A planimeter is an olde tyme device used to measure the areas of planar regions. We will explain how they work and show how one can use a planimeter to prove the isoperimetric inequality. We will then discuss spherical and hyperbolic ``Bicycle Curves'', which bear similarities to the planimeter.

Let X be the moduli of SL(n,C), SU(n), GL(n,C), or U(n) valued representations of a rank r free group. We compute the fundamental group of X and show that these four moduli otherwise have identical higher homotopy groups. We then classify the singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Lastly, we show that the moduli X are generally not topological manifolds, except for a few examples we explicitly describe.

I'll present the main ideas from Moser's paper "Minimal Solutions of Variational Problems on a Torus". The results and methods from this paper were inspired by the work of Aubry and Mather, which was discussed last week. I'll try to highlight the nice connections between the discrete problem of "minimal configurations" and the PDE problem of "minimal foliations".

The moduli spaces of curves have proven themselves to be important objects of study in algebraic geometry. One approach to studying them might be to construct maps from the moduli space to other objects that could be described explicitly. So that's what we'll do. This talk ought to be accessible to all graduate students.

The moduli spaces of curves have proven themselves to be important objects of study in algebraic geometry. One approach to studying them might be to construct maps from the moduli space to other objects that could be described explicitly. So that's what we'll do. This talk ought to be accessible to all graduate students.

One way to construct closed three manifolds is from Dehn surgery on a link in the three sphere. When Dehn surgery on a hyperbolic knot produces a non-hyperbolic 3-manifold, we call such a surgery exceptional. After going through some of the fundamental definitions, we will follow a construction of a family of knot complements due to Mario Eudave-Muoz that admit exceptional surgeries.

This will be a practice talk for the VIII Americas Conference on Differential Equations PASI 2009. Since that talk is only 30 mins I may show some additional results on the dynamics of the 3-D H\'{e}non family that I have not shown before before.

Our aim is to study elliptic reducible lower dimensional tori in Hamiltonian systems via parameterizations. We want to hightlight that the approach presents a lot of advantages in contrast with classical methods and that it produces simple and constructive proofs that lead to numerical algorithms for the computation of these objects. This is a joint work with J. Villanueva. The talk will be only 20 minutes long.

When we look out on a clear night, the universe seems infinite. Yet this infinity might be an illusion. During the first half of the presentation, computer games will introduce the concept of a “multiconnected universe”. Interactive 3D graphics will then take the viewer on a tour of several possible shapes for space. Finally, we'll see how recent satellite data provide tantalizing clues to the true shape of our universe. The only prerequisites for this talk are curiosity and imagination. For middle school and high school students, people interested in astronomy, and all members of the UT and surrounding communities. Reception to follow.

We are concerned with the solutions of BSDEs associated with a forward diffusion process. The main property is that the solution is Markovian in the sense that it can be represented as a deterministic function of time and state process. And we will introduce the important links between these FBSDEs and PDEs: Under some assumptions, the solution of the FBSDEs corresponds to the viscosity solution of some quasilinear parabolic PDEs, and conversely, in some cases, these Markovian FBSDEs give a Feynman-Kac representation of the same PDEs. These results can be applied to European option pricing in the constrained Markovian case and give a generalization of Black-Scholes Formula. This talk is based on the paper by Karoui and Peng (1997).

Radar and sonar, satellite observations CAT scans, indeed most of medical imaging, are ubiquitous examples of the recovery of a hidden or remote object by making measurements from a distance. They are wonderful case studies of the application of mathematics. In all such problems there is a partial differential equation lurking behind the scenes. If we knew the shape, location and material properties of the object then mathematics developed in the latter half of the nineteen century and first half of the twentieth would let us predict, in theory, exactly the kind of measurements we would obtain. The much harder, but interesting part, is the converse; given the measurements, where, and what, was the obstacle? This talk will explore this inverse problem, specifically the issues of uniqueness and determination. What is the least amount of measurements one can make in order to obtain a unique recovery? Are there algorithms that would let us reconstruct the object and what are their limitations? But there will be an additional feature. It is well know that such inverse problems are ill-posed - the solution does not depend continously on the measured data, but we will point out that seemingly minor changes in the partial differential equation or the boundary conditions can result in completely changed answers for the inverse problem.

A boundary slope of a knot K is the slope of a curve corresponding to the boundary of an essential surface in the complement of K. This talk will introduce these concepts and give numerous examples. We will then prove that for a certain class of knots, the extreme boundary slopes of a knot are given by its Jones polynomial and signature.

We consider the Navier-Stokes equations for a heat-conducting, compressible, viscous fluid in a bounded domain in R^3. We will discuss the existence of weak solutions to this system of equations due to E. Feireisl. We will also describe a bound from below on the temperature for these solutions, recently obtained by Mellet and Vasseur. This bound makes use of techniques related to De Giorgi's argument for the regularity of elliptic equations. This is a practice for my candidacy talk.

In statistical theory of turbulence phenomena like, instability with respect to the initial data, roughness of the solution, and dissipation of energy are very closely related. We use the shear flow to show that the situation is radically different for individual solutions of the incompressible Euler equations. The shear flow was introduced in our community by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this example to provide non-generic, yet nontrivial, examples concerning the loss of smoothness of solutions of the three-dimensional Euler equations, for initial data that do not belong to showing that the space is critical for the Euler equations. The role of this space as the critical case is underlined by an example of Pak and Park showing that the problem is well posed in the Besov space (one has and by the extension of our instability results to the Besov space which satisfies the inclusion Moreover, we show the existence of solutions with vorticity having a non trivial density on non smooth surface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which related to the Onsager conjecture. It may be interesting to compare the properties of the family of shear flow solutions with the ``wild solutions" constructed (not explicitely) by C. De Lellis and L. Szekelyhidi. There one has an infinite family of solutions for the same initial data. They are "wild" in the sense that they are limit on oscillating velocity fields and they conserve the energy. In relation view an "optimistic" construction of a measure for statistical theory of turbulence one could imagine that in both cases one has family of solutions of measure zero. (In spite of the fact that in the second case the wild solutions form a residual set)

The classical Dirichlet Approximation Theorem can be proved very simply, but it also has several more complicated proofs. We will present at least a few of these, then discuss a variety of ways to generalize the theorem. This talk will be accessible to all graduate students. First and second year students are especially welcomed, as I intend to include some discussion of how to generate research problems.

In this talk, we present a piece of joint work of Chi Hin Chan and Magdalena Czubak, in which we establish the existence of smooth solutions to fractal Burgers equation with critical exponent by applying the parabolic De Giorgi's method as developed by Luis Caffarelli and Alexis Vasseur. In the talk, we will make a parallel comparison between our work on fractal Burgers' equation with critical exponent and the work by Caffarelli and Vasseur in their paper "Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation."

In this talk, I'll tell you slightly less than everything you need to know about L-spaces, why they're interesting, and what they have to do with knot Floer homology.

We will be discussing Tate's thesis, Fourier Analysis in Number Fields and Hecke's Zeta-Functions. Please see the website: http://www.ma.utexas.edu/users/khopkins/NTJC.html for more information.

We will explore three ways in which integers arise in topology -- the Euler number, the Brouwer degree, and the index of a singularity of a vector field -- and see how the relationships among them tell us important facts about surfaces and manifolds. No prior knowledge of the subject will be assumed.

The local Langlands correspondence relates representations of (essentially) the Galois group of a local field F to complex representations of GL_n(F). An interesting question is how the objects on both sides of this correspondence vary in algebraic families, and in particular if the correspondence can be extended to such families. We discuss joint work with Matt Emerton that suggests that the answer is yes.

Protein-protein interaction belongs to the most important processes in the body. Today many diseases can be and treated at the protein, i.e. biomolecular level. Prediction and scoring of possible protein-protein interaction is an important prerequisite for efficient drug design on a computer. In recent years significant interest has been focused on the determination of electrostatic potentials of large biomolecules. However, the standard continuum approach ultimately becomes inaccurate when used to determine electrostatic properties on atomic scales [1]. In the papers [2], [4], we have proposed a novel formulation of nonlocal electrostatics allowing numerical solutions for the nontrivial molecular geometries. Some rather simple examples were solved and demonstrated correct physical behaviour. However, the mathematical model presented in these two papers is not completely equivalent to the physical model formulated in terms of Maxwell equations with nonlocal material relationship in the exterior domain, see [3]. In the recent paper [5], we present new system of four partial differential equations for nonlocal electrostatic which is formally equivalent to the physical model, shows its ellipticity, derive a simple analytical solution in the case of the unit sphere, find the fundamental solution of this system in an explicit analytical form and by the use of this, rewrite the interface problem in form of boundary integral equations. References: [1] T. Simonson. Macromolecular electrostatics: continuum models and their growing pains. Curr. Op. Struct. Biol., 11:243–252, 2001. [2] A. Hildebrandt, R. Blossey, S. Rjasanow, O. Kohlbacher, and H.-P. Lenhof. A novel formulation of nonlocal electrostatics. Phys. Rev. Let., 93(10):104–108, 2004. [3] A. A. Kornyshev, A. I. Rubinstein and M. A. Vorotyntsev. Model nonlocal electrostatics.1. J. Phys. C: Solid State Phys. 11:3307–3322, 1978. [4] A. Hildebrandt, H.-P. Lenhof, R. Blossey, S. Rjasanow, and O. Kohlbacher. Electrostatic potentials of proteins in water: A structured continuum approach. Bioinformatics, 23(2):99–103, 2007. [5] C. Fasel, S. Rjasanow, and O. Steinbach. A boundary integral formulation for nonlocal electrostatics. In K. Kunisch, G. Of, and O. Steinbach, editors, Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH 2007, pages 117–124. Springer, Heidelberg, 2008.

Abstract: Loop spaces carry many fascinating structures. These structures have long played a central role in algebraic and differential topology and have lately become central to algebraic geometry and representation theory. I will give an elementary introduction to one of these structures, the relation between the natural circle action on a loop space LX and differentiation on the base X, following my joint work with David Nadler (Northwestern). In particular I will explain how S^1-equivariant vector bundles on LX give flat vector bundles on X. One can also describe arbitrary connections in similar fashion, and the notion of curvature is captured in a universal fashion by the Hopf fibration.

We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrodinger equation. Traditional Gaussian beam type methods for the Schrodinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al. [S. Leung, J. Qian, R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics 72 (2007) SM61-SM76], we develop an Eulerian Gaussian beam method which uses global Cartesian coordinates, level-set based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semi-classical solutions to the Schrodinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi-classical solutions to the Schrodinger equation. We also highlight the importance of initializing Gaussian beam propagation. Numerical experiments indicate that this Eulerian Gaussian beam approach yields accurate semi-classical solutions even at caustics. This is a joint work with S. Leung and R. Burridge.

 Recent work has shown that fast matrix multiplication algorithms can be constructed by embedding the two input matrices into a group algebra (for this talk, a complex group algebra CG for a finite group G), applying a generalized discrete Fourier transform, and performing the multiplication in the Fourier basis.  The efficiency of this algorithm depends on the dimensions of the irreducible representations of G.  I will present some extensions of this idea using group actions and partial matrix multiplication algorithms.

Impulse control problems have wide applications in economics and engineering. Regularity study helps finding closed-form solutions and provides useful insight for numerical approximation when the former is impossible. In this talk we will derive the -regularity (in particular, smooth-fit principle) of the value functions for impulse controls of multidimensional diffusions and jump diffusions, using a viscosity solution approach and other PDE tools.

A graph G, is intrinsically knotted if every embedding of G into R^3 contains a nontrivial knot. A graph G, is intrinsically linked if every embedding of G into R^3 contains a non-split link. I will give a summary of the history of intrinsically linked and intrinsically knotted graphs. A natural generalization of intrinsic linking is intrinsic n-linking. A graph G is intrinsically n-linked if every embedding of G into R^3 contains a non-split n-component link. I will discuss some of my results about intrinsic n-linking, and look at knotting in intrinsically linked graphs.

In this talk, we will prove the existence of maximizers for Strichartz and Sobolev-Strichartz Inequalities for the wave equation. As in the work of S. Shao in the Scrodinger context, the main tool that we use is the corresponding wave profile decomposition result, extending the prior work of H. Bahouri and P. Gerard to higher dimensions.

An arithmetic toric variety is a toric variety over a field k that splits over a Galois extension K/k. We classify these, up to toric isomorphism, using Galois cohomology with coefficients in the toric automorphism group. Cox's quotient construction helps to compute this cohomology set. We use this to study the forms of projective space and toric surfaces. We also study forms of affine toric varieties and toric ideals. This is joint work in progress with Javier Elizondo, Paulo Lima-Filho, and Frank Sottile.

How likely is it that a curve is defined over the rational numbers? If a curve is defined over the rational numbers, does that force it to have unusual geometric properties? We will see that these questions are related to the unirationality of the moduli spaces of curves and discuss some of the work that has been done in this area.

The time-depended solutions of collisional electron kinetic equation with the heating term allowing the solutions in self-similar variables are considered. Our investigation is concentrated on the evolution and formation of the distribution function tails for long times and establishment of the asymptotic self-similar solutions. A broader class of the heating terms resulting in enhancement of the tail of the distribution function in comparison with Maxwellian is analyzed both analytically and numerically. Also formation of non-stationary non-equilibrium electron and phonon distribution functions in metals under action of a strong pulse electric field is briefly considered.

The Waring rank of a polynomial of degree d is the least number of terms in an expression for the polynomial as a sum of dth powers. The problem of finding the rank of a given polynomial and studying rank in general has been a central problem of classical algebraic geometry, related to secant varieties; in addition, there are applications to signal processing and computational complexity. Other than a well-known lower bound for rank in terms of catalecticant matrices, there has been relatively little progress on the problem of determining or bounding rank for a given polynomial (although related questions have proved very fruitful). I will describe new upper and lower bounds, with especially nice results for some examples including monomials and cubic polynomials. This is joint work with J.M. Landsberg.

Three dimensional cryo-electron microscopy (3D cryo-EM, for short) is the problem of determining the three dimensional structure of a macromolecular complex from the set of images, taken by an electron microscope, of randomly oriented and positioned identical molecular particles which are immobilized in a thin layer of ice. A solution to this problem is of particular interest, since it promises to be an entirely general technique which does not require crystallization or other special preparation stages. Present approaches to the problem fail with particles that are too small, cryo-EM images that are too noisy or at resolutions where the signal-to-noise ratio becomes too small. In my talk, I will describe a novel algorithm, referred to as the spectral algorithm, due to Amit Singer and Yoel Shkolnisky for the solution of the 3D cryo-EM problem. The appealing property of this new algorithm is that it exhibits remarkable numerical stability to noise. My main goal is to give a conceptual explanation for the admissibility (correctness) and the numerical stability of the spectral algorithm. In the course, I will describe a formal mathematical framework for cryo-EM which explains how the various numerical observations of Singer and Shkolnisky follow from basic representation theoretic principles.

I will show an algorithm done on braid drawings, and discuss how it solves the word problem for the braid group.

An introduction to number-theoretic functions and the principle of Mobius inversion. A little group theory, some identities/applications, and a generalization to posets. * Modern biology would likely classify Mobius as a vertebrate. Were he an invertebrate, even the claim to originality as such would be historically dubious, at best.

Two important models in the study of fluid mechanics are the incompressible Euler and Navier-Stokes equations. We will discuss some of the basic features of these models and compute a few explicit solutions for flows with a simple structure. Then we will discuss the notion of weak solutions and give a brief overview of important results and open problems related to these solutions.

Let E be an elliptic curve over Q with supersingular reduction at p and K an imaginary quadratic extension of Q. We analyze the structure of the p-primary part of the Tate-Shafarevich group of E over the anticyclotomic Z_p-extension K_{\infty}/K by viewing it as a module over Z_p[Gal(K_{\infty}/K}].

Abstract: Ricci solitons are natural extension of Einstein metrics on one hand, and arise as singularity models in the Ricci flow on the other hand. In this talk I'll discuss the link between singularities of the Ricci flow and Ricci solitons, and report on some recent progress on the classification and geometry of Ricci solitons.

With Andrew Hassell and Luc Hillairet, we extend the results on eigenfunction concentration in billiards as proved by the speaker. There, the methods developed in the works of Burq-Zworski to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood of the vertices, there is a lower bound for some and any eigenfunction .

THIS TALK IS POSTPONED. PLEASE CHECK BACK FOR THE NEW DATE AND TIME. SORRY FOR THE INCONVENIENCE. Computation of high frequency solutions to wave equations is important in many applications, and notoriously difficult in resolving wave oscillations. Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. An alternative way to compute Gaussian beam components such as phase, amplitude and Hessian of the phase, is to capture them in phase space by solving Liouville type equations on uniform grids. In this talk I shall present a systematic construction of asymptotic high frequency wave fields from computations in phase space for acoustic wave equations (also for the Schr\"&ob;o&cb;dinger equation). The k-th order Gaussian beam superposition is shown to converge to the original wave field in the energy norm, at an optimal rate in arbitrary spatial dimension. This is in collaboration with James Ralston (UCLA).

The universal enveloping algebra of a Lie algebra is an example of a Hopf algebra. One requires several definitions in order to even begin to care about the preceding sentence. I am going to start from the beginning and give lots of examples to show what kinds of funny looking things come up when you're dealing with coproducts.

The only known examples of non-arithmetic lattices in SU(2,1) are due to Picard (1880's) and Mostow (1980). We will decribe an infinite family of groups, constructed in a manner similar to Mostow's, which are potential new non-arithmetic lattices in SU(2,1). We will then outline the construction of a conjectural fundamental domain for one of them. Most of this is joint work with John Parker.

We study a class of Markovian optimal stochastic control problems in which the controlled process is constrained to satisfy an a.s.~constraint P-a.s. at some final time . When the set is of the form , with non-decreasing in , we provide a Hamilton-Jacobi-Bellman characterization of the associated value function. It gives rise to a state constraint problem where the constraint can be expressed in terms of an auxiliary value function which characterizes the set for some . Contrary to standard state constraint problems, the domain is not given a-priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function which is itself a viscosity solution of a non-linear parabolic PDE. Applying ideas recently developed in Bouchard, Elie and Touzi (2008), our general result also allows to consider optimal control problems with moment constraints of the form or . Co-authors : Romuald Elie and Cyril Imbert from CEREMADE, University Paris-Dauphine

In this talk we will describe two classes of solutions to the Ricci flow: Ricci solitons and the so-called breathers. We will then describe Perelman's variational approach to the study of the Ricci flow and prove one of his no-breathers theorems: expanding (or steady) breathers on closed Riemannian manifolds are gradient expanding (or steady) Ricci solitons.

Instabilities in fluid motion are ubiquitous and yet instabilities come in various "flavors". The partial differential equations of fluid dynamics are very challenging nonlinear systems. A classical approach to detecting instabilities is to study the spectral problem associated with the linearized equations. We will discuss how in some situations it is possible to prove that linear instability implies instability for the full nonlinear equations. Examples where this can be proved include the cases of the 2-dimensional Euler equations, the 3-dimensional Navier-Stokes equation and an interesting equation arising in oceanography called the surface quasi-geostrophic equation.

Abstract: Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and G. Mikhalkin. Tropical geometry arguments yield combinatorial descriptions of (ordinary and relative) Gromov-Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In the case of the projective plane, these descriptions can be used to obtain new formulas for the corresponding enumerative invariants. In particular, we give a proof of Goettsche's polynomiality conjecture for plane curves, and enumerate plane rational curves of given degree passing through given points and having maximal tangency to a given line. On the combinatorial side, we show that labeled floor diagrams of genus 0 are equinumerous to labeled trees, and therefore counted by the celebrated Cayley's formula. The corresponding bijections lead to interpretations of the Kontsevich numbers (the genus-0 Gromov-Witten invariants of the projective plane) in terms of certain statistics on trees. This is joint work with Grisha Mikhalkin.

We consider the wave equation in a medium with a rapidly varying speed of propagation. We construct a multiscale scheme based on the heterogeneous multiscale method, which can compute the correct coarse behavior of wave pulses traveling in the medium, at a computational cost essentially independent of the size of the small scale variations. This is verified by theoretical results and numerical examples.

The Coxeter FA conjecture states that a Coxeter group acts by isometries on a CAT(0) polyhedral cell complex of dimension without global fixed points if and only if has an infinite special subgroup of rank . This conjecture has been proven for by J. P. Serre and for by A. Barnhill. I will discuss some classes of Coxeter groups with infinite special subgroups of rank 4 that act without global fixed points on CAT(0) cell complexes of dimension 3.

I will lead a discussion of the paper, Cycle integrals of the j-function and mock modular forms, by W. Duke, O. Imamoglu, and A. Toth. See http://www.math.utexas.edu/users/khopkins/NTJC.html for paper.

Estimating the Tree of Life, an evolutionary tree describing how all life evolved from a common ancestor, is one of the major scientific objectives facing modern biologists. This estimation problem is extremely computationally intensive, given that the most accurate methods (e.g., maximum likelihood heuristics) are based upon attempts to solve NP-hard optimization problems. Most computational biologists assume that the only feasible strategy will involve a divide-and-conquer approach where the large taxon set is divided into subsets, trees are estimated on these subsets, and a supertree method is applied to assemble a tree on the entire set of taxa from the smaller "source" trees. I will present supertree methods in a mathematical context, focusing on some theoretical properties of MRP (Matrix Representation with Parsimony), the most popular supertree method, and SuperFine, a new supertree method that outperforms MRP.

Let G be the group in the title (which, upon reflection, is a vector space over the rationals), and let X be its completion with respect to the metric induced by the absolute Weil height. We will describe some results about finitely generated subgroups of algebraic numbers which can be rather easily settled by working in X. And we will discuss several open problems and conjectures about the structure of G and X.

The Lambda-Cold Dark Matter (LCDM) model of cosmology is perhaps the simplest model that best describes the makeup and evolution of the universe in accordance with physical observations. This model contains up to 20 different cosmological parameters from space and ground based surveys. These cosmological measurements have reached a remarkable level of accuracy over the last decade. Future sky surveys promise to give even more numerous and more accurate data. However, such data does not inform directly about the cosmological parameters of interest. Detailed physical simulation models are typically required to relate information from these surveys to cosmological parameters of interest. A Bayesian formulation adapted from Kennedy and O'Hagan (2001) and Higdon et al. (2008) is used to give parameter constraints from physical observations and a limited number of simulations. The framework is based on the idea of replacing the simulator by an emulator which can then be used to facilitate computations required for the analysis. In this talk I'll describe an application that uses large scale structure and Cosmic Microwave Background (CMB) data to inform about a subset of the parameters controlling the LCDM model.

Abstract: We consider an appropriate sequence of surgery constructions (introduced by Huisken-Sinestrari) for mean curvature flow of two-convex hypersurfaces in which we vary the parameters controlling the surgery procedure, and show that in the limit we retrieve the weak solution of the level set flow.

Local martingales which are not martingales (hence strict) have always had a marginal presence in Probability theory and Mathematical Finance. They do not exist in discrete Probability, and thus they are treated as technical hiccups, only to be smoothed by localization. However, several new theories in recent times have stressed on the importance of being strict: whether it is digging for relative arbitrages in Stochastic Portfolio Theory, or a foundation of the theory of financial bubbles. We intend to show that, far from being an anomaly, strict local martingales capture a fundamental probabilistic phenomenon. They appear whenever we apply a change of measure (on a filtration) which is dominated by, but not equivalent to, the existing measure. Thus they appear in such diverse areas as diffusions conditioned to have a certain boundary behavior, size-bias sampling, and non-colliding particles used in Random Matrix Theory. Is there a way to avoid being strict under a change of measure ? The answer is yes, if you do not hit zero before the martingale, representing the Radon-Nikodym derivative, hits zero.

A graph is called planar if it can be embedded onto a plane (or equivalently, a sphere) without any of its edges intersecting. It doesn't take long to convince yourself that the complete graph on five vertices and the complete bipartite graph on three and three vertices are not planar, but Kuratowski's Theorem says any non-planar graph contains a homeomorph of one of these. We'll define all this, then prove the theorem.

We will give an overview of some of the recent developments in the study of the deformations of infinite volume hyperbolic 3-manifolds. We will also provide an outline of the proof of Bers-Sullivan-Thurston's Density Conjecture.

On this talk we will explain the blow up of finite singularities of Ricci flow. There will be a discussion about convergence and compactness of flows with mention to Hamilton's Compactness Theorem. We'll also discuss the Berstein-Bando-Shi estimates which roughly says that bounds on the norm of the Riemann tensor along the flow gives raise to bounds on the norm of all its derivatives along the flow.

The moduli spaces of curves are useful and fascinating objects of study in algebraic geometry. One way to understand them is to consider their birational geometry. In this talk, we will construct rational maps from the moduli space to other spaces that have interesting geometric consequences.

In 2005 Ozsvath and Szabo developed an unknotting number one obstruction using Heegaard Floer Homology. In this talk I will explain how this technique can be used to classify a certain infinite class of 3-stranded pretzel knots with unknotting number greater than 1.

We will discuss how the algebraic K-theory of fields can be studied using only information from the representation theory of its absolute Galois group. In particular, we will discuss a context in which K-theoretic "descent" holds on the nose, and also relate the ideas with representation varieties of infinite discrete groups.

How do mathematicians, artists, and philosophers try to comprehend geometry beyond our third dimension? Modern computer graphics approaches make it possible to see and interact with four-dimensional phenomena in ways never available before. With the aid of computer graphics images and animations I will discuss the 4th dimension.

We will discuss a Brauer-Manin obstruction to a local-global principle for integral points on affine curves over number fields.

We are interested in studying geometric methods for studying high dimensional data sets. These data sets typically come equipped with a notion of distance, and can be thought of as finite metric spaces. We will describe how to apply standard methods to obtain homological signatures for finite metric spaces, as well as how to construct diagrams which help to analyze various stability properties for them. We will also discuss topological visualization techniques. All the topics will include examples.

We will go over some of the background for Dusa McDuff's November 2nd talk. Subjects include symplectic geomtry and a refresher on continued fractions.

For a decreasing, positive, divergent series {a_n} we will look at the effects of "weighing" the series with a sequence {b_n} converging to zero. Will the new series {a_n*b_n} converge or diverge?

The family of multidimensional diffusions named Volatility-stabilized market models (VSM) was introduced by Fernholz and Karatzas in 2005 as a toy model that nevertheless reflects some of the traits of a real-world equity market. These models reflect the fact that in real markets the smaller stocks tend to have a greater volatility and a greater rate of growth than the larger ones. A central question left open was to analyze the behavior of market weights, i.e., the vector of the ratios of stock capitals of individual firms over the entire market capital. In this talk we will show that the law of the market weights is actually a well-known object in another area of probability, namely, the Wright-Fisher model of population genetics. We analyze various dynamic and equilibrium properties of the market weights. In particular, we show that these weights do not exhibit a power law decay, as opposed to another class of models introduced by the same authors called the rank-based models. Power law decay is an empirically observed fact and has been a center of interest for decades of economic literature.

In this talk I will describe a new method to obtain short-time asymptotics for the heat kernel of parabolic linear operator with non-constant coefficients. Our method is based on dilation at a point z = z(x,y), Dyson and Taylor series expansions, and the Baker-Campbell-Hausdorff commutator formula. I will relate this method and the resulting expansion to other classical short-time expansions. Finally, I will present some new closed form asymptotic expansions of certain securities derivative prices using our method, and discuss some possible extensions. This based on joint works with W. Cheng, R. Constantinescu, A. Mazzucato, and V. Nistor.

We study volume-preserving maps with collections of continuous symmetries and invariants. Unlike in the Hamiltonian case, there are examples of maps with a symmetry that do not have an invariant. Conversely, a map with an invariant need not have a symmetry, and the dynamics on level surfaces of the invariant can be chaotic. We explore the consequences of symmetries in the context of volume preserving maps and exact incompressible, or Liouville vector fields. In order to simplify certain cases, we define a symmetry as a flow that has a global section. This has many consequences for the discrete map and leads to a coordinate system that simplifies the form of the map.

Continuing with the proof of the Density Conjecture, we explain the last part of the proof, exploring the notion of non-realizability of laminations. We describe how the problem can be reduced to a purely topological problem and then how we use hyperbolic geometry to solve this problem.

I will give an introduction to free boundary problems through a specific problem known as the obstacle problem. Continuity-related tools for studying the problem will be discussed. For the technical part of the talk I will show that solutions to the obstacle problem are harmonic away from the contact set. Many regularity theorems will be stated without proof.

I will talk about the derivation on a function field over a finite field, and how it helps me solve my research problem.

Schubert Calculus: what it is and what it's good for.

This work is motivated by a recent series of papers by Degond, M¨¦hats and Ringhofer about the formal derivation of Quantum Hydrodynamics models from the entropy principle. Starting from the Quantum Liouville equation, they obtain a non-closed system for the first moments of the density operator. Then, as in the Levermore moment method for kinetic equations, the system is closed by introducing the solution to the quantum moment problem which consists in minimizing the quantum free energy under some constraints of density, current or energy. The first step towards the rigorous derivation of such Quantum Hydrodynamics models is the mathematical analysis of the quantum moment problem. We will present in the talk a result of existence for the quantum moment problem under a density constraint and give some elements of the proof. The solution will also be characterized in a simplified setting. This is joint work with Florian M¨¦hats (Universit¨¦ de Rennes, France).

I'll describe new invariants of a framed link in a 3-manifold, which arise as the pages of a spectral sequence generalizing the surgery exact triangle in monopole Floer homology. The construction introduces a connection between the topology of link surgeries and the combinatorics of polytopes called graph associahedra. It also sheds light on the relationship between gauge theory and symplectic geometry. For a classical link L in S^3, we obtain a sequence of bigraded vector spaces, interpolating between the reduced, Z/2Z Khovanov homology of L and a version of the monopole Floer homology of the branched double cover. In addition, this perspective yields a simple, topological proof that odd Khovanov homology is mutation invariant. Paper reference: arxiv.org/abs/0903.3746, arxiv.org/abs/0909.0816

The study of shock statistics in Burgers equation arises as a problem of fundamental interest in mathematical physics, among other areas. In statistics, it appears in the context of limit theorems with `cube-root' asymptotics, culminating in a remarkable analysis of the parabolic hull of Brownian motion by Groeneboom (1989). More recent work in stochastic coalescence (by Bertoin, Duchon, and others) has provided another viewpoint on this problem, from which we show a Lax pair is naturally derived. This fact, in addition to the form of some exact solutions, suggests that the model is completely integrable. If time permits I will also touch upon some interesting connections with random matrices. This is joint work with Govind Menon (Brown).

The asymptotic dimension of the mapping class group dimension. A key piece of the proof is the construction of a quasi-tree that mapping class acts on. This construction works in a quite general setting for groups that have some aspect of negative curvature. We will describe this construction and explain how it relates to the asymptotic dimension of the mapping class group. This is joint work with K. Fujiwara and M. Bestvina.

We will look at the minimization of the functional J(u) = \int |\nabla u|^2 + Q^2\chi_{u0} over non negative functions with prescribed value at the boundary. We will show Lipschitz regularity for the minimizers and non degeneracy in order to prove that the free boundary has finite (n-1)-Hausdorff measure and derive the free boundary condition. Alt, H. W.; Caffarelli, L. A. Existence and Regularity for a Minimun Problem with Free Boundary, J. Reine Angew. Math., 1981 Cafarelli, L. A.; Salsa S. A Geometric approach to Free Boundary Problems, AMS 2005

I will talk about the derivation on a function field over a finite field, and how it helps me solve my research problem.

On this talk we'll talk about some curvature conditions that are preserved by Ricci flow and some that are not. We later show that non-negative Ricci curvature is not preserved in dimensions four and above.

We study the high energy limit of the scattering operator for the Schrodinger equation in the presence of an obstacle that is a disjoint union of handlebodies. From the high energy limit of the scattering operator, we recover the magnetic field and the electric potential. Our work proves that the approximate solution of the Schrodinger equation proposed by Aharanov and Bohm in their classical paper from 1959 is close to the exact solution that describes the experimental setup of Tonomura et al by which the Aharanov-Bohm effect was confirmed. This is a joint work with Ricardo Weder.

The largeness and positive virtual betti number conjectures are two active areas of research in 3-manifold topology. It is not difficult to see that largeness implies positive , but an interesting question is how much stronger the largeness conjecture is. In my talk I will discuss some characterizations of largeness and positive , and then talk about some conditions under which these two conjectures are equivalent. This is a practice candidacy talk.

The purpose of this talk is to discuss how the classical maximum value principles in PDEs be extended to their numerical approximations arising from finite element methods. In particular, the discussion will be primarily focused on the second order elliptic problems, and the finite element methods shall include standard Galerkin, P1 non-conforming, and mixed elements.

We will discuss several methods for reliable computation of invariant dynamics in conservative discrete dynamical systems. This will include some numerical techniques based on computational topology which are used to automatically compute topological horseshoe factors; and some quasi-analytical methods for computing high order expansions of stable and unstable manifolds and their intersections. This is a Ph.D. thesis defense.

Let E be an elliptic curve over Q. For a prime p of good reduction for E, let E_p be the reduction of E modulo p. In 1988, Neal Koblitz formulated a conjecture regarding the distribution of the primes p such that the number of F_p-points on E_p is also a prime. In many ways, this conjecture may be viewed as a 2-dimensional analogue of the classical twin prime conjecture. I will discuss a result saying that, for most elliptic curves in a two-dimensional family, Koblitz's conjecture is true. This is joint work with Antal Balog and Chantal David.

The cochains of a simplicial complex or singular cochains of a space can be viewed as a homotopy theoretic version of the algebra of functions. The "spectrum" or "variety" of this algebra is a homotopy theoretic completion of the space.

In this talk, we shall discuss our recent progress on acoustic and electromagnetic cloaking with transformation media. In wave scattering, a cloaking device is an artificially designed device which makes the target object invisible to wave detections. We shall present our study on both perfect cloaking with singular transformation media and near-invisibility cloaking from a regularization viewpoint.

This talk will discuss hyperbolic knots, some reasons why they're interesting, and how the hyperbolic structure given to such knot complements can be used to get topological information about the knot. In particular, we will consider how the geometry of such knot complements provides some conditions for determining when a given arc is an unknotting tunnel.

The most likely potential counterexamples to the 4D smooth Poincare Conjecture, as viewed by many 4-manifold topologists in recent years, were constructed by Cappell and Shaneson in the 1970's. These are indexed by an ifninite family of matrices. It took years of hard work with Kirby calculus in the 1980's to show that even the simplest example was standard (Akbulut, Kirby and then myself). Using further work from that decade and a clever observation, Akbulut showed last summer that a subfamily indexed by the integers is standard. This talk will discuss how to show a much larger family is standard, by a vastly simpler method that requires essentially no Kirby calculus. Instead, we exploit symmetries of a simple embedded 3-manifold.

In this seminar we will present an alternative proof of regularity for solutions of fully non-linear convex elliptic operators using the Method of Continuity. Moreover, we will compare this proof with the standard way of using this method in the case of linear second order elliptic equations.

We will describe some recent, elementary results in the theory of electromagnetic scattering. There are two classical approaches that we will review - one based on the vector and scalar potential and applicable in arbitrary geometry, and one based on two scalar potentials (due to Lorenz, Debye and Mie), valid only in the exterior of a sphere. In extending the Lorenz-Debye-Mie approach to arbitrary geometry, we have encountered some new mathematical questions involving differential geometry, partial differential equations and numerical analysis. This is joint work with Charlie Epstein.

Although the cohomology of a space is a commutative algebra, the cochains form only an "E-infinity" algebra. I will explain what this structure is, how it relates to the Steenrod operations, why it reduces to a commutative differential graded algebra in characteristic zero, and why it does not in other characteristics.

The theorem of Gross, Kohnen, and Zagier says approximately that the Heegner divisors for a weight two Hecke eigenform lie on a line in the Jacobian, and that their positions on this line are given by the coefficients of certain half-integer weight modular forms. In this talk we formulate two conjectures which partially generalize their theorem to higher weight modular forms. This is done by the construction of what we call `higher weight Heegner points'. Examples and results will be discussed. We will try to motivate this topic from the angle of the celebrated Riemann Hypothesis.

Following Davi's talk from last week, I'll discuss curvature conditions that are preserved under the Ricci flow. The focus will be a recent result of Wilking that puts the analysis of curvature conditions into a ridiculously simple algebraic framework.

Sea ice is both an indicator and agent of climate change. It also hosts extensive microbial communities which sustain life in the polar oceans. Fluid flow through porous sea ice mediates a broad range of processes such as the growth and decay of seasonal ice, the evolution of melt ponds and sea ice reflectance, and biomass build-up. We'll discuss recent mathematical advances using percolation theory, hierarchical models, and diffusion processes in understanding the fluid permeability of sea ice and the thermal evolution of its microstructure. Our work will help in predicting how global warming may affect Earth's sea ice packs and how polar ecosystems may respond. Related results on electromagnetic properties will help in monitoring ice thickness and in remote sensing of the polar marine environment. Video from a 2007 Antarctic expedition where we measured fluid and electrical transport in sea ice will be shown.

Polycrystalline materials like metals and ceramics are formed by a mesoscopic network of grains. Grain boundaries evolve by mean curvature, and the network coarsens over time. We are interested in modeling certain statistics of the network, in particular the Grain Boundary Character Distribution. I will discuss a simple birth / death model for the GBCD and progress towards a more detailed Boltzmann equation approach.

We consider the Navier-Stokes equations for a heat-conducting, compressible, viscous fluid in a bounded domain in R^3. We will discuss the existence of weak solutions to this system of equations due to E. Feireisl. We will also describe a bound from below on the temperature for these solutions, recently obtained by Mellet and Vasseur. This bound makes use of techniques related to De Giorgi's argument for the regularity of elliptic equations. This is a candidacy exam.

The largeness and positive virtual betti number conjectures are two active areas of research in 3-manifold topology. It is not difficult to see that largeness implies positive , but an interesting question is how much stronger the largeness conjecture is. In my talk I will discuss some characterizations of largeness and positive , and then talk about some conditions under which these two conjectures are equivalent. This is a PhD candidacy exam.

Constant-power loads are popular in applications that utilize distributed power architectures. They present interesting control problems. Boundary control is one method that can achieve stabilization of dc-dc converters loaded with these loads.

This is a thesis defense.

In characteristic zero, a space is said to be formal if its rational De Rham complex is equivalent to its cohomology as a commutative differential graded algebra; such spaces have all rational homotopy theory determined by their cohomology. I will explain why the analogue never holds in characteristic p and outline an approach that may work for partial information about p-adic homotopy theory.

Argumentation and experimentation are the two basic methods for pursuing truth. The importance and applicability of each method varies with the complexity and accessibility of the system under consideration. For instance, the stationarity of a physical system allows for considerable weight to be placed upon empiricism when searching for truth. In contrast, mathematics relies heavily upon argumentation when establishing truth, since it lacks a physical existence. Finding truth in mathematical finance is particularly difficult because financial systems do lend themselves to experimentation or axiomatization. Nonetheless, we often look upon finance as a science, albeit a "soft science." I suggest that even this characterization is misleading and that finance and economics would be better characterized as decision making tools. Note: this talk will be more philosophical than mathematical.

We will introduce the nonlinear Schrodinger equation, define criticality, and then survey recent progress.  We will discuss some of the technology involved using the energy-supercritical case as our model.

We improve Kolyvagin's upper bound on the order of the p-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and Swinnerton-Dyer conjectural formula.

In many of the modern biomedical imaging modalities, the measurable signal can be described as the solution of a partial differential equation that depends nonlinearly on the tissue properties (the "parameters") one would like to image. Consequently, there are typically no explicit solution formulas for these so-called "inverse problems" that can recover the parameters from the measurements, and the only way to generate body images from measurements is through numerical approximation. The resulting parameter estimation schemes have the underlying partial differential equations as side-constraints, and the solution of these optimization problems often requires solving the partial differential equation thousands or hundred of thousands of times. The development of efficient schemes is therefore of great interest for the practical use of such imaging modalities in clinical settings. In this talk, the formulation and efficient solution strategies for such inverse problems will be discussed, and we will demonstrate its efficacy using examples from our work on Optical Tomography, a novel way of imaging tumors in humans and animals. The talk will conclude with an outlook to even more complex problems that attempt to automatically optimize experimental setups to obtain better images.

With the perpetual trend towards smaller and smaller scales in science and engineering, fluid-flow problems in the transitional molecular/continuum regime have rapidly gained prominence over the past years. Accurate numerical simulation of flow problems in the transitional regime poses a fundamental challenge, on account of the large difference between the molecular free path and a typical length scale of observation. To bridge the gap between the molecular length scale and the continuum length scale in numerical simulations, many heuristic approaches have recently appeared in the literature to couple molecular-dynamics models to continuum models, such as the Navier-Stokes equations. The appropriateness of such a direct connection between a molecular model and a continuum model is arguable, however, because the range of validity of the models is highly disparate. A suitable model for transitional molecular/continuum flows is provided by the Boltzmann equation. In the Boltzmann equation, the flow is characterized by a one-particle probability-density function, which measures the probability that a molecule resides in a certain subset of the position/momentum space. The Boltzmann equation itself is an integro-differential equation that governs the evolution of the one-particle probability-density. The Boltzmann equation is in principle valid downto the molecular scale, while on the other hand it encapsulates all conventional continuum models, such as the compressible and incompressible Navier-Stokes equations, in the sense that with appropriate scalings of the macroscopic length and time scales, limit solutions of the Boltzmann equation correspond to solutions of these continuum equations. Essentially, the Boltzmann equation is connected to the continuum equations by the fact that solutions of the Boltzmann equation converge to a particular class of solutions, the so-called Maxwell-Boltzmann equilibrium distributions. Direct numerical simulation of the Boltzmann equation is prohibited by its high-dimensional setting: for a problem in d spatial dimensions, the corresponding position/momentum domain is 2d dimensional. However, it is anticipated that in many cases the computational complexity can be significantly reduced by means of adaptive low (d) dimensional approximations based on Boltzmann moment closures. In many applications, interest is in fact restricted to one particular goal functional. For instance, in micro-scale heat-transfer problems, it is ultimately only the heat flux across a certain part of the boundary that is of interest. This class of problems provides fertile ground for goal-oriented adaptive-refinement strategies. An essential impediment in the development of goal-oriented adaptive-refinement techniques for the Boltzmann equation, is the fact that the convergence-to-equilibrium property which forms the basis of the hierarchical modeling process only occurs for d=2,3. Hence, to test our ideas, we would have to consider problems in 4 or 6 dimensions. To bypass this complication, we have developed a 1D prototype of the Boltzmann equation, which exhibits all the characteristic features of the Boltzmann equation, including the weak-convergence-to-equilibrium property. The underlying molecular model is based on random collisions, which conserve energy but not momentum. In the presentation, I will give an overview of transitional molecular/continuum flows, from the perspective of hierarchical modeling and model adaptivity. I will then elaborate the 1D prototype of the Boltzmann equation that we have developed, and derive its characteristic properties, such as an entropy inequality. Finally, I will present numerical result obtained by a discontinuous Galerkin finite-element discretization of the prototype, including recent results for goal-oriented error estimation.

We will discuss the proofs of the various classification theorems for real division algebras, including Frobenius' Theorem, Hopf's Theorem and Hurwitz's Theorem, along with the rich history underlying them.