In this talk we address the issue of global regularity for active scalar equations arising in fluid dynamics. We present a non-linear lower bound for the fractional Laplacian when evaluated at extrema of a function, and give applications to critical equations. We also obtain the global regularity of solutions to equations which are slightly-supercritical. The later results turn out to be sharp in certain scenarios.

We will explain closed hyperbolic 3-manifolds with constrained rank and injectivity radius have a geometric decomposition with bounded complexity. Applications to arithmetic manifolds, rank vs Heegaard genus, and the group theory of Kleinian groups will be given.

Finite element exterior calculus (FEEC) analyzes mixed variational problems and their numerical approximation properties using the ideas and tools of Hilbert complexes. This framework, developed by Arnold, Falk and Winther for elliptic and elasticity problems [1], has been extended by Holst and Stern [2,3] to include variational crimes, allowing applications to linear and nonlinear geometric elliptic partial differential equations and surface finite element problems. In this talk, I will present a new extension of FEEC to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Using the framework of Bochner spaces and their associated norms, I will show how FEEC can be applied to establish a priori error estimates for Galerkin finite element approximation of parabolic problems. This approach recovers the results of Thomee [4] for two-dimensional domains and the lowest-order mixed method as a special case, effectively extending these results to arbitrary spatial dimension and an entire family of mixed methods. I will also discuss how the Holst and Stern results aid in novel analysis and error estimates of non-linear evolution problems and evolution problems involving variational crimes. This is joint work with Michael Holst. [1] Arnold, Falk, and Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (2010), 281--354 [2] Holst and Stern, Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces, arXiv:1005.4455. [3] Holst and Stern, Semilinear mixed problems on Hilbert complexes and their numerical approximation, Found. Comput. Math., in press, 2010. [4] Thomee, Galerkin finite element methods for parabolic problems, Springer Verlag, 2006.

We examine the theory and numerical treatment of coupled nonlinear elliptic geometric PDE containing critical exponents. A motivating example is the conformal formulation of the Einstein constraint equations. We first outline some new results on the existence of solutions to the constraint equations for rough metrics and arbitrarily prescribed mean extrinsic curvature. We then develop some new a priori error estimates for Galerkin finite element approximation; the estimates have the surprising feature that no angle conditions are involved in the case of critical and subcritical nonlinearity. Moreover, it then becomes possible to turn the a priori estimates into pointwise control of discrete solutions, without the need for a discrete maximum principle, and hence again without the need for angle conditions. We then briefly describe a new approach to analyzing the geometric error made if the domain is a smooth Riemannian manifold rather than a polyhedral domain. The approach involves the development of variational crimes analysis in Hilbert complexes, and then using the abstract framework to develop analogues of the Strang Lemmas for the Finite Element Exterior Calculus (FEEC). We indicate how this new variational crimes framework in FEEC recovers the classical a priori surface finite element estimates of Dziuk and Demlow, and allows for substantial generalizations, including hypersurfaces of arbitrary spatial dimension, the Hodge Laplacean, nonlinear problems, and evolution problems.

Title 1: Cluster algebras from surfaces I will start by giving a gentle introduction to the cluster algebras of Fomin and Zelevinsky. Then I'll describe the class of cluster algebras coming from triangulated surfaces. These cluster algebras are interesting for several reasons: they are closely connected to the decorated Teichmuller space associated to the corresponding surface, and they comprise most of the "tame" cluster algebras. I'll then explain joint work with Musiker and Schiffler, in which we give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type. If time permits, I'll also describe our work constructing (vector-space) bases of these cluster algebras.

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that one can use the Wronskian method to construct a soliton solution to the KP equation from each point of the real Grassmannian. In my talk I'll describe the beautiful combinatorics (permutations, triangulations, etc) that arises when one studies the regular soliton solutions that come from the Grassmannian. We'll also see how the theory of total positivity and cluster algebras provide a natural framework for studying KP solitons. This work can be used to solve the inverse problem for soliton solutions coming from the positive Grassmannian. Regular KP soliton solutions provide a good model for shallow water waves (like beach waves), and I'll end the talk with some pictures and movies. This is joint work with Yuji Kodama.

We will discuss our goals for Junior Number Theory seminar this semester, schedule talks, etc.

The Uniformization Theorem implies that any compact Riemann surface has a constant curvature metric. Kahler-Einstein (KE) metrics are a natural generalization of such metrics, and the search for them has a long and rich history, going back to Schouten, Kahler (30's), Calabi (50's), Aubin, Yau (70's) and Tian (90's), among others. Yet, despite much progress, a complete picture is available only in complex dimension 2. In contrast to such smooth KE metrics, in the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface), motivated by applications to algebraic geometry and Calabi-Yau manifolds. More recently, Donaldson suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk we will try to give an introduction to Kahler-Einstein geometry and briefly describe some recent work mostly joint with R. Mazzeo that resolves some of these conjectures. One key ingredient is a new C^{2,\alpha} a priori estimate and continuity method for the complex Monge-Ampere equation. It follows that many algebraic varieties that may not admit smooth KE metrics (e.g., Fano or minimal varieties) nevertheless admit KE metrics bent along a divisor.

A key problem in biomedical research is to elucidate the complex gene regulatory network underlying complex traits such as common human diseases. In genetical genomics (eQTL) experiments, gene expression levels are often treated as quantitative traits that are subject to genetic analysis. These data can also provide important information on gene regulation and genetic networks. In this talk, I introduce a sparse high dimensional multivariate regression model for studying the conditional independent relationships among a set of genes adjusting for possible genetic effects, as well as the genetic architecture that influences the gene expression. I present a covariate adjusted precision matrix estimation method (CAPME), which can be easily implemented by linear programming. Asymptotic convergence rates and sign consistency are established for estimators of the regression coefficients and the precision matrix. Numerical performance of the estimator is investigated using both simulated and real data sets. Simulation results have shown that the CAPME results in great improvements in both estimation and graph structure selection. We apply CAPME to analysis of a yeast eQTL data in order to identify the gene regulatory network among a set of genes in the MAPK pathway. In addition, I will discuss analysis of multi-tissue eQTL data and simultaneous estimation of multiple precision matrices with similar structures.

Special seminar for graduate students, by today's colloquium speaker, with pizza provided by the math department.

I will show that a pair of alternating links have homeomorphic branched double-covers if and only if they are mutants. The proof involves some interesting combinatorial / lattice theoretic invariants coming from Heegaard Floer homology.

I will give an overview of the ongoing joint work with Gopal Prasad on length-commensurable and isospectral locally symmetric spaces. The centerpiece of this work is the new notion of weak commensurability for Zariski-dense subgroups. We have been able to determine when two arithmetic subgroups are weakly commensurable. This leads to various geometric results, some of which are related to the famous question "Can one hear the shape of a drum?"

Adaptive finite element methods (AFEM) are standard tools in the numerical treatment of partial differential equations (PDE). The convergence analysis of AFEM for linear elliptic PDE is now well understood; however, the case of nonlinear PDE has not yet been well-studied. In this talk, I will develop an AFEM convergence theory for a large class of semilinear PDE having important applications in biophysics and astrophysics. We first develop an abstract contraction framework, which generalizes the recent contraction results [Cascon, Kreuzer, Nochetto and Siebert 2008] for linear elliptic PDE to the nonlinear setting under a type of weak Lipschitz assumption on the nonlinearity. We then show that this assumption is satisfied for the case of nonlinearities satisfying critical and subcritical growth conditions, without the need for unrealistic angle conditions necessary for deriving discrete maximum principles. Moreover, we also show how to obtain discrete pointwise control for this class of critical and subcritical nonlinearities without introducing angle conditions beyond shape regularity. I will also present a weak-* convergence framework of AFEM for Petrov-Galerkin approximation of very general class of nonlinear PDE, and survey a number of our supporting projects on fast solvers that are key to practical use of adaptive methods.

The talk will be built around the following question: let and be two central quaternion division algebras over the same field ; when does the fact that and have the same maximal subfields imply that and are actually isomorphic over ? I will discuss the motivation for this question that comes from the joint work with G.~Prasad on length-commensurable locally symmetric spaces, and will then talk about some available results. One of the results states that if the answer to the above question is positive over a field (of characteristic not 2) then it is also positive over any finitely generated purely transcendental extension of . I will also discuss some generalizations to algebras of degree and some recent finiteness results. This is a joint work with V.Chernousov and I.Rapinchuk.

In 1958, Kolmogorov defined the entropy of a probability measure preserving transformation. Entropy has since been central to the classification theory of measurable dynamics. In the 70s and 80s researchers extended entropy theory to measure preserving actions of amenable groups (Kieffer, Ornstein-Weiss). My recent work generalizes the entropy concept to actions of sofic groups; a class of groups that contains for example, all subgroups of GL(n,C). Applications include the classification of Bernoulli shifts over a free group. This answers a question of Ornstein and Weiss.

I will introduce basic terminology and tools in the study of infinitesimal deformations of objects in algebraic geometry. The talk will begin with the definition of a deformation functor and discuss how one studies obstructions to lifting a deformation, and move on to demonstrate some geometric information one can gain. The focus will be on simple examples and intuition, in preparation for the following talks on more advanced deformation questions.

We consider the problem of finding the optimal constant for the embedding of the space into the space , where is a bounded domain with boundary of class . This is equivalent to find the first eigenvalue of the 1-biharmonic operator under (generalized) Navier boundary conditions. In this talk we provide an interpretation for the eigenvalue problem, we show some properties of the first eigenfunction, we p rove an inequality of Faber-Krahn type, and we compute the first eigenvalue and the associated eigenfunction explicitly for a ball, and in terms of the torsion function for general domains.

The Einstein constraints equations are of fundamental interest in mathematical physics in the study of Einstein's theory of general relativity. This coupled nonlinear elliptic system must also be solved in various forms for gravitational wave simulation. The constraint equations have been studied intensively for half a century; in this lecture we consider two closely-related problems: (1) The development of new existence results for the constraint equations that violate the so-called "near-CMC condition"; (2) The development of "provably good" approximation methods, with a focus on adaptive finite element methods. The first problem has been open since the "conformal method" for solving the Einstein constraints was first proposed in 1973. We recently developed a new analysis approach based on combining topological fixed point arguments with the construction of "global barriers"; this framework made it possible in 2009 to establish the first existence results for the Einstein constraints without using the near-CMC condition. We will give an overview of these new results, and also outline some ongoing projects to further extend the solution theory. (Various parts of this part of the work are joint work with G. Nagy, and G. Tsogtgerel.) Once the solution theory is in place, we can address the second problem, which splits into two clearly stated, fundamental, numerical analysis questions: (2a) A priori error estimates for finite element-type methods for nonlinear elliptic systems posed on 2- and 3-surfaces; (2b) Establishing proofs of contraction for adaptive finite element methods for nonlinear elliptic systems of this type. Both problems are at the heart of two extremely actively areas of research in numerical analysis; we will give just a brief overview of some of our recent relevant results in each of these areas, in an effort to produce an efficient (low complexity), robust (always works), and high-fidelity (highly accurate) numerical approximation scheme. Solving Problem (2a) will involving exploiting very recent work on finite element exterior calculus, and problem (2b) will involve extending recent work on convergence theory for linear problems to the nonlinear setting. (Various portions of this part of the work are joint with Y. Zhu, G. Tsogtgerel, R. Bank, R. Szypowski, and S. Pollack.)

The knot complement problem was proven in 1989 by Cameron Gordon and John Luecke. It states that two knots with homeomorphic complements are themselves equivalent. In this talk, we'll work through some relevant background material, make sense of the theorem statement, discuss a alternative formulation using the language of Dehn Surgery, and give a sketch of the proof.

Abstract:An invertible weighted bilateral shift T is constructed such that T and it's inverse have no common nontrivial invariant subspace. Thus T has no non-trivial hyper-invariant subspace.

A d-dimensional variety is a complete intersection if it is the intersection of n-d hypersurfaces in n-space. There is a natural corresponding property for the local rings associated to a variety which can be generalized to all local rings - in two ways.

Join us for our first session of the spring semester as we learn about mass point geometry, and how to solve problems using this powerful tool. We will work through a collection of problems that get progressively more challenging.

A two-player zero-sum stochastic differential game, defined in terms of an m-dimensional state process that is driven by a one-dimensional Brownian motion, played until the state exits the domain, is studied. The players controls enter in a diffusion coefficient and in an unbounded drift coefficient of the state process. We show that the game has value, and characterize the value function as the unique viscosity solution of an inhomogeneous infinity Laplace equation. Joint work with R. Atar

Atmospheric entry simulations are complex problems due to their intense multiphysics and multiscale nature. The conventional physico-chemical nonequilibrium models used for these simulations are often derived "correlation-based" from experiments and stretched out of the validity range for which they have been conceived. We propose to use kinetic theory as a powerful tool to derive macroscopic conservation equations, transport fluxes, energy exchange terms and chemical production rates for atmospheric entry plasmas. Two approaches are followed: a multiscale Chapman-Enskog perturbative method and Boltzmann-moment system method with Grad closure. The following nonequilibrium effects are studied: mass disparity, electromagnetic field influence, ionization reactions, internal energy excitation, and rarefied gas effects.

An analytic function in the upper half plane is said to be of Nevanlinna class if it can be expressed as the quotient of two bounded analytic functions in . This space is an integral domain with a naturally occurring valuation. Its field of fractions is a space of meromorphic functions called functions of Bounded type. Indeed the valuation has a canonical extension to this field of fractions, it induces a non-Archimedean absolute value and hence a metric. We show that this metric space is not complete and describe some of the objects in the completion.

We will discuss the basics of quantum mechanics (in 1 dimension) and discuss the mathematical formulations of QM systems.

One of the fundamental problems in 3-dimensional contact geometry is the construction of tight contact structures on closed manifolds. Two obvious ways to try to construct such structures are via Legendrian surgery and admissible transverse surgery. It was long thought that when performed on a closed tight contact manifold these operations would yield a tight contact manifold. We show that this is not true for admissible transverse surgery. Along the way we discuss the relations between these two surgery operations and construct some contact structures with interesting properties.

The Fourier Transform (FT) and its 'fast' cousin the FFT are the cornerstone of computational Harmonic Analysis. Molecular recognition or docking is the search for the relative transformation between two (or more) molecules which yields the minimum energy configuration of a stable molecular complex. In this talk I shall first describe the use of the FFT (Fast Fourier Transform) over motion groups, and then its application for efficient molecular docking algorithms which includes improved energetic scoring based on van der Waal and electrostatics interactions, as well as a new on-the-fly affinity functions based on hydrophobicity and hydrogen bonding. I shall also introduce the use of irregular (non-uniform) versions of cartesian and spherical FFT methods, in combination with adaptive sampling for speeding up the search and scoring of the top solutions to the molecular docking problem. These algorithms are being implemented and encapsulated in our freely available CVC software MVP (Modeling Visualization Processing).

There is an existential connection between databases and categories. A database, which consists of a "schema" and some "conforming data", is tightly analogous with a category C and a set-valued functor C--Set. With such a simple connection between seemingly disparate worlds, we can compare the standard operations from category theory with those from database theory. The correspondence is quite strong, and the categorical viewpoint offers a simplification and standardization of ideas such as "schema mapping" (functor), data migration (pullbacks and push-forwards), and others. From a mathematical standpoint, the ideas in this talk will be fairly unsophisticated; the main point is to offer a simple connection between two fields and suggest that categories provide the "right way" to think about databases. I'll also discuss how techniques from homotopy theory relate to the study of information.

I will continue the introduction to deformation theory initiated last week. Topics will include tangent-obstruction theories in general, (pro-)representability of deformation functors, and discussion of a concrete example or two.

In a recent paper Y.Brenier (2009) obtained a new kinetic formulation for weak entropy solutions of scalar conservation laws. In this formulation entropy solutions (their graphs) are represented as level curves of a kinetic density function. We will show that this approach naturally extends to the class of entropy measure-valued solutions. Such solutions are unique, regular, stable and exists for all times. We discuss a number of interesting properties of the solutions and, in particular, the geometric interpretation of solutions as curves in a suitable Hilbert space for which the tangent vector minimizes an interaction functional. In the second part of the talk we will describe a kinetic formulation for smooth solutions of the Euler equations of gas dynamics with the geometric structure similar to the kinetic scalar conservation law.

Theorems of Bousfield-Gugenheim and Sullivan from the 1970s show that there is an equivalence between two categories: the rational homotopy category of nilpotent spaces with rational homology of finite type and the homotopy category of rational CDGAs with minimal models of finite type. In this talk we will discuss what all of the above means, sketch a proof of this equivalence, and show some applications of the equivalence that highlight its usefulness. This is a candidacy talk, and some knowledge of algebraic topology will be assumed.

The parity conjecture is a weak version of Birch-Swinnerton-Dyer Conjecture or more generally, Beilinson-Bloch-Kato Conjecture. It is conjectured that the vanishing order of the L-function at the central point has the same parity as the dimension of the Bloch-Kato Selmer group. I will explain an approach to this conjecture for modular forms by varying the modular forms in a p-adic family. This is a joint work with Kiran Kedlaya and Jay Pottharst.

Discretizations for heterogeneous 3d Helmholtz equations over cubic domains typically require grid points, where is the frequency of the problem. Thus, 3d problems quickly become challenging as the frequency is increased. In particular, for over a decade, the best-known methods involved preconditioning with an Incomplete LU (ILU) factorization of a diagonally shifted version of the discrete Helmholtz operator. While the memory usage of said approaches is typically only with a small constant, the number of iterations is known to grow linearly with .

Modern problems across science and engineering increasingly require high-dimensional models; with more parameters than observations. It is now well understood that statistically reliable inference is still possible under such high-dimensional settings, provided one restricts to constrained subclasses of models with particular low-dimensional structure. Examples include linear regression with sparsity constraints (compressed sensing), estimation of covariance or inverse covariance matrices, sparse principal component analysis, low-rank matrix estimation, and sparse additive non-parametric models. Over the past decade, there has been a strong body of work that have proposed statistical estimators for inferring such structurally constrained high-dimensional models, with strong statistical guarantees. In this talk, we consider the computational facet of such estimation: could we provide a general computational scheme to solve any of the convex optimization problems that arise in such high-dimensional inference? We find that such a general computational scheme is indeed possible: specifically, we discuss and analyze a scheme based on a greedy strategy. Our framework not only unifies existing greedy algorithms that have been proposed for such high-dimensional problems by recovering them as special cases but also yields novel ones.

Number theory is usually thought by those who do not do it to be some sort of mystical force that you can only do if you've sold your soul to the devil. The aim of this talk is breaking this mythos and showing via the power of examples the joys of number theory and convincing you that even YOU can get into the magic that is the theory of numbers, algebraic, analytic, and classical. Common sense and perhaps a few tricks from modular arithmetic are about all you should need to get into this talk. Admission is free, but if you feel like leaving a tasty soul afterwards, I won't complain. ;-)

Regression problems with a convexity constraint on the mean function are common in economics, financial engineering, operations research and electrical engineering. In a purely regression setting, convexity constraints can increase predictive accuracy compared to unconstrained regression. In a convex optimization setting, convex regression can be used to approximate objective functions and constraints. However, current convex regression methods are computationally infeasible for moderate to large problems in a multivariate setting. We introduce two new methods for multivariate convex regression, a Bayesian and a frequentist version. We give consistency results for both methods and show adaptive convergence rates for the Bayesian method. We apply the methods to value function approximation for sequential decision problems including response surface estimation and pricing American basket options.

The goal of this talk is to unfold the strong affiliation of round handles with smooth four-manifolds. Various fundamental topics that appear in the study of four-manifolds, such as handlebodies, cobordisms, logarithmic transforms along tori, exotic smooth structures, and broken Lefschetz fibrations, will come into play as we discuss the relevant interactions between them.

Floer cohomology is a refined measurement of the persistence of intersections between Lagrangian submanifolds in a symplectic manifold. It has a multitude of applications within symplectic topology, but it also connects to 3-manifold topology, via Heegaard Floer theory, and to algebraic geometry, via homological mirror symmetry (HMS). In this talk, I'll present joint work with Y. Lekili; the subject is HMS, but the ideas emerged from our work on Heegaard Floer theory. We argue that HMS can be refined so as to connect symplectic topology to arithmetic geometry. Specifically, we prove a version of HMS for the punctured 2-torus, a derived equivalence between its Fukaya category (over the integers) and sheaves on a nodal elliptic curve defined over the integers. This is concrete mathematics; we are asserting a relation between certain simple closed curves inscribed on the punctured 2-torus and vector bundles over the arithmetic curve Y^2 + XYZ = X^3, also known as the central fiber of the Tate curve.

I will discuss moduli problems, which are global analogs of deformation functors. My goal will be to outline a proof of the uniformization theorem for the moduli space of bundles in topological and algebraic settings.

Building predictive interaction models is a challenging problem, especially when the number of variables is large. Statisticians commonly demand that an interaction only be included in a model if both variables are marginally important. We study the problem of identifying hierarchical two-way interaction models from the viewpoint of the Lasso (i.e., L1-penalized regression). We show that by adding a set of convex constraints to the Lasso problem, we can produce sparse interaction models that honor the hierarchy restriction. In contrast to stepwise procedures that are most commonly used for building interaction models, our formulation is convex, and its solution is completely characterized by a set of optimality conditions. This makes it easier to study as a statistical estimator. We argue that restricting to hierarchical interactions can be advantageous both statistically and computationally. Our proposal extends more generally to any problem in which "hierarchical sparsity" is desired (i.e., one parameter is forced to be zero if another is zero). For example, in (univariate) polynomial regression, hierarchical sparsity yields low-order polynomial fits. Underlying our work is the observation that there is more to interpretability and simplicity in a model than sparsity alone.

A compact manifold with boundary components and is an -cobordism if deformation retracts onto both and . The -cobordism theorem states that if in addition , , are simply connected and has dimension greater than 5, then is diffeomorphic to and so is diffeomorphic to . We will discuss some ideas from the proof and some of its applications including the Generalized Poincare conjecture in dimensions greater than 4.

Given two Riemannian manifolds, one may measure the distance between them using the Lipschitz, Gromov-Hausdorff and Intrinsic Flat distances. We will present new approaches to estimating these distances given information on subdomains in the manifolds. Then we will apply these results to understand the Gromov-Hausdorff and Intrinsic Flat limits of sequences of Riemannian manifolds which converge smoothly away from singular sets.

Abstract:An invertible weighted bilateral shift T is constructed such that it has a bounded log S and for which it seems T and it's inverse have no common nontrivial invariant subspace. Thus S has no nontrivial invariant subspace.

We confirm the global Gan-Gross-Prasad conjecture for unitary groups under some local restrictions for the automorphic representations. If time permits, we will also discuss the arithmetic version.

I will discuss the problem of fitting a homogenized single scale model to observations from a multiscale diffusion process, and demonstrate how subsampling of the observed data can be used to estimate the effective parameters. Also I will describe a method for investigating the microscopic behavior of a multiscale process using local estimators.

We prove homological mirror symmetry for a smooth Calabi-Yau hypersurface in projective space. In the one-dimensional case, this is the elliptic curve, and our result is related to that of Polishchuk-Zaslow; in the two-dimensional case, it is the K3 quartic surface, and our result reproduces that of Seidel; and in the three-dimensional case, it is the quintic three-fold. We will spend the first hour introducing homological mirror symmetry, leading up to a precise statement of our result. In the second hour, we will describe some of the techniques used in its proof, and draw many pictures in the one-dimensional case.

The speaker will reformulate the definitions and ideas behind covering spaces in the language of categories and functors as presented in May's book. If there is time, he will also do the same for Galois Theory. The talk is meant to give an example of what category theory feels like when it is put to use. The speaker asks that the audience be familiar with the basic definitions and examples of categories and functors. Familiarity with groups and G-sets at the level of Dummit and Foote as well as an acquaintance with covering spaces at the level of Hatcher will be expected.

We study dispersive estimates for the Klein-Gordon equation on a square lattice with non-isotropic coupling between nodes. Areas of minimal dispersion occus at critical values of the velocity function for plane waves, with the time-decay exponent determined by the type of degeneracy present at the critical value. For the two-dimensional lattice, there exists a unique velocity (up to mirror symmetries) along which wavefronts propagate with amplitude |t|^{-3/4}. Velocities along the light cone experience dispersion at the faster rate |t|^{-5/6}. As time permits, we will examine dependence on the mass and propagation speed parameters, including the small mass (wave equation) and large mass (Schrodinger equation, after scaling) limits.

We shall first discuss a novel prior construction for Gaussian factor models, with applications in prediction and variable selection with high-dimensional correlated predictors. While Gaussian factor models can be easily generalized to accommodate binary, count and ordered categorical variables, they lead to challenging computation and complex modeling structure in case of unordered categorical data. We propose a novel class of simplex factor model as an alternative. Akin to Gaussian factor models, the joint pmf of the high-dimensional categorical observations factorizes conditional on fewer latent factors, leading to a parsimonious decomposition of the joint probability.

Our paper contributes to the literature of technology adoption. In most of these models it is assumed that after the arrival of a new technology the probability of the next arrival is constant. We extend this approach by assuming that after the last technology jump the probability of a new arrival can change. Right after the arrival of a new technology the intensity equals a specific value that switches if no new technology arrival has taken place within a certain period after the last technology arrival. We look at different scenarios, dependent on whether the firm is threatened by a drop in the arrival rate after a certain time period or expects the rate of new arrivals to rise. We analyze the effect of variance of time between two consecutive arrivals on the optimal investment timing and show that larger variance accelerates investment in a new technology in case the arrival rate increases if no new arrival takes place within a specific time period after the last arrival. For the case that the arrival rate is supposed to decrease, increasing variance has a non-monotonic effect on investment timing. We find that firms often adopt a new technology a time lag after its introduction, which is a phenomenon frequently observed in practice. Regarding a firm's technology releasing strategy we explain why clear signals set by regular and steady release of new product generations stimulates customers buying behavior. Depending on whether the arrival rate is assumed to change or be constant over time, the optimal technology adoption timing changes significantly. In a further step we add an additional source of uncertainty to the problem and assume that the length of the time period after which the arrival intensity changes is not known to the firm in advance. Here, we find that increasing uncertainty accelerates investment, a result that is opposite to the standard real options theory. Joint work with with V. Hagspiel (from IST and Tilburg University) and K. Huisman (from ASML and Tilburg University)

The Lindemann-Weierstrass Theorem (concerning algebraic independence of values of the exponential function) is one of the most celebrated results in transcendental number theory. I'll discuss the historical development, sketch the proof, and talk about some of the related literature (for example, Mahler's effective version).

Higgs bundles can be viewed as certain reductions of flat bundles. The moduli space of Higgs bundles enjoys many geometric properties: it has a hyperKahler structure; moreover, it is an integrable system called the Hitchin system. I will describe the fibers of the Hitchin's Hamiltonians in the cases when and . Time permitting, I will explain how to understand the hyperKahler structure from the point of view of nonabelian Hodge theory.

I'll review the basic elements of Seiberg & Witten's solution of N=2 SUSY gauge theories and discuss ways in which one may want to test their proposal.

I will present two algorithms for efficient Bayesian inference: online variational Bayes (VB) and the no-U-turn sampler (NUTS). Online VB incrementally fits an approximation to the posterior, considering only a subset of the full dataset at each iteration. When applied to the latent Dirichlet allocation model, online VB is able to discover a set of topics from millions of Wikipedia documents in a fraction of the time needed by batch algorithms. NUTS is an MCMC algorithm that extends the Hamiltonian Monte Carlo (HMC) algorithm.

The thin obstacle problem can be reformulated in terms of minimizing a new functional. A two-phase problem then naturally arises which has applications in heat control at the boundary of a domain. This is analagous to the two phase classical obstacle problem which has applications in heat control through the interior of a domain. We study both the regularity of the solutions and the free boundaries.

Projective geometry contains many copies of some of our favorite geometries. For example spherical, Euclidean and hyperbolic geometry are all subgeometries. For this reason it is an ideal setting to examine various deformation phenomena. After reviewing some elementary projective geometry we will see how it helps us to understand various types of deformations.

We will define the finite descent obstructions for existence of integral points on varieties over global fields and talk about some recent results on their sufficiency.

Dr. Ben Rhodes will lead a discussion of various problems. Some of the problems are Math Counts problems that have connections to college level courses and current topics of research. Some have an interesting history. We will try to make connections with problems considered in prior meetings this year. Join us to think about these problems and share your ideas. (A word of advice from the speaker: “The more you participate, the less you will have to listen to my stories and experiences.”)

At the beginning of the '90, DiPerna and Lions studied in detail the connection between transport equations and ordinary differential equations. In particular, by proving an existence and uniqueness result at the level of the transport equation, they obtained (roughly speaking) existence and uniqueness of solutions for ODEs with Sobolev vector-fields for a.e. initial condition. Ten years later, Ambrosio has been able to extend such a result to BV vector fields. In some recent works we have investigated this theory in a more general setting, which allows us to show the semiclassical convergence of the quantum dynamics to the Liouville dynamics for the linear Schrodinger equations, under very weak regularity assumptions on the potential. In analogy to the classical DiPerna-Lions' theory, the price to pay for allowing singular potential is that the convergence result holds true only for "a.e. initial data", where "a.e." is with respect to a suitable family of reference measures in the space of the initial data. The aim of this talk is to give an overview of these results.

I will show that every knot bounds at least one Seifert surface, and use this fact to define the genus of a knot. Then I'll show that the genus is additive and use this to construct countably many nonequivalent knots. There are no prerequisites although it may be easier to understand for someone who is familiar with the first semester of topology.

In this talk we will present a study on the kernel functions of the Dirichlet-­‐Neumann maps for dissipative systems in a half space. We start from the consideration of the Green’s function for an initial-­‐boundary value problems for linear dissipative systems. With the fundamental solutions of the dissipative systems, one can reduce the initial-­‐boundary value problems into boundary value problems so that the well-­‐posedness of the system gives linear algebraic systems over the polynomials in the Fourier and Laplace variables for the Dirichlet-­‐Neumann datum at the boundary, where Fourier variables are in the directions of boundary, and the Laplace is for the time variable. In order to invert the Dirichlet-­‐Neumann map from the transformation variables to the space-­‐time variables we introduce a path, which contains the spectral information of the systems, in the complex plan for the time Laplace variable. On this path, the Laplace-­‐Fourier variables can be recombined, through the Cauchy’s complex contour integral, into a form resemble to that for a whole space problem. Thus, the classical results for the whole space problem can be used to obtain the pointwise spae-­‐time structure for long wave components of the kernel function of the Dirichlet-­‐Neumann map for points within a finite Mach region. We also apply direct energy estimates to yield the pointwise structure of the kernel functions in any high Mach number region. Finally, we have obtained exponentially sharp estimates for the kernel function in the space-­‐time variables. For example, the kernel functions for both D’Alermbert wave equation with dissipation and a linearized compressible Navier-­‐Stokes equation can be expressed explicitly in space-­‐time variables with errors which decay exponentially in both space-­‐time variables. This gives a globally quantitative and qualitative wave propagations at boundary.

We introduce a new and general set of identifiability conditions for factor models which handles the ordering problem associated with current common practice. In addition, the new class of parsimonious Bayesian factor analysis (PBFA) leads to a factor loading matrix representation which is an intuitive and easy to implement factor selection scheme. The PBFA is applied to the 1970 British Cohort Study within a life course framework to analyze the effect of child cognition, mental and physical health on education and adult economic and health outcomes in a model where individuals select into education on the basis of their gains.

An L-space is a rational homology 3-sphere Y whose Heegaard Floer homology has rank equal to |H^2(Y;Z)|, the smallest rank possible. We call Y a strong L-space if it admits a Heegaard splitting whose associated Heegaard Floer chain complex has exactly |H^2(Y;Z)| generators -- an elementary combinatorial condition that does not require any knowledge of Floer theory. I will show that if Y is a strong L-space, then the fundamental group of Y is not left-orderable, a special case of a conjecture of Boyer, Gordon, and Watson. The branched double cover of any alternating link in S^3 is a strong L-space, and it is tempting to conjecture that the converse holds as well. I will discuss the evidence for this conjecture and for strategies for classifying strong L-spaces. This is joint work in progress with Sam Lewallen and Josh Greene.

In this talk we will discuss the classical definition of the Brauer group, Br(k), of a field k and relate it with its definition as H^2 of the multiplicative group of the separable closure of k. We will then look at examples of different kinds of fields and their Brauer groups, and briefly discuss the connections Br(k) has to class field theory and algebraic geometry.

In this talk I'll continue last week's discussion of the moduli space of stable Higgs bundles over a smooth projective curve. In particular, I'll give a more detailed construction of the Hitchin fibration, explaining how its fibers can be described concretely via the spectral cover construction. Although I'll initially focus on the case of vector bundles (i.e. structure group ), following a paper by Hausel and Thaddeus I'll also describe the case of the (Langlands dual) structure groups and . In this case one obtains "dual" special Lagrangian fibrations, giving a direct connection between Langlands duality and the SYZ picture of mirror symmetry.

We present the classical theorem of Stallings that gives a simple condition on the fundamental group of a 3-manifold which determines whether the manifold fibers over the circle. The proof makes use of standard 3-manifolds arguments as well as introduces ideas that have become central in geometric group theory. The talk will be almost entirely self-contained and should be accessible to everyone.

Due to rapid technological advances, various types of genomic, epigenomic, transcriptomic and proteomic data with different sizes, formats, and structures have become available. Using genome continuum models as a general principle we present a class of Bayesian methods to model these genomic profiles using functional data analysis approaches. Our methods allow for simultaneous characterization of these high-dimensional functions using non-parametric basis functions, joint modeling of spatially correlated functional data and detection of local features in spatially heterogeneous functional data.

Discrete images, composed of patches of slowly-varying pixel values, have sparse or compressible representations with respect to their discrete derivative. Although image compression is a primary motivation for compressed sensing, stability results for total-variation minimization do not follow directly from the standard theory. In this talk, we present near-optimal reconstruction guarantees for total-variation minimization using properties of the bivariate Haar transform; we end by discussing several related open problems. This is joint work with Deanna Needell.

The outer automorphism group Out(F) of a non-abelian free group F of finite rank shares many properties with linear groups and the mapping class group Mod(S) of a surface, although the techniques for studying Out(F) are often quite different from the latter two. Motivated by analogy, I will present some results about Out(F) previously well-known for the mapping class group, and highlight some of the features in the proofs which distinguish it from Mod(S).

Homogeneous dynamics is another name for the theory of flows on homogeneous spaces, or homogeneous flows. The study of homogenous flows has been attracting considerable attention for the last 40-50 years. During the last three decades, it has been realized that some problems in number theory and, in particular in Diophantine approximation, can be solved using methods from the theory of homogeneous flows. The purpose of the first lecture is to give examples of interactions between number theory and homogenous dynamics; mostly only formulations will be given, but there will be also very brief description of some proofs.

TBA

The left tail of the implied volatility skew, coming from quotes on out-of-the-money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations (BSDEs), to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear PDE and provide a small time-to-maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parametrized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data. Joint work with Stephan Sturm (Princeton University).

Thurston showed the complement of the figure eight knot in S^3 was hyperbolic by explicitly decomposing it into two tetrahedra. I will describe this construction to get a finite-volume hyperbolic 3-manifold with a cusp. If I have time, I will discuss the cusp shape and the fundamental group of the manifold as a subgroup of isometries of H^3. No background necessary.

The second and third lectures will be the continuation, on a more technical level, of the first lecture. Topics to be (very briefly) covered include: (1) recurrence to compact sets and Diophantine approximation on manifolds; (2) orbit closures, the Oppenheim conjecture and the Littlewood conjecture; (3) classification of ergodic invariant measures and equidistribution; (4) quantitative Oppenheim conjecture and counting of integral points on homogenous varieties; (5) effective results in homogeneous dynamics.

Much work by many authors has revealed the densest lattices (discrete subgroups with minimal-volume quotient) in many Lie groups. We solved this problem for PGL3(Q2) while working on a problem in algebraic geometry, the explicit construction of fake projective planes. Here Q2 is the 2-adic completion of the rational numbers. We showed that a lattice in PGL3(Q2) that Mumford had constructed is in fact densest possible, and that there is exactly one more of that density. Neither lattice is an obvious one to write down, and the key parts of the proofs are geometric, building fundamental domains in the 2-dimensional building on which PGL3(Q2) acts. (Joint work with Fumiharu Kato.)

According to a classical theorem of Bieberbach, if G is a discrete subgroup of the group of isometries of R^n then G contains a subgroup of finite index consisting of parallel translations. An analogous statement for discrete groups of affine transformations is not true. Nevertheless there is Auslander's conjecture which atates that if G is a subgroup of the group of affine transformations of R^n which properly discontinuously and cocompactly on R^n then G contains a solvable subgroup of finite index. I will describe some partial results related to this conjecture and will also talk about what happens if the assumption of cocompactness of R^n/G is dropped.

Singular symplectic varieties and their resolutions of singularities lie at the crossroads of algebraic and symplectic geometry, representation theory, and integrable systems. Central examples include the nilpotent cone of a complex semisimple Lie algebra and its resolution by the cotangent bundle of the flag variety (the Springer resolution); the nth symmetric product of the affine plane and its resolution by the Hilbert scheme of points; and a Kleinian surface singularity and its minimal resolution. A singular variety and its resolution never have equivalent geometry (as encoded, for example, in their derived categories). Replacing a symplectic variety by a quantization, however---an algebro-geometric analog of passing to a Fukaya-type category---one miraculously finds that such equivalences are common. I'll discuss singular symplectic varieties and their resolutions, examples, quantization, and a general criterion for such geometric equivalences that extends classical results (for example, Beilinson-Bernstein localization). Time permitting, I'll also discuss some additional features of these quantizations that parallel emerging structures in the (much more complicated) world of Fukaya categories. This is based on joint work with K. McGerty.

The goal of this talk is to familiarize the audience with some basic properties of quadratic forms on fields of characteristic different from 2. I will discuss results such as Witt's theorem and Hasse Minkowski, and briefly recall some classical theorems on sums of squares.

Just like the Poincare group, (a cover of) the isometry group of affine Minkowski space, is the symmetry group for classical theories, the Super Poincare group is the symmetry group for super spacetime, and thus plays an important role in super-symmetric quantum mechanics. We will discuss the construction of the super Poincare group by first describing Clifford algebras and spinors in signature metrics and then construct the super Poincare group and its group law via its algebra.

I will finish describing the "easy" part of the AGT conjecture, which gives a rather unexpected relation between moduli spaces of instantons in four dimensions and the Virasoro algebra.

Using reverse phase protein arrays (RPPA) we measure time-course protein expression for a set of selected markers that are known to co-regulate biological functions in a pathway structure. To accommodate the complex dependent nature of the data, including temporal correlation and pathway dependence for the protein markers, we propose a mixed effects model with temporal and protein-specific components. We develop a sequence of random probability measures (RPM) to account for the dependence in time of the protein expression measurements.

The second and third lectures will be the continuation, on a more technical level, of the first lecture. Topics to be (very briefly) covered include: (1) recurrence to compact sets and Diophantine approximation on manifolds; (2) orbit closures, the Oppenheim conjecture and the Littlewood conjecture; (3) classification of ergodic invariant measures and equidistribution; (4) quantitative Oppenheim conjecture and counting of integral points on homogenous varieties; (5) effective results in homogeneous dynamics.

We shall discuss the number theoretic ingredients that go into the proof of equidistribution of mass for classical Hecke eigenforms. This is joint work with Kannan Soundararajan. We shall also present a revision to the argument by Henryk Iwaniec.

We are developing a two-scale mortar iterative method for problems in porous media. We iterate the fine scale mortar by PCG and use coarse mortar to damp out the smooth error. The numerical examples show the condition number is O( log(H/h) ).

This will be a presentation of our recent results joint with J. Friedlander, B. Mazur and K. Rubin, which concern a new symbol associated to ideals in Galois number fields, the spin. The spin takes two values +1, -1 and the question is, how often does it change over prime ideals. The answer is (in case of cubic fields) it assumes each value half of the time. The result has applications for asymptotic distribution of the rank of 2-Selmer group of families of elliptic curves.

In the classical Obstacle Problem, one has the added luxury of having an obstacle that is apriori very smooth. There are situations where this assumption has to be dropped and the most one can hope for is some weaker notion of concavity or convexity. In this talk I hope to present some tools to understand obstacle problems with implicit constraints, when the obstacle depends on the solution you are trying to find! This will lead us into the realm of subdifferentials and nonlinear elliptic PDE theory. I hope to show how some of these tools are used, and why one should care about implicit constraint obstacle problems.

A contact manifold with boundary naturally gives rise to a sutured manifold, as defined by Gabai. Honda, Kazez and Matic have used this relationship to define an invariant of contact manifolds with boundary in sutured Floer homology, a Heegaard Floer invariant of sutured manifolds developed by Juhasz. More recently, Kronheimer and Mrowka have defined an invariant of sutured manifolds in the setting of monopole Floer homology. In this talk, I'll describe the construction of an invariant of contact manifolds with boundary in sutured monopole Floer homology. If time permits, I'll talk about analogues of Juhasz' sutured cobordism maps and the Honda-Kazez-Matic gluing maps in the monopole setting. This is joint work with Steven Sivek.

Understanding the classification and properties of contact structures on a given 3-manifold is typically a difficult endeavor. Eliashberg classified all overtwisted contact structures, but classifications of tight contact structures has only been achieved for certain limited classes of 3-manifolds. Understanding whether there are symplectically or Stein fillable tight contact structures, adds another layer of difficulty. In this talk, we will discuss the existence of fillable contact structures in the case where the 3-manifolds are Seifert-fibered spaces (results due to Gompf and Lecuona-Lisca). We will also discuss examples due to Lisca and Stipsicz of Seifert fibered 3-manifolds with arbitrarily many non-isotopic tight contact structures, none of which are fillable. This is a candidacy talk.