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).

The "Robin problem", also known as the "third boundary problem", is a model for the concentration of a substance in a domain delimited by a semi-permeable membrane. I will address the problem of the geometric characterization of domains where the concentration of the substance is bounded below by a non-zero constant.

This is a doctoral dissertation defense.

We will discuss some properties of the eigenvalues of the Hessian of the distance function to a manifold . This was motivated by the implementation of the level set method for motion by mean curvature in codimension greater then one.

The problem is to understand the sequence of (random) probability measures, on a space of density functions, which correspond to Bayesian posterior distributions. Samples are assumed to come from some fixed but unknown (the true) density function and the posterior is derived via the likelihood function and prior distribution. As the sample size grows to infinity it is required that the posterior accumulates in suitable neighbourhoods of the true density function. Sufficient conditions for this to happen in $L_1$ neighbourhoods are provided.

I will discuss the twin prime problem at a really informal level. I promise that everyone who comes will understand everything I say. Unsurprisingly, this talk will be devoid of proofs. Maybe you can be the first to supply one!

We will construct $\overline{M}_{0,n}(\mathbb{P}^r,d)$ as a GIT quotient of the graph space $\overline{M}_{0,n}(\mathbb{P^r} \times \mathbb{P}^1.(d,1))$. As a consequence we will describe a different construction for $n=0$, analogous to the Fulton MacPherson blow-up $\mathbb{P}^1[n] \to (\mathbb{P}^1)^n$.

Consider a simply connected complete smooth Riemannian surface. Hadamard's famous theorem states that if we inject photonic gas (=light) into one source point at a constant rate, then no two photons will ever cross paths, and the wave front boundary will be a smooth curve without cusps, and the evolution goes on forever. Here, we consider the analogous situation with injection of two-dimensional viscous fluid instead of photonic gas. Again, the boundary of the injected fluid domain is a single smooth curve without cusps, and the evolution goes on forever. However, so far, we only know this is true under the additional assumption that the surface is real-analytically smooth. The injection domains have a mean-value property. This is joint work with S. Shimorin.

I will discuss some results joint with Chris Leininger on separable subgroups of the mapping class groups.

This is a nice little Homology result for flows. I'll give a proof and some examples.

thesis defense

Space-time localization of a class of Constantin-Fefferman-type geometric criteria for preventing singularity formation in the 3D Navier-Stokes equations will be discussed. The localization is expected to hold from the point of view of fluid response--on the other hand, the velocity field is recovered from the vorticity field by a non-local formula, and that springs a number of technical difficulties. An additional, and perhaps a bit esoteric, motivation for localization here is that the existing geometric scenarios for avoiding finite time blow-up are somewhat complimentary--in particular, some are anisotropic, and some are isotropic.

The p-adic numbers are usually defined analytically, as a completion of the rationals. They can also be constructed using the language of categories and inverse limits. We will discuss both approaches and use this as motivation to learn about these “categories” that keep coming up in class.

The is a Ph. D qualifying exam

I will discuss Witten's proof of the positive energy theorem of general relativity, involving Dirac operators and spinors. Most of the talk is based on Parker and Taubes' rigorous account. Emphasis will be on the geometric structures involved. (If you liked Dr. Freed's course on the index theorem last spring, you will love this.)

Obstacle problems consist of studying the least supersolution of some equation which stays above a given function. We study the corresponding obstacle problem for the fractional laplacian and achieve regularity results for the solution that are almost optimal. This problem has applications to option pricing in mathematical finance. This is a thesis defense.

The inner ear (cochlea) can detect pressure fluctuations that are 1/2,000,000,000 that of atmospheric pressure. How does one model such a sensitive organ? We will first look at the physiology of the ear and discuss the current models, most of which are linear. We then introduce a nonlinear and nonlocal treatment of the cochlea. There will be simulations, a discussion of numerical methods, and a plethora of results to discuss. This is a thesis defense.

This is work done in a VIGRE research project.

The quot scheme parameterizes quotient sheaves of the rank n free sheaf over P^1 of given rank k and degree d. This smooth variety is a compactification of the space of rational curves of degree d in the Grassmannain of rank k quotients of C^n. The ranks of its homology groups were determined by Stroome, who used the action of a natural torus on the quot scheme. In joint work with Linda Chen and Tom Braden, we give a presentation for the equivariant Chow ring of the quot scheme. While we use the formalism of Goresky, Kottwicz, and Macpherson, this formalism does not strictly apply, as the 1-dimensional torus orbits on the quot scheme are not isolated. In this talk, I will describe some of the geometry of the quot scheme and this torus action, and then explain how to extend the formalism of Goresky-Kottwicz-MacPherson to obtain a presentation of the equivariant Chow ring of the quot scheme.

Let K be an imaginary quadratic field. Modular forms for K are related to the cohomology of arithmetic 3-manifolds. This absence of algebraic geometry makes calculating the dimensions of such cohomology groups difficult. Using Galois representations, we give an approach to calculate such spaces in certain explicit towers of congruence subgroups. (Joint work with Nathan Dunfield)

A crossed product of an algebra with a group of automorphisms encodes the group action in a larger algebra, and deformations of these algebras have been of much recent interest. For example, Drinfel'd's graded Hecke algebras are deformations of crossed products of polynomial rings with finite groups. These have been studied for particular types of groups by many people, for example Lusztig (real reflection groups), Ram-Shepler (complex reflection groups), and Etingof-Ginzburg (symplectic reflection groups). We will discuss some of these examples and generalizations.

The market of Derivatives was created thirty years ago when Black and Scholes proposed the risk replication approach. Probability, Analysis and Mathematical Physics contributed to its further development. A new academic field, Mathematical Finance, was born. We will go through its major milestones and we will discuss some of the current challenges in global markets beyond Derivatives.

We will discuss the "Nearest Point Map". First, we will build an intuiton of all the necessary conditions in order to guarantee the existence and uniqueness of such map. Then we will talk about some properties of the function, according to some characteristics of the space and the set itself. We finish with a more general notion: retraction.

I will explain how I solved a 20 year old problem in computational complexity using elementary differential and algebraic geometry. The problem in question was to determine if the best known algorithm for multiplying two by two matrices is the best possible, even after allowing for a small error. I will then explain a more general program for studying the asymptotic exponent for matrix multiplication - the number x such that asymptotically one can multiply n by n matrices using n^x multiplications. (Using naive row-column multiplication technique we know x is at most 3 and much work over the years has brought it close to 2.) This program involves studying higher secant varieties of compact Hermitian symmetric spaces, a subject of interest in its own right. Most of the talk will be elementary, acessible to anyone who knows how to multiply matrices.

Many people are familiar with the Fourier Transform over the real numbers. I will talk about how the Fourier Transform can be applied to finite abelian groups and discuss some of the results that follows from this. I will also talk about the Reed-Muller codes and their connection to the Fourier Transform. Snacks by Silvia.

I will discuss the problem of propagation of bounds for solutions of the nonlinear Boltzmann equation which have one-dimensional (plane-wave) symmetry. Such bounds are sufficient to guarantee uniqueness and propagation of smoothness and to obtain classical solutions if the initial data are smooth. The results are based on a certain regularization property of the averaged "gain" term and on integral inequalities techniques that allow us to restrict the blowup behavior of the pointwise bounds. I will present two approaches: one for large data and one for small data, and discuss their advantages and limitations.

This is a qualifying exam

The talk will describe recent results by Y. Pinchover (Technion, Israel) and the speaker. Liouville properties of elliptic equations on non-compact Riemannian manifolds have been studied extensively for the past 30 years, since the work done by S. T. Yau. Here by the Liouville property we mean finite dimensionality of the space of solutions of a given polynomial growth (in particular, of the space of bounded solutions). This is a non-generic property in the sense that small variations in the operator's coefficients destroy it, so the study must be operator specific. Major results have been achieved in work by T. Colding and W. Minicozzi, as well as by P. Li with co-authors. A parallel activity has existed in complex analysis, where one is interested in finite dimensionality of the spaces of holomorphic functions of a polynomial growth with respect to a suitable Riemannian metric. An interesting case was discovered more than a decade ago by M. Avellaneda and F. Lin and by J. Moser and M. Struwe, who showed that for divergent type $\bf periodic$ elliptic second order equations in $\Bbb R^n$ the Liouville property holds, and the corresponding dimensions are the same as for the Laplace equation. We will present the recent results that provide necessary and sufficient conditions for the Liouville theorem to hold for periodic elliptic equations (including overdeterminate systems), as well as explicit formulas for the dimensions of the spaces of solutions. The answers are given in terms of the notions borrowed from the solid state physics: dispersion relations and Fermi surfaces. New results are also obtained concerning Liouville theorems for holomorphic functions on abelian coverings of compact complex manifolds and for solutions of periodic equations on abelian coverings of finite combinatorial and quantum graphs.

Dilute force-free systems of dissipatively colliding particles - Granular Gases - reveal interesting phenomena such as non-Maxwellian velocity distribution, anomalous diffusion, self-organized vortexes and, at late stages of their evolution, density inhomogeneities and clusters. Similar to molecular gases, Granular Gases may be described by the concepts of classical Statistical Mechanics, such as temperature, velocity distribution function etc. Once initialized with a certain velocity distribution, during their evolution Granular Gases cool down due to inelastic collisions of their particles, characterized by the coefficient of restitution $\varepsilon$. We consider the first step of the evolution of a Granular Gas, where no spatial structures have emerged yet, starting from an initially homogeneous distribution. For simplicity of the mathematical analysis, in most studies the coefficient of restitution is assumed to be constant. It is known, however, that this assumption contradicts experiments and even a dimension analysis. Instead, for realistic materials it is a function on the impact velocity, $\varepsilon=\varepsilon(g)$ It is shown that the model assumptions $\varepsilon=$const and $\varepsilon=\varepsilon(g)$ (viscoelastic particles) lead to qualitatively different results. Whereas the velocity distribution function for the case $\varepsilon=$const. has a simple scaling form, the distribution function of gases of viscoelastic particles depends explicitely on time. This property allows to assign a Granular Gas an age. Moreover, in the long-time behavior, for viscoelastic particles the formation of clusters is suppressed or may occur only as a transient phenomenon. Ref: N.V. Brilliantov and T. P\"oschel: "Kinetic Theory of Granular Gases", Oxford University Press (2004)