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.

Given an open smooth variety M there is (conjecturally) a tautological compactification with a nice boundary, the so called log canonical compactification. I'll recall the (entirely elementary) construction, point out examples of known compactifications of moduli spaces that turn out to be of this sort, and then mention an example, moduli of hyperplane arrangments, where the compactification is not know, and indicate why finding it is of compelling interest.

In general, one can associate an Eisenstein series to each cusp of the Hilbert-Blumenthal modular group $PSL_{2}(O_{K})$, where K is any algebraic number field. Using the zero Fourier coefficients of the expansions of these Eisenstein series at the various cusps, one can form the N by N scattering matrix $\Phi(s)$, where N is the order of the ideal class group of K. In the classical case K=Q, this scattering matrix plays a fundamental role in the Lax-Phillips theory of the automorphic wave equation. In addition, the logarithmic derivative of the determinant of $\Phi(s)$ accounts for the contribution of the continuous spectrum to the Selberg trace formula. In this talk, I will prove that this determinant can be given in terms of the zeta function of the Hilbert class field of K, and thus the analytic properties of the Eisenstein series are controlled by number theory.

This will be the second of a series of talks on paracomposition, a technique introduced by Alinhac in the mid-1980s. This week, I hope to present an application to the Euler equation.

I'll attempt to answer the following questions: When are hyperbolic cone deformations useful? Why are they useful? What are they? How do they work? And, Can I use them? (I.e., what's in it for me?)

TBA

This is a two day workshop, please visit the conference website at http://www.ices.utexas.edu/~gamba/workshop05/ for more information.

The ABC Conjecture has been called amazing. In this talk, I will present the conjecture in perhaps several different formulations. I will discuss its implications for number theory and perhaps attempt to prove it in the function field case.

This paper extends the deterministic endogenous R&D growth model to a stochastic endogenous growth model, which is used to study growth stocks. The model provides an understanding of the links between economic growth, monopolistic competition in R&D, and the valuation of growth stocks. With the presence of stochastic shocks, the model leads to a decomposition of the value of growth stocks. The decomposition implies that the value of growth stocks should be very volatile, while the long-run average return is roughly equal to the growth rate of R&D labor. The model also explains an empirical size distribution puzzle observed for the cross-sectional study of growth stocks. http://www.ma.utexas.edu/Seminars/MathFin/

The general construction of symplectic quotients is based on the observation that every coisotropic submanifold of a symplectic manifold is foliated by isotropic leaves and that the quotient, if it exists, is again a symplectic manifold. The Marsden-Weinstein quotient is one important example of a symplectic quotient which arises from a Hamiltonian action of a compact Lie group. I will introduce many of these concepts and prove the existence of the Marsden-Weinstein quotient under the correct assumptions.

This is a two day workshop, please visit the conference website at http://www.ices.utexas.edu/~gamba/workshop05/ for more information.

Much of the data for the ocean is Lagrangian in that it is derived from subsurface floats drifting with the flow. At the same time, many questions about the ocean involve transport and mixing, and therefore concern Lagrangian dynamics. A strategy for model analysis, based on dynamical systems ideas, at both model input and output will be described. This will include Lagrangian data assimilation, optimal float placement design and an approach to model evaluation.

I'll discuss how to use discrete-time games to approximate the motion of a front propagating with a given normal velocity. When the related PDE is of first-order, this approximation is well-known; perhaps less well-known is how to handle a second-order geometric PDE, such as the level-set formulation of mean curvature motion.

A conic bundle is a 3-fold fibred over a surface with general fibre a smooth rational curve. These varieties are important in the classification of 3-folds as described by Mori's minimal model program. A conic bundle corresponds to an algebra over the surface with general fibre 2x2 matrices. We use a version of the MMP for this noncommutative surface to construct simple birational models of conic bundles. The noncommutative singularities which arise have many nice properties, for example, they are simultaneously Clifford algebras and quotients of matrix algebras. This is joint work with Daniel Chan and Colin Ingalls

The Cauchy integral formula and some tricks from Algebra can be used to prove that if A and B are algebraic numbers such that A is not zero or one and B is irrational then A^B is transcendental. We will review a proof and consider some examples.

In the first part of the talk we introduce the Boltzmann equation and discuss its properties. The existence of a solution to the spatially homogeneous Boltzmann equation was usually studied in physically relevant weighted $L_1$ spaces. In contrast, we discuss in the second part of the talk the solvability and uniqueness of the solution to the Boltzmann equation in Sobolev spaces for $s<-3/2$. These results are valid for a wide range of physical interaction models like hard and soft potentials, cutoff and non-cutoff kernels, etc.

The dynamics of fluids is described by complex non-linear differential equations, the Navier-Stokes equations if viscosity is taken into account, Euler's equations if not. We are unable to solve them in the general case, or even with sufficient enough generality to feel that we understand them. One central issue was raised by Leray long ago, namely the existence or not of solutions becoming singular after a finite time starting from smooth intial data with finite energy. I'll present some new ideas on this question, for the case of Euler's inviscid equations. I'll show how the various constraints (conservation of energy and of circulation) limit the possible scenarios of finite time singularity, excluding in particular singularity at a single point.

We will look at examples of Montesinos knots with boundary slopes that are not detected by the character variety.

I'll show how topological conjugacies of tiling spaces need not be sliding block codes, and discuss some of the implications.

Let E be an elliptic curve over a global field K and fix a torsion-free subgroup G of the Mordell-Weil group E(K). For almost all non-archimedean primes v of K, the reduction of E over the residue field k(v) is an elliptic curve, and we write G_v for the image of G in the 'special fiber' E(k(v)). The Lang-Trotter conjecture asserts that there is a finite set of rational primes S and a constant d(E,G,S) such that the set of v for which the index [E(k(v)):G_v] is divisible by some prime not in S has density d(E,G,S). That is, the set of v such that G_v contains the prime-to-S part of E(k(v)) has density 1-d(E,G,S). We'll present joint work with J.F. Voloch in which we prove the conjecture if K is a global field of char p, E/K is an elliptic curve with non-constant j-invariant, and rank(G) >= 6.

A quiver is simply a directed graph. By a representation of a quiver we mean a collection of vector spaces indexed by the nodes of the graph, together with linear maps corresponding to the arrows of the graph. We will introduce the notions of simple and indecomposable representations of a quiver (which may be considered part of the Abelian category structure of quiver representations). Gabriel asked which quivers have "finite representation type", that is, which quivers have finitely many indecomposables, and discovered a beautiful connection to Lie theory: The quivers with this property are precisely those whose underlying undirected graph is the Dynkin diagram of a Lie algebra. Moreover the indecomposable objects themselves are indexed by the roots of the associated Lie algebra. In the first part of the talk we will give an account of this theorem. In the latter part we will discuss connections between the representation theory of the Kronecker quiver - the most manageable quiver which lies outside of Gabriel's theorem - and quantum affine sl_2 (the "quantum loop group" of sl_2) and its finite dimensional representation theory.

One problem of interest to geometric group theorists is the following: given a lattice in a geometry, does the group of automorphisms of the lattice contain a subgroup of finite index generated by reflections in elements of the lattice? I'll discuss some of the ideas, techniques, and pretty pictures that come in handy when one tries to solve this problem. If I'm feeling particularly ambitious, I'll talk about a result by E. B. Vinberg in 1972 that partially determines which integral quadratic forms of signature (n,1) have reflection groups of finite index.

Spring Break- No Seminar

I will discuss the Brauer-Siegel theorem. Assuming there is no Siegel zero, I will explain how zeta functions of positive quadratic forms over number fields can be used to strengthen a certain constant appearing in Siegel's lower bound for the residue of the Dedekind zeta function at . Time permiting, I will discuss Phragmen-Lindeloff type results for growth on vertical lines of these zeta functions.

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.

Discrete structures arise in a vast array of continuous geometric contexts. The aim of this talk is to convey a glimpse of the possibilities, by describing some recent work and open problems in connection with cohomological data on manifolds, properties of convex polytopes, and mixtures of the two.

The definition of viscosity solutions allows non-smooth functions to be solutions of certain fully nonlinear partial differential equations, and it also provides a general setting in which one may consider comparison, uniqueness, and existence theorems. In my talk, I will motivate the definition of viscosity solutions of first order partial differential equations by using the "vanishing viscosity" method to the Hamilton-Jacobi Equations, and sketch the proof of consistency and uniqueness theorems. Then, I will build up the notion for the second order partial differential equations and briefly describe the maximal principle and Perron's method to give comparison, uniqueness, and existence assertions. Some interesting examples will also be shown. This is my option talk in analysis.

One requirement of a polynomial that is to be used as an S-box in an AES-like cryptosystem is that it be APN (almost perfectly non-linear). A polynomial being APN comes down to looking at the factorization of a plane curve and an absolutely irreducible plane curve signifies that the polynomial is not APN. I will discuss which monomials are APN and which are not.

I will introduce the audience to the theory of modular forms. Basic definitions, results and examples will be discussed. (snacks by BK)

Abstract: I will discuss solutions of a general class of fully nonlinear equations in conformal geometry with isolated singularities, in the case of non-negative Ricci curvature. We prove that such solutions either extend to a H\"older continuous function across the singularity, or else have the same singular behavior as the fundamental solution of the conformal Laplacian.

Originally concieved by Kazhdan, Property T is an analytic property of unitary actions of groups on Hilbert spaces with many interesting geometric implications. We will define Property T, look at some consequences, define real, complex, and quaternionic hyperbolic space, and see that Property T shows us a curious difference between quaternionic hyperbolic manifolds and their real and complex hyperbolic cousins. (This is an analysis option talk.)

We will first explain the action of G=Aut(P^1) (which is basically SL_2(C) ) on X= the pointed Linear Sigma Model (which is basically a product of projective spaces). Given something called a linearized ample line bundle on X we can form a nice quotient X/G. We will examine how the quotients are related for different choice of the line bundes, and conclude with a calculation of the (ranks of ) the cohomology rings of X/G. Go Washington!

The isometry group of a closed hyperbolic n-dimensional manifold is finite. We prove that for every n>1 and every finite group G there exists an n-dimensional closed hyperbolic manifold whose isometry group is isomorphic to G. The cases n=2 and n=3 were proved by Greenberg (1974) and by Kojima (1988), resp. Our proof is non constructive; it uses a 'probablistic approach', i.e., counting results from the theory of 'subgroup growth'. The talk is based on a joint paper with M. Belolipetsky with the same title- to appear in Invent. Math.

We'll first review some classical geometry of affine spheres and Tzitzeica equation. After formulating the structure equations as an integrable system associated to twisted loop algebra of sl(3), we were able to find simple rational elements in this loop algebra whose dressing actions give exactly the classical Tzitzeica transformations. (They are then parallel to the example of pseudospheres, sine-Gordon equation and classical Backlund transformation.)

This talk will be highly related to my (approaching) defense. Expect to hear about Mahler measure, polylogarithmic motivic complexes, and stuff like that.

Fewnomial theory starts with the principle that ``simple'' equations should have ``simple'' solutions. The first example, going back to work of Descartes in 1637, is the fact that a univariate real polynomial with exactly m monomial terms has no more than m-1 positive roots. This has since been generalized to arbitrary systems of real equations by Khovanski, and to arbitrary systems of polynomials over p-adic fields by Rojas. (The latter result generalizes earlier work of Lenstra in the univariate case, and makes effective an earlier result of Denef and van den Dries.) We discuss the following developments: (1) Improved upper bounds on the number of connected components of certain real ``fewnomial'' hypersurfaces. (2) Thesholds for the computational hardness of deciding the existence of a real root for an n-variate m-nomial. (3) Applications of arithmetic fewnomial theory to torsion points on curves and computational complexity. Along the way, we'll see circuits (in the sense of combinatorics), A-discriminants, and Baker's Theorem on linear forms in logarithms. This talk is self-contained and I assume very little background, so graduate students are most welcome.

I will describe a model for a chemical reaction (KPP reaction diffusion advection equation) propagating through a cylindrical domain under the influence of a shear flow which oscillates randomly in time. I'll describe some probabilistic techniques for characterizing the asymptotic speed of propagation.

In many diffusive settings, initial disturbances will gradually disappear and all but their crudest features --- such as size and location --- will eventually be forgotten. Quantifying the rate at which this information is lost is sometimes a question of central interest. Joint work with Yong Jung Kim (University of California at Riverside and KAIST) addresses this issue for the fastest conservative nonlinearities in a model problem known as the fast diffusion equation $ u_t = \Delta(u^m), \qquad (n-2)_+/n < m \le n/(n+2), \quad u,t \ge 0, \quad x \in &ob;\bf R&cb;^n, $ which governs the decay of any integrable, compactly supported initial density towards a characteristically spreading self-similar profile. For other values of the parameter , this equation has been used to model heat transport, population spreading, fluid seepage, curvature flows, and avalanches in sandpiles. For the fastest conservative nonlinearities, we develop a potential theoretic comparison technique which establishes the sharp conjectured power law rate of decay $1/t$ uniformly in relative error, and in weaker norms such as $L^1(&ob;\bf R&cb;^n)$.

I am going to talk about triangle groups and their specific property. Next, I am going to show how to apply the property on Dehn surgery.

Consider the Torelli group of a genus g surface, that is, the group of mapping classes which act trivially on the homology of the surface. Denote by K_g the subgroup generated by Dehn twists about separating curves. In joint work with Benson Farb, we show that K_g is not finitely generated.

The law of quadratic reciprocity is a beautiful theorem relating two primes and their symmetry in relation to the squares. In this talk we define a quadratic character for a finite group and prove a law of reciprocity for its value. (snacks by mark)

We propose explicit and more realistic models for the pricing of a firm's risky debt based on information reduction. Via the study of a standard Merton (1974) type structural model for credit risk, we show how structural models are transformed into reduced-form intensity based models when the information set is restricted to that available to market participants. Analytic solutions are provided for the reduced-form intensity process and risky debt prices. This talk is based on joint works with R. Jarrow and Y. Zeng of Cornell University. http://www.ma.utexas.edu/Seminars/MathFin/

Most known smoothable simply connected 4--manifolds admit infinitely many different smooth structures (distinguished, for example, by Seiberg--Witten invariants). There are some 4--manifolds, though, for which the existence of such 'exotic' structures is still open, the most notable examples being the 4--dimensional sphere S^4 and the complex projective plane CP^2. In a recent project with Z. Szabo and J. Park we found constructions of exotic smooth structures on the five- and six-fold blow--up of CP^2. In the lecture we describe the construction of these 4--manifolds and indicate the necessary input from Seiberg--Witten theory for proving their exoticness.

The Bellman equation is a famous and useful type of nonlinear first order PDE. We will show where it comes from in optimal control theory and discuss some of its properties.

This is the first of two talks in conformal geometry. We will explain first the classical Yamabe problem (construct a constant scalar curvature metric in the same conformal class as a given metric), for a compact manifold without boundary. Then we will consider the same problem on a complete, non-compact manifold. PDE's play an important role in both, however, the talk will be Geometry oriented.

An *Elliptic Divisibility Sequence* is a sequence of integers described by the nonlinear recursion $W_{m+n}W_{m-n}=W_{m+1}W_{m-1}W_n^2-W_{n+1}W_{n-1}W_m^2.$ This is the same recurrence that is satisfied by the division polynomials of an elliptic curve, so (almost) equivalently one can attach an elliptic divisibility sequence to a point of infinite order $P\in E(\mathbb{Q})$ on an elliptic curve by writing $x(nP)=A_n^{\phantom2}/D_n^2$ and setting $W_n=\pm D_n$. In this talk I will discuss various properties of elliptic divisibility sequences, including their sign variation over (joint work with N. Stephens), convergence properties of subsequences in , conjectural gcd growth bounds over , and if time permits, results over function fields.

A new family of exact solutions is analyzed, which model 2D circulations of an ideal fluid in a uniformly rotating elliptical tank, under the semi-geostrophic approximation from meteorology and oceanography. The fluid pressure and stream function remain quadratic functions of space at each instant in time, whose fluctuations are described by a single degree of freedom Hamiltonian system depending on two conserved parameters: domain eccentricity and the constant value of potential vorticity. These parameters determine the presence or absence of periodic orbits with arbitrarily long periods, fixed points of the dynamics, and aperiodic homoclinic orbits linking hyperbolic saddle points. The energy relative to these parameters selects the frequency and direction in which isobars nutate or precess, as well as the steady circulation direction of the fluctuating flow. The canonically conjugate variables are the moment of inertia and angle of inclination of an elliptical inverse-potential-vorticity patch evolving in dual coordinates.

An automatic group is a group whose Cayley graph you can "see". We will formalize this notion, give examples and non-examples of automatic groups, and prove that the three-dimensional integral Heisenberg group is not automatic by considering disks in its Cayley graph. The argument will involve one "proof by machines" and at least two proofs by pictures.
You are invited to view the teaser trailer here:
http://math.utexas.edu/~katerman/topology.mov

Bob is going to tell us a bit about the Lorenz Template.

As a generalization of Falting's Theorem (Mordell's Conjecture), Bombieri and Lang conjectured that the rational points on varieties of general type are not Zariski dense. This has deep consequences to the distribution of rational points in families of curves of genus . In this talk, we'll discuss an analogous conjecture concerning integral points on varieties of logarithmic general type; in particular we'll prove a result by Abramovich saying that the number of integral points on the quadratic twists of an elliptic curve is uniformly bounded if one assumes the so-called Lang-Vojta conjecture. (candidacy talk)

We prove a conjecture of Buchstaber and Krichever that their addition formula for theta functions characterizes Jacobians among all principally polarized abelian varieties, and use this to obtain cubic equations for the hyperelliptic locus. We further explain how this is related to Fay's trisecants, the secants of Kummer varieties studied by Gunning, and to the analogue of Castelnuovo-Mumford regularity - the number conditions imposed by sets of points on translates of the 2\Theta linear system on an abelian variety. We then discuss further applications to constructing curves of small class on abelian varieties.

Variational methods are used to describe intresting orbits of a dynamical system that are related with its global behavior. Join us in this adventure into the secrets of chaos and tasty treats (provided by elizabeth).

A whirlwind tour of the mathematics aspects of Mirror Symmetry based mainly on Morrison's survey (alg-geom/9609021). In a vain attempt at being self-contained it is sure to include vague references to correlation functions, pseudoholomorphic curves, Gromov-Witten invariants and moduli spaces. Actual calculations may also be featured.

We construct infinitely many Coxeter groups with finite covolume in hyperbolic n-space for every n<20. For n in this range (except n=16,17), the number of such polyhedra grows at least exponentially with respect to volume.

Recently, I discussed the classification of arithmetic lattices in the special unitary groups SU(n,1). In this talk, I will give a specialized discussion in the setting of SU(2,1), focussing on geometric and algebraic questions for the lattices and their associated complex hyperbolic orbifolds.

This is the continuation of the Gadget talk last week. Here we will consider the curvatures of a manifold, a generalization of the scalar curvature. They are defined as the symmetric functions of the eigenvalues of the Schouten tensor and give rise to interesting problems in conformal geometry. The end of the talk will try to relate to the material from the previous week.

Multiplicative distance functions are measures of complexity of polynomials. Like the prototypical Mahler's measure, multiplicative distance functions are uniquely determined by their action on the roots of polynomials. For each multiplicative distance function and $N > 0$ we will associate two analytic functions $F_N(s)$ and $H_N(s)$ which encode information about the the range of values taken on by polynomials of degree in $\mathbb{R}[x]$ and $\mathbb{C}[x]$. We will show that $H_N(s)$ can be represented as the determinant of a Gram matrix with respect to an inner product determined by the multiplicative distance function. Similarly, we will see that $F_N(s)$ can be represented as the Pfaffian of a Gram matrix with respect to a skew-symmetric inner product determined by the multiplicative distance function. By using orthogonal and skew-orthogonal polynomials we will then show that $F_N(s)$ and $H_N(s)$ have simple product representations (examples will be provided). This talk may be of interest beyond just number theory, since certain multiplicative distance functions arise from potentials in two-dimensional electrostatics. Moreover, in this context, $H_N(s)$ and $F_N(s)$ are essentially the partition function with respect to certain physical systems.

This is part 5 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 central limit theorem has been obtained as a consequence of rapid mixing for certain compact group extensions. However mixing is not a prerequisite for the central limit theorem. We will discuss a simple example where the extension is not mixing but where the CLT for smooth observables still holds. This is joint work with Ian Melbourne (University of Surrey).

The unofficial title is "An algebraic integration for Mahler measure". I am going to discuss how one could possibly obtain/explain some formulas for Mahler measure of multivariable polynomials in terms of values of regulators. This is my thesis defense (please come!). Cookies will be provided.

Usually, bounds toward the Ramanujan conjecture play a crucial role in getting a subconvexity estimate for automorphic -functions. It is believed, however, that the Ramanujan conjecture is irrelevant to the Lindelöf hypothesis. We give an evidence to this, i.e., we get subconvexity which does not rely on bounds toward the Ramanujan conjecture. Let be either a holomorphic cusp form of weight , or a Maass cusp form with Laplace eigenvalue $1/4+k^2$, for . Let be a fixed holomorphic or Maass cusp form. We obtain the following bound for the Rankin-Selberg -function $L(s,f\otimes g)$ in the aspect: $ L(1/2+it, f\otimes g) \ll k^{1 - 1/(8+4\theta) + \varepsilon}$ where is from bounds toward the generalized Ramanujan conjecture. Now a trivial still yields a subconvexity bound.
This is a joint work with Yuk-Kam Lau and Yangbo Ye.

We'll discuss Van der Waerden's Theorem and present a proof. If time permits, we'll expand the discussion into the theorem's application in Ramsey theory.

We study an optimal investment problem under incomplete information for an investor with constant relative risk aversion. We assume that the investor can only observe asset prices, but not the instantaneous returns. Furthermore, we assume that the instantaneous returns follow an Ornstein--Uhlenbeck process, and that their initial distribution is Gaussian. We analytically solve the Bellman equation for this problem, and identify the optimal investment strategy under incomplete information. We explore the relationship between the value function under partial observation and the value function under full observation, and derive a formula for the economic value of information. Furthermore, we discuss how the optimal strategy under partial observation can be computed from the optimal strategy for an investor with full observation. Explicit solutions are presented in a model with only one risky asset. http://www.ma.utexas.edu/Seminars/MathFin/

Heath-Jarrow-Morton Model introduced in year 1992 provides a general theory and a unifying framework for valuing contingent claims under a stochastic term structure of interest rates. In my talk, I will present the one-facor and more general multi-factor model and then demonstrate how to value contingent claims in that economy which do not explicitly depend on the market prices of risk. Finally, I will compare the equilibrium pricing with arbitrage pricing methodology. This is my option talk in Applied Math.

In this talk, we investigate some group theoretic properties of the fundamental groups of closed Kahler manifolds.

In the last twenty years, stock options have become an important device for firms to compensate and incentivize their employees. Motivated by recent recommendations by both European and American accounting standards boards that these options be expensed in firms' statements, we provide an analytical and flexible framework for their evaluation. Instead of looking at single options in isolation, we consider the the flow of options an employee can expect to receive throughout his/her employment. This includes features such as vesting, possibility of reset if the firm's stock value diminishes, suboptimal exercise, reload potential and trading and hedging restrictions on the employee. http://www.ma.utexas.edu/Seminars/MathFin/

We show that every finite-volume, 1-cusp hyperbolic 3-manifold admits an essential, quasi-Fuchsian immersion of a closed hyperbolic surface. (Joint with Xingru Zhang)

We begin with the description of the Fourier Restriction Problem: given a hypersurface in for which values of $p, q, f \in L^p(\mathbb R^n)$ has a well defined restriction of its Fourier transform to and such that $\hat f \in L^q(S)$. We discuss some of the results, then we move to Strichartz estimates explaining the close connection between the two problems and the wave equation and (if time permits) any dispersive PDE.

Given a field extension L over K, a scheme over L is defined over K if it is a fibered product of a scheme over K and Spec L. I'll go through some basic definitions and examples.

In this talk we'll go over some of the main results in Diophantine approximation and how one can use it to prove some facts about integral points on affine varieties. I'll discuss briefly a new approach to Siegel's theorem, due to Pietro Corvaja and Umberto Zannier, but in the special case of the projective line minus 3 points.

Serving as a warm-up for my candidacy exam next week, this talk addresses Boyer and Zhang's proof of the Finite Filling Conjecture which determines constraints on the set of slopes on the boundary torus of a hyperbolic knot exterior whose associated Dehn fillings have finite or cyclic fundamental groups.

This is a new trick whose objective is to lower the total energy of a bundle or a sheaf by performing cut and glue operations. I will explain the surgery operation and give applications to the topology of moduli spaces, construction of sheaves on singular varieties and the existence of instantons.

thesis defense

Metaheuristics is a class of optimization methods that eats combinatorally complex problems for breakfast (almost). The traveling salesman problem is the canonical example. An intrepid salesman must find the shortest route to visit N cities and return home; surprisingly difficult for N as small as 20 (even with Google maps). There is no shortage of equally difficult problems involving routing, scheduling, bin packing, etc. I will introduce three 'proven' metaheuristic methods: simulated annealing, genetic algorithms, and tabu search in the context of an easy to understand but difficult to solve problem in geology.

ABSTRACT: I will dicuss the convergence of weak solutions of the Navier-Stokes equations to a solution of the Euler equations in the limit as the viscosity goes to zero. The setting will be the entire plane, or a bounded domain in the plane (with nonstandard boundary conditions for the Navier-Stokes equations), with unbounded initial vorticity. This is a thesis defense.

We present a new approach to solve optimal control problems of the monotone follower type. The key feature of our approach is that it allows to include an arbitrary dynamic fuel constraint. Instead of dynamic programming, we use the convexity of our cost functional to derive a first order characterization of optimal policies based on the Snell envelope of the objective functional’s gradient at the optimum. The optimal control policy is constructed explicitly in terms of the solution to a representation theorem for stochastic processes obtained in Bank and El Karoui (2004). As an illustration, we show how our methodology allows to extend the scope of the explicit solutions obtained for the classical monotone follower problem and for an irreversible investment problem arising in economics. http://www.ma.utexas.edu/Seminars/MathFin/

We study the existence and geometric properties of an optimal configurations to a variational problem with free boundary. More specifically we analyze the nonlinear optimization problem in heat conduction where the quantity to be minimized, the flow of heat, is given by $\int_{\partial D} \Gamma (x,u_\mu) d\sigma, $ where is a fixed domain. A nonconstant temperature distribution is prescribed on $\partial D$ and a volume constraint on the set where the temperature is positive is imposed. Among other regularity properties of an optimal configuration, we prove analyticity of the free boundary up to a small singular set. \par We also establish uniqueness and symmetry results when $\partial D$ has a given symmetry. Full regularity of the free boundary is obtained under these symmetry conditions imposed on the fixed boundary. This is a thesis defense.

In this talk I'd like to introduce the idea of treating schemes as functors from the category of schemes to the category of sets. What information do we gain by doing so? Come to the talk and see. I will also briefly review some basics of categories and schemes (for my own benefit).