I will talk about loop groups and soliton equations. This is a Ph.D. candidacy exam.

I will talk about loop groups and soliton equations. This is a Ph.D. candidacy exam.

Let F_q be a finite field with q elements, F_qbar an algebraic closure of F_q, and A^n an n-dimensional affine space over F_qbar. Let C be an affine absolutely irreducible curve in A^n. We interpret the points of C over F_q as points in the cube [-1,1]^&ob;n-1&cb;. The main result of this paper is an asymptotic formula for the distribution of points of C in [-1,1]^&ob;n-1&cb; provided the characteristic p of F_q is large, while n, log_p(q) are fixed, and the degree of C is bounded. When p = q, this becomes a recent result of Cobeli and Zaharescu.

(Speaker is an assistant professor candidate.) I will describe a program, joint with D. Gaitsgory (U. of Chicago), devoted to the topology of the loop spaces of spherical varieties. Spherical varieties form a class of varieties - including symmetric varieties, flag varieties, and toric varieties - which have many beautiful properties. A consequence of our work is a new characterization of spherical varieties in terms of their loop spaces. Even when a spherical variety is smooth, natural stratifications of its loop space have singularities. Our work describes these singularities using perverse sheaves - the geometric objects on a singular space which generalize the notion of a flat vector bundle on a manifold. Given a spherical variety X, we describe perverse sheaves on its loop space in terms of representations of a complex group H. The structure of H governs much of the geometry of X. The association of H to the space X generalizes Langlands' association of a dual group to a complex group G.

I will discuss how to show existence of a minimizer of an energy functional associated with a drop of fluid on a membrane.

The agent controls a state process by controling a process and receives a compensation $C_T$ from the principal at time . Given $C_T$ which depends on (and thus on implicitly or explicitly), the agent problem is to find optimal so as to maximize the agent's utility. The principal's problem is to find the optimal $C_T$ to maximize the principal's utility. We use stochastic maximum principle to solve this "two step" optimization problem. Under certain technical conditions, we find that the optimal contract $C_T^*$ and the optimal control $u^*$ can be characterized by the solution to a Forward-Backward Stochastic Differential Equation. http://www.ma.utexas.edu/Seminars/MathFin/

In an experiment conducted by biologists from the University of Texas, unknotted circular strands of DNA were put into a solution containing specific proteins that created the transpososome complex. The enzyme CRE was then introduced, which cut the DNA at certain sites and reattached the strands in a different manner. The resulting product as the experiment was repeated many times over was a set of knotted products – the trefoil, the Hopf link, and others. From these results, biologists and mathematicians were able to infer the manner in which the DNA is tangled within the transpososome, and uncovered some other interesting properties. More specifically, it was discovered that the DNA within the transpososome was in the form of a 3-string tangle, and that there existed a unique solution with only 6 crossings. Furthermore, Darcy, Luecke, and Vasquez have shown that this is the only solution with 9 or less crossings, and that after free isotopy it is the only solution with 7 or less crossing. We are currently in the process of bringing those numbers up, and proving the same property for 10 crossings and 8 up to free isotopy.

I will describe how a Morse-Bott function on a manifold gives rise to a differential complex isomorphic to the de Rham complex. (This is a candidacy exam.)

The pressure term has always created difficulties in treating the Navier-Stokes equations of incompressible flow, reflected in the lack of a useful evolution equation or boundary conditions to determine it. In joint work with Jian-Guo Liu and Jie Liu, we show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity. As a consequence, in a general domain with no-slip boundary conditions, we can treat the Navier-Stokes equations as a perturbed vector diffusion equation instead of as a perturbed Stokes system. We illustrate the advantages of this view by providing simple proofs of (i) the stability of a time-differencing scheme that is implicit only in viscosity and explicit in both pressure and convection terms, requiring no solutions of stationary Stokes systems, and (ii) existence and uniqueness of strong solutions based on the difference scheme.

http://www.ma.utexas.edu/Seminars/MathFin/

http://www.ma.utexas.edu/Seminars/MathFin/

Lyapunov exponents provide a measure of the long-term geometric mean rate of expansion of a vector under the iteration of a measure-preserving diffeomorphism of a compact Riemannian manifold. We will discuss how this interpretation comes about, and how Oseledec's multiplicative ergodic theorem ensures the existence (almost everywhere) of Lyapunov exponents.

http://www.ma.utexas.edu/Seminars/MathFin/

The speaker is an Assistant Professor candidate

The speaker is an Assistant Professor candidate

We will create the schedule of talks for this semester.

A partition of a number is a representation of this number as the sum of positive integers. For example, the five partitions of 4 are 4, 3+1, 2+2, 2+1+1 and 1+1+1+1. Through the utilization of generating functions, partition theory can be used to prove complex series-product identities. This talk will consist of a short introduction to partition theory and presentations of pretty combinatorial proofs of two ugly series-product identities (including Fabian Franklin's remarkable proof of Euler's pentagonal number theorem).

This paper estimates recent default risk premia for U.S. corporate debt, based on a close relationship between default probabilities, as estimated by Moody's KMV EDFs, and default swap (CDS) market rates. The default-swap data, obtained through CIBC from 27 banks and specialty dealers, allow us to establish a strong link between actual and risk-neutral default probabilities for the 69 firms in the three sectors that we analyze: broadcasting and entertainment, healthcare, and oil and gas. We find dramatic variation over time in risk premia, from peaks in the third quarter of 2002, dropping by roughly 50% to late 2003. This is joint work with Rohan Douglas, Darrell Duffie, Mark Ferguson, and David Schranz. http://www.ma.utexas.edu/Seminars/MathFin/

In the 1920’s Franel and Landau observed that the Riemann Hypothesis for $\zeta (s)$ is equivalent to a simple statement about the distribution of the Farey fractions (fractions in lowest terms in [0,1]). Davenport proposed that an analogous result should hold for $L(s,\chi )$, and Huxley demonstrated in a 1971 paper that this is indeed the case. In this talk I will begin by discussing Huxley’s theorem and explain how it is a generalization of the results of Franel and Landau. In the case of an L-function it turns out that the Riemann hypothesis is equivalent to a statement about the distribution of a subset of the Farey fractions with congruence restrictions on the denominators. Subsets of this kind do not share the basic properties that make the ordinary Farey series so predictable. However in recent years some progress has been made in their study and much of the talk will be directed toward discussing some of these results. This is a self-contained talk (not dependent on having attended Part 1).

I will outline some of the geometric and topological tools used obtain lower volume bounds for hyperbolic 3-manifolds with boundary. I'll focus on a few specific examples which should elucidate the origin of these lower bounds. The talk will be aimed at a general mathematical audience. The speaker is an Assistant Professor candidate.

This is part 1 of a series of talks on Raphael Danchin's 1997 paper, "Poches de Tourbillon Visqueuses." This paper addresses the vanishing viscosity limit of the Navier-Stokes equations in two dimensions for smooth vortex patches.

The recent proof of the Waldhausen conjecture has shed some light on addressing the problem of how many different Heegaard splittings a 3-manifold contains. In particular, if a 3-manifold contains no incompressible tori, there are only a finite number of Heegaard splittings of a given genus. Motivated by a desire to understand the converse of this statement, i.e. when a toroidal 3-manifold has an inifinte number of Heegaard splittings, we will discuss the various ways a Heegaard splitting and an incompressible torus are related in a 3-manifold.

(candidacy talk) The Weil pairing was first used as a cryptanalytic tool in 1993 when the MOV reduction attack used the pairing to transport the discrete log problem on supersingular elliptic curves to the discrete log problem on a multiplicative group, where there are more efficient algorithms for solving it. Since supersingular curves are susceptible to this attack, they were considered "weak" and avoided. However, in recent years, the pairings on supersingular curves have been used for constructive cryptographic purposes, such as identity based encryption, short signatures, and one-way tripartite key exchange. I will define the Weil and Tate pairings, discuss their properties, and look at these recent cryptographic applications that are attracting much attention.

Public key cryptography is used billions of times daily, most commonly in the form of transactions where parties exchange information over the internet. This talk will focus on the most well known public key cryptosystem, RSA, developed by Rivest, Shamir, and Adleman as well as the associated problems of factoring large composite numbers and authenticity. This talk is "diet" because I will not talk about elliptic curves and other advanced topics usually associated with the subject.

When using the Black-Scholes hedging strategy under stochastic volatility, what level of volatility should be used? Can this strategy be corrected to take into account volatility fluctuations? We answer these questions within a class of models with volatility running on fast and slow time scales. After introducing the notion of effective volatility, we compute the corrected hedging ratios, and we show how the parameters needed in the hedging strategy can be efficiently calibrated from the term structure of implied volatilities. Finally we will discuss an application to defaultable bonds. http://www.ma.utexas.edu/Seminars/MathFin/

I will describe some recent results on the analysis and computation of ideal knot shapes, i.e. the shapes that allow a given knot to be tied in the shortest possible piece of rope of prescribed diameter. The analysis pertains to other optimal packing problems involving thick tubes. The computations exploit an unusual spline for Hermite interpolation of space curves called biarcs, which are particularly well adapted to the computation of thickness.

Generalizations of the continued fraction algorithm, to simultaneously approximate 2 (or more) numbers were given by Euler 1849, Jacobi 1868, elaborated by Perron 1907 later by many others. Others including Lagarius have tried to make a more systematic study of such algorithms, especially almost everywhere convergence results. Closer to our work is the paper of Brentjes, 'A two-dimensional continued fraction algorithm', Crelle, 1981. With a geometric algorithm, Brentjes finds all the 'best approximants' in a lattice to a line in the positive part of $R^3,$ via Euclidian geometry of their projections onto a plane. Our approach overlaps with Brentjes, and was motivated by the Theorem: given a "Pisot" matrix there is a integral matrix with determinant $\pm 1$ such that $B^&ob;-1&cb;AB$ has all entries Here "Pisot" means the entries are integral, $detA=1,$ one eigen value is $\ge 0,$ and the other 2 are inside the unit circle.

In this talk I will show how one can attach a zeta function to a positive quadratic form over a number field K. I will explain how to prove a functional equation for this zeta function in the case K is imaginary quadratic, and indicate how this can be used to compute the scattering matrix of a certain Eisenstein series.

This is part 2 of a series of talks on Raphael Danchin's 1997 paper, "Poches de Tourbillon Visqueuses." This paper addresses the vanishing viscosity limit of the Navier-Stokes equations in two dimensions for smooth vortex patches.

I will construct an example of a hyperbolic rational homology sphere containing an embedded totally geodesic surface.

We will discuss the concept of "minimal discrete Riesz s-energy" of a finite set of points on a sphere (or some other manifold) in a Euclidean space. An important problem is to discover the configurations of the point sets where the minimal energy is attained. In particular, we will discuss whether or not those points are well-separated.

This is the first talk in a series aimed at explaining Mirzakhani's proof of the Kontsevich-Witten Conjecture on intersection numbers of tautological classes on moduli spaces of pointed curves. In this first talk we will explain the general Virasoro Conjecture (of which the Kontsevich-Witten Conjecture is a particular case) and then develop the Teichmueller theory backround required in the proof.

The irrationality of zeta(3) was first proved by Apery in 1978; shortly thereafter, Beukers came up with a much simplified proof. I will describe Beukers' proof, and then discuss a few related methods for getting good irrationality measures, in particular ideas of Hata and Rhin & Viola. If time permits, I will discuss some ideas for extending the proof to other odd zeta values, the irrationality of which is an open problem. (candidacy talk)

I will describe recent progress on the multi-scale modelling of the sequence-dependent mechanical properties of DNA. It is generally believed that the mechanics of DNA on the scale of a few hundred base pairs is highly pertinent to its biological function. Various experimental techniques can access information at this scale, but for simulation with full atomistic Molecular Dynamics such length scales are only just accessible, and even then only for very short lengths of time and at a high computational cost. Thus the objective is to pass constitutive information from fine grain descriptions to coarser grained models that are computationally more efficient and yet retain sufficient detail to be able to model the experimental data.

We can talk about two kinds of stabilities of the Ricci flow at Ricci flat metrics. One of them is a linear stability, defined with respect to Perelman's functional $\mathcal{F}$. The other one is a dynamical stability and it refers to a convergence of a Ricci flow starting at any metric in a neighbourhood of a considered Ricci flat metric. We will show that dynamical stability implies linear stability. We also show that a linear stability together with the integrability assumption imply dynamical stability. As a corollary we get a stability result for $K3$ surfaces part of which has been done by Guenther, Knopf and Isenberg.

The linearization coefficients for a polynomial basis $\{P_n\}$ are the coefficients in the expansion $P_n P_k = \sum c_{n,k,l} P_l$. Surprisingly, these coefficients turn out to be positive integers for a number of classical families. For the Hermite polynomials, they can be found explicitly using probability theory. For other families, such as the Chebyshev polynomials of the 2nd kind, one can instead use a different ``free'' probability theory. In this talk, we will outline the proof of this result, introducing all the necessary ingredients along the way. Note: The speaker is an Assistant Professor candidate.

The Kauffman-Harary Conjecture: if K is an alternating, reduced knot or link diagram with prime determinant p and crossing number c(K), then every p-coloring of K uses exactly c(K) distinct colors. I will introduce some knot theory concepts and present a proof of this conjecture for the case of rational knot diagrams.

In plain vanilla Hodge theory, one defines the cohomology groups of a Kahler manifold with coefficients in C from three different perspectives (Betti, DeRham, and Dolbeault), and proves that they are all isomorphic. The idea of nonabelian Hodge theory is to replace the additive group C with the nonabelian group GL(n,C); you'll find that what you get is cooler than rocky road. This talk will be all motivation and no proofs. In particular, I'll spend a good chunk of time reviewing ordinary Hodge theory before moving on to the nonabelian case.

Many widely used statistical models for discrete data are algebraic varieties, that is, solution sets to systems of polynomial equations. We discuss this algebraic representation and its implications for statistical problems such as maximum likelihood estimation. Special emphasis is placed on models which are used in computational biology.

We will present some basic concepts in diophantine approximation in projective space over a local field. In particular, we will introduce the projective metric, explain how to do linear algebra in this setting, and describe a projective analogue of Dirichlet's Theorem for diophantine approximation. The proof in the archimedean case is a consequence of Vaaler's generalization of Dirichlet's Theorem to compact abelian groups endowed with a norm. Mahler measure will not make an appearance.

This is part 3 of a series of talks on Raphael Danchin's 1997 paper, "Poches de Tourbillon Visqueuses." This paper addresses the vanishing viscosity limit of the Navier-Stokes equations in two dimensions for smooth vortex patches.

Abstract: We investigate the homogenization of a free boundary problem describing the combustion of a premixed gas. We study the existence of Travelling Waves-like solutions describing fronts propagation in periodic medium, and show that the homogenized speed of propagation (and effective combustion rate) depends on the extremal properties of the medium, and not on its average properties.

I will discuss various aspects of the objects mentioned in the title.

The Waring-Goldbach problem studies the additive representations of positive integers as sums of a fixed number of k-th powers of primes. In this talk I will try to describe the current state of affairs in the area by presenting a mixture of classical results and some recent developments.

The speaker is an Assistant Professor candidate. A quiver variety is a general type of degeneracy locus associated to a quiver of vector bundles and bundle maps over a variety. Examples include Schubert varieties in flag varieties and determinantal varieties, and the study of formulas for quiver varieties tends to produce very interesting combinatorics. In work with Fulton, I have proved a formula for the cohomology class of a quiver variety when the underlying quiver is equioriented of type A, and I have later generalized this to a formula for the Grothendieck class of the quiver variety, i.e. the class of its structure sheaf in the Grothendieck ring of the embedding variety. These formulas are stated in terms of integers called quiver coefficients. Knutson, Miller, and Shimozono have shown that the coefficients in the cohomology formula are non-negative. I will speak about a proof that the more general quiver coefficients in the K-theory formula have signs that alternate with codimension. I will also explain a cohomology formula for non-equioriented quiver varieties, which I have recently proved with R. Rimanyi.

Besides the fact that the security of much encrypted data today lies in its difficulty, the problem of 'quickly' factoring composite numbers has generated some interesting applications of some more advanced mathematics. We will present an easy introduction to Lenstra's elliptic curve method of factorization, starting from the more elementary Pollard p-1 method on which it is based. (Baked Goods by Pippa)

We present various sharp norm estimates for some of the most central objects in harmonic analysis - Calderon Zygmund operators. We discuss their applications, for example to quasiregular maps in the plane as well as related subjects, such as commutators and elliptic PDE. Analysis is the study of complicated objects through its building blocks and its core lies in the creation of new techniques to obtain the desired estimates. We give a non-technical overview of the new methods used. The speaker is an Assistant Professor candidate.

A firm issues a convertible bond. At each subsequent time, the bondholder must decide whether to keep the bond, thereby collecting coupons, or to convert it to stock. The bondholder wishes to choose a conversion strategy to maximize the bond value. Subject to some restrictions, a convertible bond can be called by the issuing firm, which presumably acts to maximize equity value and thus to minimize the bond value. This creates a two-person game, and we model the bond price as the value of this game. We show, however, that under our standing assumption (dividends are paid at a lower rate than the money market rate) this game reduces to one of two optimal stopping problems, and the relevant stopping problem can be determined a priori, i.e., without first solving the convertible bond pricing problem. Because of dividend payments, the partial differential equation describing the pricing function becomes nonlinear. This means that our analysis involves a fixed point problem. We also prove that for large time to maturity the value of the convertible bond approaches the value of the perpetual convertible bond. The presentation is based on joint work with Steven E. Shreve. http://www.ma.utexas.edu/Seminars/MathFin/

...and he didn't even know it. This talk should be accessible to the speaker.

Each year the MASS program brings together and supports a select group of undergraduates from around the country. They receive intensive training in mathematics which prepares them for advanced research and graduate school. Two of the principal architects of this program will give an informative presentation and be on hand to answer your questions. Further information on the 2005 MASS program is available from the website http://www.math.psu.edu/mass

I describe geometric splittings of cusped hyperbolic 3-manifolds associated to ideal points of the character variety using elementary and (almost) computational methods. Character varieties allow the study of representations into SL(2,C) and PSL(2,C) up to conjugacy, and provide a link between the topology and the geometry of a 3-manifold. I describe this link for cusped hyperbolic 3-manifolds with suitable ideal triangulations. The techniques used are geometric and combinatorial; normal surface theory, standard spines, angle structures and hyperbolic geometry are involved.

Let's have a look at a few of the derivations from physics and engineering that give us some of the partial differential equations that we care so much about.

Due to Cohen's structure theorems, the structure of complete local rings is well-known. The properties of the relationship between a local ring and its completion, however, are not as well understood. We will introduce the concept of completions of local rings and discuss some suprising recent discoveries about the behavior of prime ideals under this operation.

An extended abstract (pdf file) is available at the link http://www.ma.utexas.edu/~og/colloq/.

Consider a packing of spheres in R^3 at close to maximum density. Do such packings tend to be perturbations of face-centered cubic (fcc) lattices, perturbations of hexagonal-close-packed (hcp) lattices, perturbations of other lattices, or something altogether different? I'll present the current evidence, including recent computer experiments by Hans Koch, Charles Radin and myself.

This will be the first of a series of talks on paracomposition, a technique introduced by Alinhac in the mid-1980s. The idea is that one can write the composition of two sufficiently nice functions f and g as the sum of three terms. The first term bears the singularities of f, while the second bears the singularities of g. Finally, one can, in some sense, sum up the regularities of f and g to get that of the remainder term.

Abstract: The model under study is a coupled system made of two parabolic equations and an ODE (or a transport equation) describing the propagation of a flame in a spray of liquid droplets. The physics is the following: the droplets evaporate into a gaseous reactant, which itself is able to burn. When the droplets have a nonhomogeneous spatial distribution in the unburnt zone, the propagation should be looked for under the form of pulsating waves - i.e. solutions that are time-periodic in a Galilean reference frame. We will in this talk discuss some existence results for pulsating waves, according to the properties of the properties of the evaporation or chemical kinetics. Joint work with P. Constantin, K. Domelevo and L. Ryzhik.

We consider tiling spaces generated by geometric substitutions. Examples to be described include the Penrose and pinwheel tilings, as well as the dual tilings constructed from symbolic substitutions of unimodular Pisot type. With Radin and Sadun, we proved an analog, valid for most substitution tiling spaces, of the Curtis-Lyndon-Hedlund theorem in symbolic dynamics. Under some additional conditions this implies a kind of rigidity for these spaces. I'll outline a proof of this and some related results.

We will present some basic concepts in diophantine approximation in projective space over a local field. In particular, we will introduce the projective metric, explain how to do linear algebra in this setting, and describe a projective analogue of Dirichlet's Theorem for diophantine approximation. The proof in the archemedian case is a consequence of Vaaler's generalization of Dirichlet's Theorem to compact abelian groups endowed with a norm. (this is a candidacy talk.)

We determine the optimal investment strategy of an individual who targets a given rate of consumption and who seeks to minimize the probability of going bankrupt before she dies, also known as {\it lifetime ruin}. We assume that the financial market is a Black-Scholes market with a single riskless asset with constant return and a single risky asset following geometric Brownian motion. We impose two types of borrowing constraints: First, we do not allow the individual to borrow money to invest in the risky asset nor to sell the risky asset short. Second, we allow the individual to borrow money but only at a rate that is higher than the rate earned on the riskless asset. We solve the two minimization problems via associated differential equations. For this talk, I will assume that the individual consumes at a constant (real) dollar rate and will include a numerical example to illustrate our results. http://www.ma.utexas.edu/Seminars/MathFin/

The discovery of calculus far predates the formal definition of a limit. In the seventeenth century, calculus was done with loosely defined infinitessimal numbers. Using some ideas from set theory, we will construct an extension of the real numbers, known as the hyperreals, in which this notion of infinitessimals makes sense. We will then show how to use this set to reprove some basic facts from calculus without the use of a limit.

Abstract: We study a Hele-Shaw problem with a mushy region obtained as a mesa type limit of one phase Stefan problems in exterior domains. We study the convergence, determine some of the qualitative properties and regularity of the unique limiting solution, and prove regularity of the free boundary of this limit under very general conditions on the initial data. Indeed, our results handle changes in topology and multiple injection slots. This paper is joint work with Marianne Korten and Chuck Moore.

On a symplectic manifold with a circle action, there are formulas giving the value of certain oscillatory integrals in terms of fixed points of the circle action. Atiyah generalizes this result to the loop space of a spin manifold X, and uses this to calculate the index of the Dirac operator on X. I'll try and explain this.

This will be an introductory talk about one of the world's favorite free boundary problems.

It is conjectured that the Riemann-Hilbert monodromy map between certain quiver varieties and character varieties of a punctured Riemann sphere induces a map on cohomology which preserves mixed Hodge structures and induces an isomorphism on the pure parts. Doing arithmetics and representation theory on the varieties I will exhibit some striking evidence in favor of this conjecture, with a more recent technique applying Letellier's calculation of Fourier transforms of invariant functions on a finite Lie algebra. This gives a powerful method to calculate the Poincare polynomial of Nakajima's starshaped quiver varieties and together with the conjecture hints at a deep connection between the character tables of finite Lie groups and algebras.

Suppose s lies in an abelian extension of the rationals. In a recent paper, Amoroso and Dvornicich give an effective lower bound for the Weil height of s. We will use this result and a little Galois Theory to give a lower bound for the height of r-z, where r is any algebraic number and z runs through all roots of unity.

E. Bombieri and J. D. Vaaler obtained two-sided bounds for the Mahler measure of a monic polynomial, expressed via a weighted Vandermonde determinant. Thus one is naturally led to the questions on asymptotics for such determinants, which is directly related to the electrostatic problem of distributing charge on the unit circle in the presence of an external field. Potential theory with external fields found numerous applications in approximation theory, orthogonal polynomials, random matrices, combinatorics, and number theory. In this joint work with J. D. Vaaler, we show that weighted potentials are also useful for the study of heights of some polynomial subspaces.

inpsired by some of the questions at Dave's talk last week (snacks by Sal)

To make mathematical models of problems in finance, usually probability theory provides nice tools. But to solve the problem one will often start by finding an equivalent PDE problem and solve that one instead. We'll look at one of the most classical examples of how this is done.