolds in the late 1970s, W. Thurston proposed a conjectural classification and parameterization for infinite volume hyperbolic 3-manifolds with finitely generated fundamental group. His conjectural picture, which has topological and geometric components, predicts that such a hyperbolic 3-manifold is determined by simple topological, combinatorial, and conformal data. I will discuss recent progress on the classification problem from various points of view, including topological, geometric, and dynamical perspectives. The speaker is an Assistant Professor candidate