In my talk I will discuss some topics in hyperbolic geometry, including models for the hyperbolic plane, hyperbolic isometries, and various ways to represent them via matrix groups.

We will talk about a "boundary obstacle problem", i.e. the minimization of the Dirichlet integral among all functions that lie above a given obstacle in the boundary of the domain. Such a problem appears in topics such as elasticity, temperature control and modeling of semi-permeable membranes. The optimal regularity of the solution was proven by Caffarelli and Athanasopoulos by exploiting the quasi-convexity of the solutions and a monotonicity formula, this problem is also closely related to the fractional Laplacian. We will discuss the proof of the optimal regularity and current work in this topic.

The aim of this talk is to describe several methods for numerically approximating the integral of a multivariate highly oscillatory function. We begin with a review of the asymptotic and Filon-type methods developed by Iserles and Nørsett. Using a method developed by Levin as a point of departure we will construct a new method that uses the same information as a Filon-type method, and obtains the same asymptotic order, while not requiring moments. This allows us to integrate over nonsimplicial domains, and with complicated oscillators. We also present a method for approximating oscillatory integrals with stationary points.

Though modular forms are perhaps most important to number theorists, they appear as well in topology, physics, and nature, and their theory uses topology, complex analysis, and both the analytic and algebraic sides of number theory. In this talk we will give an introduction to modular forms. We will begin with their definitions, and cover the central elements of the theory up to and including Hecke operators. There will be key motivating examples throughout. This talk is self contained and is intended to be an introduction. This is the first talk of two and will serve as an introduction for the background that will be assumed in the second.

A pdf of the paper is available here: http://www.ma.utexas.edu/users/jmann/LMZC_JMB06.pdf

In this talk, we are going to discuss the deformation of the surface in the inward normal direction with the speed proportional to Guass or Scalar Curvature. The equation for the graph will be a fully nonlinear and degenerate when one of the principle curvature becomes zero. We will discuss the optimal regularity and the existence of classical solutions.

Dunfield proved that a small hyperbolic knot in a homotopy sphere has a non-integral boundary slope whenever there is a framing $<\alpha, \beta>$ for such that is cyclic. While this result may be categorized as classical 3-manifold topology, the proof relies on ideas from algebraic geometry. In my talk I will discuss character varieties and prove Dunfield's theorem.

I will present a computational method for simulating the dynamics of atomic systems on time scales much longer than can be simulated with classical dynamics. In this method, possible reaction mechanisms available to the system are found with a minimum-mode following algorithm. Rates are estimated using harmonic transition state theory, and the system is propagated in time according to the kinetic Monte Carlo algorithm. The method can be run in parallel, and a distributed computing system has allowed us to simulate the deposition and growth of metal surfaces over experimental time scales using an embedded atom potential. Our algorithm is efficient enough to model the evolution of systems with ab-initio forces as well, for which I will show several examples, including metal cluster formation on oxide surfaces.

A problem in twisted K-theory is to realize twisted K-groups generally by means of finite-dimensional geometric objects, like vector bundles. I will talk about an answer to the problem by using a twisted version of Furuta's generalized vector bundle and a notion of a finite-dimensional approximation of Fredholm operators.

Volume-preserving maps arise from the flow of an incompressible fluid or of a magnetic field. An understanding of their dynamics is important for the study of the mixing of passive tracers, but is also an interesting generalization of the dynamics of area-preserving or symplectic maps. I will show that a quadratic, volume-preserving map arises as a fundamental model analogous to Henon's two-dimensional map in three distinct situations: this model is the "hydrogen atom" of three- dimensional dynamics. Recently we obtained it as the normal form for a bifurcation of a fixed point with a triple-one multiplier. I will discuss dynamical phenomena such as the formation and break-up of invariant circles and tori, the creation of "strings of pearls," and chaotic transport due to heteroclinic intersection of two dimensional invariant manifolds.

We will introduce the Szemeredi Regularity Lemma and go through an example of how it is used to find substructures in large graphs.

(This is joint with Nonlinear Dynamics.)

Viscous flows in thin domains are encountered in a great number of practical situations. Think, for example, of flows in gas or pipelines, in fluid film bearings, in capillaries, or in oceans and in atmosphere. It is natural to expect that in these situations the governing equations of motion are approximated by simpler, lower-dimensional models leading to solutions which are precise only asymptotically as the width of the domain tends to zero. In his seminal paper in 1886, Osborne Reynolds proposed a model for the main pressure distribution in lubrication. This model, the classical Reynolds equation, even though derived under the assumption that the inertial and curvature effects are negligible, makes a reasonable (first-order) approximation for many, but not all, problems in hydrodynamic lubrication. In our work, we derive an equation, referred to as modified nonlinear Reynolds equation, for higher-order approximation of pressure (and consequently of velocity field) in hydrodynamic lubrication. This equation is a nonlinear second-order differential equation for a scalar unknown, which approximates the first two terms in the asymptotic expansion of the pressure field, and, in contrast to the classical Reynolds' equation, takes into account both the inertial and the curvature effects. The equation is justified with optimal integral and pointwise error estimates. Up to now, this kind of optimality has only been achieved for small fluxes and in simpler geometrical situations. Here, the flux, equivalently the Reynolds number, can be as large as possible within the limits of dimension reduction and the optimal estimates result from an accurate analysis of the auxiliary divergence equation.

Bridge number for knots in the 3-sphere is a classical knot invariant introduced by Schubert more than 50 years ago. More recently it has been generalized to knots in any manifold by using a Heegaard surface for the manifold in the place of a bridge sphere. We will discuss some results about bridge surfaces that are analogous to results about Heegaard splittings as well as some of the behaviors of bridge surfaces that seem to have no Heegaard splitting analogs.

We will discuss again the Signorini problem and its optimal regularity in a more rigorous manner than we did last week. The optimal regularity follows from two important lemmas that we will discuss in detail: first that solutions to the thin obstacle problem are semi-convex in the tangential directions to the free boundary and then that the normal derivative of the solution satisfies a monotonicity formula.

is the (1D) continued fraction. The 's can be connected to geometry, via approximating a line L of slope by a sequence of bases of , with 'update matrices' : that is, so that ;. One need only use 2 such 's, to wit and . Then each is just the number of one of these that are needed before the other occurs. CF's are useful for many problems in number theory, in particular, using the ideas of 'best approximants' and 'poorly approximable' numbers lead
1) to special types of (real) numbers useful in mathematics.
2) As another example, one can decide whether 2 'Pisot' 2 by 2 matrices are conjugate over the integers; and
3) Can show that given A as in (2) there is a B in GL(z,2) such that ; has all entries of the same sign. Though mostly other authors have focused on problems of type (1) in their generalizations to ``higher dimensional continued fractions'', we propose here a theory of HDCF useful in problems of the second two types. In particular, this solves these problems in dimension 3. The fundamental construction is a cell complex homotopy equivalent to the 2 skeleton of the 3 dimensional torus, tying arithmetic to geometry.

At the center of modular forms is the theory of Hecke operators. Hecke operators are a commutative algebra of operators acting on the space of modular forms of weight . This space has a canonical basis of simultaneous eigenvectors of all the and these particular modular forms have very special Fourier coefficients, L functions, and connections to elliptic curves. We will begin with the definition of the Hecke operators and explain some key results. If time permits, we will explain how this connects to the Taniyama-Shimura-Weil Theorem proved by Wiles for Fermat’s Last Theorem, and show how to get an elliptic curve from a cusp form. There will be a recap of what we discussed last week as well. The notes from last week's talk can be found at the following link: http://www.math.utexas.edu/users/khopkins/JNTTalkModFormsFirstHalf.pdf

We will discuss the regularity theory of the Monge-Ampere equation det D^2u=f(x) where f(x) is in VMO^\psi spaces defined in terms of sections.

After reviewing last week's talk, I will finish what I started.

We consider a few modeling and computation problems that is related to incoherent imaging in highly heterogeneous (random) media. We show that such imaging problems can be treated as inverse problems related to the linear transport equations. Numerical solutions of those inverse problems will then be discussed. Reconstructions from both synthetic and experimental data will be presented.

Among the simplest of the classical polynomials are the Chebyshev polynomials of the first and second kind, and . In our normalization, the indices are allowed to be half- integers as well as integers, and the ``polynomials'' actually live in . We will show that the rational functions and are very remarkable from the point of view of Grothendieck's dessins d'enfants. The fibers of these rational functions are likewise very remarkable from the point of view of algebraic number theory. For example, for the fiber of the second function above is given by a degree 15875 polynomial in with discriminant and Galois group the entire symmetric group .

Articles: Aspinwall, "D-branes on Calabi-Yau manifolds", Progress in String Theory, TASI 2003 Lecture. Notes (2005), World Scientific. Bridgeland, "Stability conditions on triangulated categories", Ann. of Math. 2007, Vol 166, No.2. Kontsevich, "Donaldson-Thomas invariants", 2007 MPIM Mathematische Arbeitstagung, http://www.mpim-bonn.mpg.de/preprints/send?bid=3352.

In this talk will define strongly -regular rings and show how that every strongly -regular element can be written as the sum of an idempotent, a unit and a nilpotent element, all of which commute with one another. That every strongly -regular element is strongly clean is a consequence of this result.

Building convergent numerical schemes for fully nonlinear partial differential equations is a challenge. In the case of first order equations, schemes have been built based on ideas from Conservation Laws. Recently several authors have tried adapting techniques from fluid mechanics or finite elements to solve second order equations. These attempts have had limited success. We will present recent work on bulding convergent difference schemes for degenerate and fully nonlinear elliptic Partial Differential equations. These equations include: the obstacle problem, the infinity Laplacian, the equation for motion of level sets by mean curvature, the Monge-Ampere equation, the Pucci maximal and minimial equations, an equation for the convex envelope. Earlier work emphasized building convergent schemes over fast or accurate solvers. Since the resulting algebraic equations are nonlinear and non-differentiable, attempts to build implicit solvers have met with little success. As a result, the solution methods have been iterative, and consequently suffer from CFL type time step limitations. Recent work has explored technique to improve accuracy as solution speed. Remarkably, many of the fully nonlinear schemes can be reformulated as Linear Programs (LPs). We will discuss hybrid schemes for better accuracy, and implicit solvers for fully nonlinear equations.

TBA

We will provide a construction of traveling waves in a lattice model through a reduction to a center manifold technique. These models coming from solid state physics are known to play an important role in mechanisms involved in the propagation of energy. This is joint work with Yannick Sire.

I will give an overview of Guass' class number conjecture and present Estermann's proof of Siegel's Theorem. Special attention will be paid to the entrance of the ineffective constant in the proof.

(Part 1 of 2.) These talks will feature a colorful and interrelated miscellany of items that have made repeated appearances in the seminar this semester, including K-theory, Clifford algebras, spinors, and characteristic classes.

I'll explain some definitions of a geometrically finite Fuchsian group, and why they're equivalent. I'll extend them to higher dimensions, and give examples showing why some formulations can be more useful than others there.

In this talk, I will discuss three topics about the DG method. I will describe two recently developed discontinuous Galerkin schemes for solving the Hamilton-Jacobi equation and equations with higher order derivatives, respectively. Both schemes do not need to rewrite the original equation into several other equations using auxiliary variables to compute the solution. For the Hamilton-Jacobi equation, our scheme works well when the Hamiltonian is convex or linear. An entropy correction is needed when the entropy condition is violated. For equations with higher order spatial derivatives, our scheme can give provable L2 stability and accuracy for a very general class of equations. Numerical tests show that this scheme is (k+1)-th order accurate when using piecewise k-th polynomials, under the condition that k+1 is greater than or equal to the order of the equation. I will also present some recent work on the superconvergence and time evolution of DG method for conservation laws when essentially upwind fluxes are used. This is a joint work with Professor Chi-Wang Shu.

Singular vectors in R^d correspond to divergent trajectories of the homogeneous flow on SL(d+1,R)/SL(d+1,Z) induced by the one parameter subgroup diag(e^t,...,e^t,e^&ob;-dt&cb;) acting by left multiplication. In this talk, I will sketch a proof of the following result: the Hausdorff dimension of the set of singular vectors in R^2 is 4/3. (Alternatively, the set of points lying on divergent trajectories of the homogeneous flow on SL(3,R)/SL(3,Z) has Hausdorff dimension 7 and 1/3.) The main idea involves a multi-dimensional generalisation of continued fraction theory from the perspective of the best approximation properties of convergents. As an application, we answer a question of A.N. Starkov regarding the existence of slowly divergent trajectories.

For non-negative integers , let be the unique modular function that is holomorphic on the upper half-plane and with Fourier expansion . Asai, Kaneko, and Ninomiya showed that , where as usual is the weight Eisenstein series and is the unique normalized weight cusp form. Ahlgren obtained similar formulas on higher level, genus , congruence subgroups, with replaced by the corresponding Hauptmodul. We generalize these results to congruence subgroups for which the underlying modular curve is hyperelliptic. In all cases, we also show how to obtain similar results starting from any meromorphic modular form of weight . (From this point of view, the above formula corresponds to ).

At the beginning of 80's Simon Donaldson introduced new invariants of 4-dimensional Rimannian manifolds by counting critical points of the Yang-Mills functional on the space of connections. At the end of 90's his student Richard Thomas constructed new invariants of 3-dimensional complex Calabi-Yau manifolds by counting critical points of holomorphic Chern-Simons functional. The latter invariants and their versions for other gauge theories are called Donaldson-Thomas invariants. > From the algebraic point of view Donaldson-Thomas invariants should ``count" stable objects in triangulated categories. In my talk I plan to discuss our recent work with Maxim Kontsevich, where we made this idea more precise. We introduced "motivic" version of Donaldson-Thomas invariants by encoding them as invertible elements of certain non-commutative algebras similar to quantum tori of Connes. As a byproduct we obtained a new type of ``wall-crossing" formula, which describes the behavior of the invariants when the stability structure on the category gets changed. If time permits I will discuss the relation to cluster transformations as well as to the count of BPS states in N=2 4d super Yang-Mills theory.

Steganography is the method of hiding information beyond the standard level of encryption so that the presence of the hidden data is not detectable. I will present an overview of a few topics in Harmonic Analysis and will discuss how Fourier transforms and wavelets pertain to steganography of digital information.

In the theory of differential games with asymmetric information the value function u(x,t,p,q) (where p and q are parameters) has to "satisfy" a parabolic equation in (x,t) for any value of the parameters and at the same time remain convex in and concave in . We shall explain the proper framework of analysis for this equation, give some representation formula of the solution, and discuss open problems.

We discuss several one-dimensional symmetry results for global solutions of (possibly singular or degenerate in some cases) elliptic equations, inspired by a conjecture of De Giorgi. In particular, symmetry results are given under the assumption that either the profiles at infinity are two-dimensional, or that one level set is a complete graph, or that the solution is quasi-minimal. In dimension two and three, symmetry results will be obtained under very mild assumptions.

In this talk, I will introduce J.M. Bony's notion of a paraproduct between two functions, and present several linearization results of Y. Meyer which utilize this construction.

We will briefly introduce the concept of arcs and complete arcs. We will then talk about the Giulietti-Ughi and state some results we obtained. If time permits we will state and prove a theorem.

This lecture blends ideas from probability theory, PDE’s, numerical analysis and physical reasoning in the style of modern applied mathematics. First elementary concepts of information theory involving the lack of information in one probability measure relative to another are introduced. These ideas are then applied to give a unified explanation of the competing equilibrium statistical theories for coherent structures in fluids. The potential application of these ideas to predict coherent structures in flows with random forcing and dissipation is then discussed briefly including the first theorem (Xiaoming Wang and Majda, CPAM 2006) justifying the approach. All of this is utilized to develop a successful prediction for the exact location and structure of Jupiter’s Red Spot in agreement with the Voyager Mission of the 1970’s by utilizing a physically based prior encoding the key features of small scale observations from the Galileo mission of the 1990’s. An apriori independent justification of this overall strategy is presented through novel numerical methods for detecting statistically relevant conserved quantities in fluid flow (Abramov and Majda, 2003, 2004.) The lecturer’s recent book with X. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flow, Cambridge Univ. Press (2006) discusses in detail most of the material presented here.

Geophysical flows are a rich source of novel problems for applied mathematics and the contemporary theory of partial differential equations. The reason for this is that many physically important geophysical flows involve complex nonlinear interaction over multi-scales in both time and space so developing simplified reduced models which are simpler yet capture key physical phenomena is of central importance. In mid-latitudes, the fact that the rotational Coriolis terms are bounded away from zero leads to a strict temporal frequency scale separation between slow potential vorticity dynamics and fast gravity waves; this physical fact leads to new theorems justifying the quasi-geostrophic midlatitude dynamics even with general unbalanced initial data for both rapidly rotating shallow water equations and completely stratified flows. At the equator, the tangential projection of the Coriolis force from rotation vanishes identically so that there is no longer a time scale separation between potential vortical flows and gravity waves. This has profound consequences physically that allow the tropics to behave as a waveguide with extremely warm surface temperatures. The resulting behavior profoundly influences longer term mid-latitude weather prediction and climate change through hurricanes, monsoons, El Nino, and global teleconnections with the mid-latitude atmosphere. How this happens through detailed physical mechanisms is one of the most important contemporary problems in the atmosphere-ocean science community with a central role played by nonlinear interactive heating involving the interaction of clouds, moisture, and convection. The variable coefficient degeneracy of the Coriolis term at the equator alluded to earlier leads to both important new physical effects as well as fascinating new mathematical phenomena and PDE’s. In this equatorial context, new multi-scale reduced dynamical PDE models are even relatively recent in origin. After a brief discussion of the observational record as background, this lecturer surveys the remarkable new hyperbolic systems that have emerged recently in applications including their physical properties, applied mathematical and rigorous mathematical theory. These last topics include novel relaxation limits for climate models with active moisture and new singular limits for hyperbolic PDE’s with variable coefficients. All of the references in this lecture can be found at http://www.math.nyu.edu/faculty/majda/.

We'll talk about what spectral sequences are, how they arise, and why we might care about them. Then we'll discuss a particular example from Khovanov homology, used by Rasmussen to give a new proof of a conjecture of Milnor.

Whether smooth solutions of the 3D incompressible Euler equations spontaneously develop finite time singularities is a long-standing open problem of great interest to both mathematical and fluid mechanical communities. In this talk, I will discuss recent results on this problem obtained with T. Y. Hou, J. Deng, and R. Li. We investigate possible cancellations in the nonlinear term that prevent the formation of finite time singularities. I will also show numerical results supporting the existence of such cancellations.

We discuss ergodic theorems for semisimple groups and lattices and apply them to derive some effective counting and equidistribution results. Based on joint work with Amos Nevo.

This year's crossword puzzle: "GETTING TO KNOW YOU - WHAT'S IN A NAME?" Prizes to the person or team that solves the puzzle first. A special thank you to Dr. Karen Uhlenbeck for the prizes. MOVIE: A HILL ON THE DARK SIDE OF THE MOON THE STORY OF SONYA KOVALEVSKAYA Host: Dr. Fernando Rodriguez-Villegas Special Introduction by: Dr. Natasa Pavlovic RSVP TO NANCY LAMM.

It would be nice to think that arithmetic algebraic geometry had something useful to say about the world we live in. The talk will concern the combinatorics of graphs and the physics of Feynman amplitudes

This lecture discusses systematic mathematical strategies for low-dimensional stochastic mode reduction for turbulent large dimensional dynamical systems and their application to modelling low frequency weather dynamics and climate change. A remarkable fact of Northern Hemisphere low frequency variability is that it can be efficiently described by only a few teleconnection patterns that explain most of the total variance. These few teleconnection patterns not only exert a strong influence on regional climate and weather, they are also related to climate change. These properties of teleconnection patterns make them an attractive choice as basis functions for climate models with a highly reduced number of degrees of freedom. The development of such reduced climate models involves the solution of two major issues: 1) how to properly account for the unresolved modes, also known as the closure problem; and 2) how to define a small set of basis functions that optimally represent the dynamics of the major teleconnection patterns. In this lecture examples of stochastic mode reduction are discussed ranging from an explicit solvable pedagogical example with three modes to a prototype atmospheric general circulation model with a thousand degrees of freedom where an effective reduced stochastic model with only ten low frequency modes captures the statistical dynamical behavior. A controversial topic in the recent climate modeling literature is the fashion in which metastable low-frequency regimes in the atmosphere occur despite nearly Gaussian statistics for these planetary waves. Here a simple 57-mode paradigm model for such metastable atmospheric regime behavior is introduced and analyzed through hidden Markov model (HMM) analysis of the time series of suitable low-frequency planetary waves. The analysis of this paradigm model elucidates how statistically significant metastable regime transitions between blocked and zonal statistical states occur despite nearly Gaussian behavior in the associated probability distribution function and without a significant role for the low-order truncated nonlinear dynamics alone; turbulent backscatter onto the three-dimensional subspace of low-frequency modes is responsible for these effects. It also is demonstrated that suitable stochastic mode reduction strategies, which include both augmented cubic nonlinearity and multiplicative noise, are also capable of capturing the metastable low-frequency regime behavior through a single stochastic differential equation compared with the full turbulent chaotic 57-mode model. This feature is attractive for issues such as long-term weather predictability. Research papers regarding most of the research here can be found on Majda’s faculty website: http://www.math.nyu.edu/faculty/majda

We will prove the Gauss-Bonnet Theorem for orientable surfaces, which is a famous and remarkable local-global result in differential geometry that relates Gaussian curvature of a surface to its Euler characteristic.

Ordinary continued fractions CF have been generalized to higher dimensions, HDCF's since Jacobi and Perron's thesis, and in many different ways. No one way is as useful as CF's. We start with showing how 2 of these ways are generalizations of CF's and talk about a way that is useful for solving the following problems: call a matrix 'Pisot' if it has a real eigenvalue and all others inside the unit circle. 1. If is a Pisot matrix there exists such that has all entries 2. Given 2 Pisot matrices, there is a finite procedure for deciding whether they are conjugate over the integers.

I will first introduce the notion of stability for steady solutions to Euler's equation, then discuss how looking at fluid motion from a geometric point of view inspired some questions that (I think) could only be answered analytically. This will motivate separately looking for instabilities in each of the two pieces of an orthogonal decomposition of the solution space of Euler's equation.

In recent years there have been a number of substantial advances in numerical techniques for solving the wave equation and Maxwell's equations in the time domain. These include: (i) Efficient methods for truncating unbounded domains which can provide arbitrary accuracy at low cost; (ii) Fast methods for evaluating the standard integral solution formulas, akin to the fast multipole method in the frequency domain; (iii) Discretization strategies focusing on techniques for using high order/high resolution approximations in complex geometry. I will review these new developments and discuss ideas for their improvement and extension to other wave propagation problems.

We will begin by introducing the Weil height of algebraic numbers and the Mahler measure of polynomials in one variable. We will then describe Lehmer's problem and the most important existing results relating to it. We will then review normalizers of stabilizers desribe their relationships to the measure and height. Time permitting, we will describe my research results.

In the 1960's "classical", or "zero-th order" knot signatures were defined and shown to have special behavior on slice knots. In the 1970's Casson and Gordon defined what we shall call "first-order" signatures that were also shown to obstruct a knots being a slice knot. We define a partial derivative (and antiderivative) of a knot. In terms of this notion we define "second-order" signatures of a knot and show that they obstruct a knots being a slice knot. In some situations we can take more derivatives and define "n-th order" signature obstructions. These are applied to show that certain increasingly subtle families of knots contain no slice knots.

I will present Korner's recent construction of Besicovtich sets via Baire's Theorem and give an overview of applications and related open questions in Harmonic analysis. This talk will assume little beyond introductory real analysis.

Cochran, Orr and Teichner defined a filtration of the smooth and topological knot concordance groups. The quotients of successive terms of these filtrations are abelian groups that reflect the existence of increasingly subtle non-slice knots. We prove that each such quotient has infinite rank.

Abstract: Yau's theorem on Kahler manifolds states that there exists a unique Kahler metric in every Kahler class with prescribed volume form. This has many applications in complex geometry. I will discuss results on Donaldson's program of extending the Calabi-Yau theory to symplectic manifolds. In a different direction, I will talk about the problem of existence of constant scalar curvature Kahler metrics, which can also be considered a generalization of Yau's theorem.

Given a separable polynomial q with coefficients in a field K, we study the trace form on the vector space K[x]/q. When it is real and definite, its generating function w is a rational function that gives rise to some nice results on the interlacing of the real roots of q and qw. This opens the door to the mystery of the p-adic world: how would this phenomena generalize over there?

This will be more conjecture than theorem meant to bring together different ideas, including applications to field theory and string theory. I will explain how the "derived philosophy" in this context yields stronger information than the usual theorems.

We will analyze the evolution of weakly convex surfaces in with flat sides by the harmonic mean curvature flow. We establish short time existence as well as the optimal regularity of the surface and show that the boundaries of the flat sides evolve by the curve shortening flow. It follows from our results that a weakly convex surface with flat sides of class , for some integer and , remains in the same class.

Today will be our end of the semester problem session. This is an opportunity for anyone to get up and speak briefly about a problem, big or small, the junior topology audience may appreciate.

We will first prove some general existence theorems about "heavy points" in a measure preserving system: points whose ergodic averages consistently dominate their expected value. We will then proceed to some examples of this behavior, contrasting the existence of such behavior in dynamical systems with the absence of such objects in other settings, and contrasting the ease of an "existence proof" with the difficulty of a constructive proof.

Moment zeta functions provide a diophantine reformulation for the distribution of rational points on a family of algebraic varieties over finite fields. They also form algebraic approximations to Dwork's transcendental p-adic unit root zeta functions. In this talk, we focus on the important example of the mirror family of Calabi-Yau hypersurfaces and discuss the recent joint work with Rojas-Leon on the higher moment zeta functions for this family.

The most infamous axiom in all of mathematics has plagued mathematicians by producing bizarre and pathological counter examples in almost every branch of math. It turns out that the Axiom of Choice is actually equivalent to several other equally innocuous theorems, many of which are fundamental in set theory, abstract algebra, analysis and topology. We will prove these equivalences for several well known theorems and discuss some other interesting implications.

We study nonlinear integro-differential equations. Typical examples are the ones that arise from stochastic control problems with discontinuous Levy processes. We can think of these as nonlinear equations of fractional order. Indeed, second order elliptic PDEs are limit cases for integro-differential equations. Our aim is to extend the theory of fully nonlinear elliptic equations to this class of equations. We are able to obtain a result analogous to the Alexandroff estimate, Harnack inequality and regularity. As the order of the equation approaches two, in the limit our estimates become the usual regularity estimates for second order elliptic pdes. This is a joint work with Luis Caffarelli.

An averaging principle is applied to construct a numerical method for integrating stiff or highly oscillatory systems of ordinary differential equations (ODEs) that may include components that are in resonance. Resonances may cause different oscillators to synchronize and generate some hidden slow dynamics. The method relies on correctly tracking a set of slow variables whose dynamics is closed up to a small perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. In this talk I will describe the algorithm and the theory underlining it. A few systems that exhibit resonances and entrainment are studied as benchmark models.

Abstract: Shafareviches well-known hyperbolicity conjecture asserts that a family of curves over a quasi-projective 1-dimensional base is isotrivial unless the logarithmic Kodaira dimension of the base is positive. More generally it has been conjectured by Viehweg that the base of a smooth family of canonically polarized varieties is of log general type if the family is of maximal variation. In this talk, we relate the variation of a family to the logarithmic Kodaira dimension of the base and give an affirmative answer to Viehweg's conjecture for families over surfaces. Methods used include the log minimal model program, a study of foliations on singular spaces, and extension theorems for logarithmic differential forms. Possible generalizations to higher dimensions will be discussed.

In 1770, Waring asked whether given k there is a fixed value g(k) such that every positive integer may be written as a sum of g(k) kth powers. (That same year, Lagrange had demonstrated the elusive conjecture that g(2) = 4.) In 1909, Hilbert answered Waring's problem in the affirmative. In this talk, a version of Hilbert's proof that has been significantly streamlined by Hausdorff, Stridsberg, and Ellison is presented. While the proof is dryly technical in spots, it comprises a collection of interesting techniques.

I'm new to this area of mathematics, so I'm taking this opportunity for us to go through the basics of tropical geometry together. I hope to leave you with a few basic examples, an inkling of why this topic is interesting, and enough understanding to have a fighting chance at understanding the tropical geometry talks in the Sr Geometry seminar.

I will describe a new framework for the representation theory of real and complex semisimple Lie groups, developed in joint work with David Nadler. We show how different aspects of representation theory (including the theory of intertwining operators, Lusztig's character sheaves and character varieties for surface groups) fit into the structure of a topological field theory (the character theory) which we associate to a Lie group. The main result is an equivalence between the character theories for Langlands dual groups, which may be considered as a three-dimensional aspect of the electric-magnetic duality of four-dimensional gauge theories.

The exterior, M, of a small hyperbolic knot in a homotopy sphere has a non-integral boundary slope whenever there is a framing for such that is cyclic. Dunfield's proof of this theorem relies on basic ideas from algebraic geometry. In my talk I will discuss character varieties and present a version of Dunfield's proof. This is a candidacy talk.

In this talk I will show that the geometric mechanism for detecting global instability introduced in a recent paper by Delshams, de la Llave and Seara can be applied to a general class of a-priori unstable Hamiltonian systems. I will also consider a particular example and I will show that it admits diffusing orbits that follow this geometric mechanism. This is joint work with Amadeu Delshams.

In recent years, techniques from semidefinite programming have produced numerical algorithms for finding representations of positive semidefinite polynomials as sums of squares. These algorithms have many applications in optimization, control theory, quadratic programming, and matrix analysis. One major drawback with these algorithms is that their output is, in general, numerical. For many applications, however, exact polynomial identities are needed. In this regard, Sturmfels has asked whether a representation with real coefficients implies one over the rationals. We discuss recent progress on this problem; in particular, we outline a positive answer to the question for totally real number fields.

We will explore everyone's favorite non-commutative group of order 43,252,003,274,489,856,000 ... the rubix cube! We will prove that the cube has certain invariants in order to compute its order and to give a nice representation for it. I will also introduce a 4-dimensional version of the world's most popular puzzle (which has order 1,756,772,880,709,135,843,168,526,079,081,025,059,614,484,630,149,557,651,47 7,156,021,733,236,798,970,168,550,600,274,887,650,082,354,207,129,600,000,00 0,000,000)

The goal of this talk is to prove that the dimensional reduction of the extended 3+2+1 Topological Quantum Field Theory given by a Modular Tensor Category associated to a Quantum Group specialized at a root of unity only depends on the order of that root.

We will introduce methods for numerical computation of KAM quasi-periodic equilibrium solutions in statistical mechanics models. For our methods we use an extension of the Lagrangian proof of KAM for twist mappings. Then we will use these computations to estimate the breakup of quasi-periodic solutions. This is joint work with R. de la Llave.

******PLEASE NOTE TIME CHANGE FOR TODAY'S GEOMETRY SEMINAR********

Spectra of triangulated categories. I'll discuss the general framework of studying triangulated categories via geometric ideas as introduced by Paul Balmer. Two specific examples/applications will be explained along the way: the bounded derived category of a quotient stack and the stable category of modular representations of a finite group scheme, with special attention given to the latter.

A curve in space can be characterized either by its parametrization or by the set of equations defining it. The number of the parametrized curves of fixed degree and genus passing through some collection of lines and points can be computed as an integral over the completed space of the parametrized curves. This number is called a Gromov-Witten (GW) invariant of the space. Analogously, if we use the completed space of the equations to count curves we obtain a Donaldson-Thomas (DT) invariant. These two invariants are very far from being equal but the recent conjecture by Maulik, Nekrasov, Okounkov and Pandharipande gives a way to obtain GW invariants from DT invariants and vice versa. In my talk I will discuss the motivations for definitions of the invariants and describe recent progress in the proof of the conjecture which is due to Maulik, Okounkov, Pandharipande and the speaker.

(Please note the speaker is an Assistant Professor Candidate.)

It is known that finite-index subgroups of the arithmetic group SL(3,Z) have no (orientation-preserving) actions on the real line. This naturally led to the conjecture that most other arithmetic groups (of higher real rank) also cannot act on the line. This problem remains open, but joint work with Lucy Lifschitz verifies the conjecture for many examples. This includes all finite-index subgroups of SL(2,Z[alpha]), where alpha is any irrational, real algebraic integer. The proof is based on the fact, proved by D.Carter, G.Keller, and E.Paige, that every element of these groups is a product of a bounded number of elementary matrices. No familiarity with arithmetic groups will be assumed.

The relationship between the geometry and topology of 3- manifolds has been the focus of a large amount of research in low dimensional topology. We discuss an approach to use combinatorial information provided by a Heegaard splitting and obtain many geometric and topological information about 3-manifolds. We show how this approach has been used in a joint work with Brock, Minsky and Souto to construct geometric structures which are described by combinatorial models.

Gyongy has shown that it is possible to construct a diffusion process with the same one-dimensional marginal distributions as a given initial Ito process. This is closely related to work by Dupire that shows how to construct a local volatility model such that the European option prices implied by the model agree with a given set of market prices. In this talk, we review this connection and then provide a generalization of Gyongy's Theorem that shows how one can match more properties of an initial Ito process by carrying more information about the path history in the mimicking Markov process. Finally, we suggest how such a construction might allow one to extend the local volatility methodology and construct price processes which fit the market prices of mildy path-dependent options.

This is the first meeting of the now-weekly intro to research seminar, aimed at first and second-year graduate students.