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A live feed of seminars and special events in the upcoming week.

- April 1st, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**Fiber-bunched cocycles: cohomology and periodic data.**Speaker:**Victoria Sadovskaya, Penn State**Location:**MB114We consider Holder continuous fiber-bunched GL(d,R)-valued cocycles over a hyperbolic diffeomorphism. We show that if two cocycles have equal periodic data, then they are Holder continuously cohomologous. We obtain similar conclusions under weaker assumptions on the periodic data. We apply these results to the question when a hyperbolic toral automorphism is smoothly conjugate to a C^1-small perturbation.

- April 2nd, 2013 (02:30pm - 03:30pm)
**Seminar:**GAP Seminar**Title:**Geometry and tensor networks**Speaker:**Jason Morton, Penn State**Location:**MB106Tensor networks -- or more generally, diagrams in monoidal categories with various additional properties -- arise constantly in applications, particularly those involving networks used to process information in some way. Aided by the easy interpretation of the graphical language, they have played an important role in computer science, statistics and machine learning, and quantum information and many-body systems. Tools from algebraic geometry, representation theory, and category theory have recently been applied to problems arising from such networks. Basic questions about each type of information-processing system, such as what probability distributions or quantum states can be represented, turn out to lead to interesting problems in algebraic geometry, representation theory, and category theory.

- April 2nd, 2013 (02:30pm - 03:45pm)
**Seminar:**Logic Seminar**Title:**Implicit definability in arithmetic, part 3.**Speaker:**Stephen G. Simpson, Pennsylvania State University**Location:**MB315In this talk I present a new proof of Harrington's theorems concerning implicit definability over the integers. Unlike previous proofs, the new proof is easy and does not involve priority arguments. The theorems read as follows. (1) There exist two sets of integers, each of which is implictly definable yet neither of which is explicitly definable from the other. (2) There exists a set of integers which belongs to an implicitly definable countable set of sets but is not by itself implicitly definable.

- April 2nd, 2013 (03:30pm - 06:00pm)
**Seminar:**Working Seminar: Dynamics and its Working Tools**Title:**Algebraic K-theory and its applications to dynamics, III**Speaker:**Kurt Vinhage, Penn State**Location:**MB216- April 2nd, 2013 (03:30pm - 04:30pm)
**Seminar:**CCMA PDEs and Numerical Methods Seminar Series**Title:**Cascadic Multilevel Algorithms for Saddle Point Systems**Speaker:**Constantin Bacuta, Department of Mathematical Sciences , University of Delaware**Location:**MB114Based on the cascade principle, we consider multilevel type of algorithms for discretizing saddle point problems. The finite element spaces are associated with successively finer and finer grids. On each fixed level a standard Uzawa, gradient Uzawa, or conjugate gradient Uzawa method is implemented. The algorithms are designed to perform more iterations on the coarse grids and fewer on the fine grids. The level change criteria is based on keeping the iteration error close to expected discretization error. To decrease the running time, the iteration on each new level starts with the best approximation from the previous level. Numerical results supporting the efficiency of the algorithms are presented for the Stokes system. We use our general theory to introduce/review the ``saddle point least-squares'' method and relate it with the Bramble-Pasciak's least square approach. As a consequence, we design a least-squares iterative solver for the time harmonic Maxwell equations. This is joint work with Francisco Sayas, and Lu Shu.

- April 2nd, 2013 (05:00pm - 06:30pm)
**Seminar:**SIAM Student Chapter Seminar**Title:**Tensor Product and its Applications in Physical Algorithms**Speaker:**Yufei Shen, Pennsylvania State University**Location:**MB106Quantum theory, in essence, is solving a differential equation called Schrödinger's Equation. For a strongly correlated system, this differential equation is notoriously hard to solve since the eigen-space becomes too big even for supercomputers. One method to solve it numerically is called Time Evolving Block Decimation (TEBD), which use Matrix Product State (MPS) to represent the space of eigenvectors. This method, by embracing quantum entanglement in the approximation, only keeps those eigenstates which are physically significant, thus greatly reduces computational cost. I will also show some results of this method by solving the Bose-Hubbard Hamiltonian.

- April 3rd, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**Growth estimates for orbits of self-adjoint groups**Speaker:**Pat Eberlein, UNC**Location:**MB114Let G denote a closed, connected, noncompact subgroup of GL(n,R) that is self-adjoint (invariant under the transpose operation). Let d_{R} denote the right invariant Riemannian distance function defined by the Euclidean inner product on M(n,R). We study a kind of metric entropy for group actions in which the real numbers R are replaced by a Lie group G and Diff(R^{n}) is replaced by GL(n,R). More specifically, for a fixed vector v in R^{n} we obtained algebraically defined upper and lower bounds L^{-}(v) and L^{+}(v) for the asymptotic growth rate of the function g → log |g(v)|/d_{R}(g,G_{v}) as d_{R}(g, G_{v}) → \infty, where G_{v} denotes the subgroup of G that fixes v. If G_{v} is compact, then we may replace d_{R}(g, G_{v}) by d_{R}(g,Id) in the result above. We compute L^{-}(v) and L^{+}(v) for the irreducible, real, linear representations of G = SL(2,R). If the dimension of the G-module V is odd, then L^{_}(v) = L^{+}(v) for all v in a nonempty open subset O of V. In particular, the function g → log |g(v)|/d_{R}(g,G_{v}) has a limit for vectors v in O as d_{R}(g, G_{v}) → \infty. These examples show that the functions L^{-}(v) and L^{+}(v) are SO(2,R)-invariant but not SL(2,R) invariant.

- April 3rd, 2013 (04:40pm - 05:30pm)
**Seminar:**Applied Algebra Seminar**Title:**Polyhedral combinatorics of conformal blocks and fusion algebras**Speaker:**Chris Manon, U.C. Berkeley**Location:**MB106**Abstract:**http://math.berkeley.edu/~manonc/Research.htmlFor a simple complex Lie algebra $\mathfrak{g}$ and a non-negative integer $L$, the fusion algebra or Verlinde algebra $\mathcal{F}_L(\mathfrak{g})$ is an elegant finite dimensional algebra which encodes the dimensions of the spaces of partition functions for the Wess-Zumino-Witten model of conformal field theory. When $\mathfrak{g} = sl_m(\\mathbb{C})$, this algebra also makes an appearance as the small quantum cohomology ring of the Grassmannian variety $Gr_m(\mathbb{C}^{m+L}).$ We will describe what we have been calling a polyhedral presentation of this algebra for $sl_2(\mathbb{C})$ and $sl_3(\mathbb{C})$, and how such a presentation is given by combinatorics related to moduli spaces of vector bundles of rank $2$ and rank $3$. We will further explain how such a presentation for $sl_m(\mathbb{C})$ with $m > 3$ would be related to questions about moduli of higher rank vector bundles. Time permitting, we will also give some remarks on how these constructions are related to phylogenetics.

- April 4th, 2013 (10:00am - 10:50am)
**Seminar:**Hyperbolic and Mixed Type PDEs Seminar**Title:**Existence and Stability of Traveling Waves for an Integro-differential Equation for Slow Erosion**Speaker:**Wen Shen, Penn State**Location:**MB216We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order non-linear conservation law where the flux function includes an integral term. We show that there exist unique traveling wave solutions that connect profiles with equilibrium slope at $\pm\infty$. Such traveling waves take very different forms from those in standard conservation laws. Furthermore, we prove that the traveling wave profiles are locally stable, i.e., solutions with monotone initial data approaches the traveling waves asymptotically as $t\to+\infty$.

- April 4th, 2013 (11:15am - 12:05pm)
**Seminar:**Algebra and Number Theory Seminar**Title:**Small Height and Infinite Non-abelian Extensions**Speaker:**Philipp Habegger, IAS, Princeton**Location:**MB106The absolute, logarithmic Weil height is non-negative and vanishes precisely at 0 and at the roots of unity. Moreover, when restricted to a number field there are no arbitrarily small, positive heights. Amoroso, Bombieri, David, Dvornicich, Schinzel, Zannier and others exhibited many infinite extensions of the rationals with a height gap. For example, the maximal abelian extension of any number field has this property. To see a non-abelian example, let E be an elliptic curve defined over the rationals without complex multiplication. The field K generated by all complex points of E with finite order is an infinite extension of the rationals. Its Galois group contains no commutative subgroup of finite index. In the talk, I will sketch a proof that K contains no elements of sufficiently small, positive height.

- April 4th, 2013 (02:30pm - 03:30pm)
**Seminar:**Noncommutative Geometry Seminar**Title:**Ghostbusting and property A**Speaker:**John Roe, Penn State**Location:**MB106I'll talk about recent work with Rufus Willett about "ghosts" in the Roe algebra, which gives a necessary and sufficient condition for the existence of (non-compact) ghosts.

- April 4th, 2013 (03:35pm - 04:25pm)
**Seminar:**Department of Mathematics Colloquium**Title:**Symplectic geometry of the moduli of ordinary differential equations**Speaker:**Leon Takhtajan, Stony Brook University**Location:**MB114I will review old and new results on the symplectic structure of 2nd order ordinary differential equations, its connection to quasi-fuchsian deformation spaces and Weil-Petersson geometry of moduli spaces. If time permits, I will explain the relation with the so-called conformal blocks of quantum Liouville theory.

- April 5th, 2013 (12:20pm - 01:30pm)
**Seminar:**CCMA Luncheon Seminar**Title:**Introduction to the regularity problem for the 3D Navier-Stokes and related equations**Speaker:**Alexey Cheskidov, University of Illinois at Chicago**Location:**MB114I will give a brief introduction to the regularity problem for the 3D Navier-Stokes equations and simpler models, such as the dyadic equations. Its relation to Onsager's conjecture will also be discussed.

- April 5th, 2013 (03:35pm - 04:25pm)
**Seminar:**Computational and Applied Mathematics Colloquium**Title:**On the regularity of the 3D Navier-Stokes equations in the largest critical space**Speaker:**Alexey Cheskidov, University of Illinois at Chicago**Location:**MB106In this talk I will discuss various well-posedeness and ill-posedness results for the 3D Navier-Stokes equations in the largest critical space. In particular, I will describe a recent norm inflation result for a hyperdissipative Navier-Stokes equations that occurs in critical and supercritical spaces even in the case where the global regularity is known. We will see how a new scaling arises that prevents us from proving a small initial data result in the largest critical space.

- April 8th, 2013 (03:30pm - 05:00pm)
**Seminar:**CCMA PDEs and Numerical Methods Seminar Series**Title:**Uniqueness questions for the Navier-Stokes equation in the hyperbolic setting**Speaker:**Magdalena Czubak, SUNY, Binghamton**Location:**MB106The solutions to the Navier-Stokes equation are unique in two dimensions in the Euclidean space. In this talk, we will consider the Navier-Stokes equation on two dimensional negatively curved manifolds and construct a family of non-unique solutions. The solutions we obtain have finite energy and dissipation and satisfy the global energy inequality. Then we will discuss possible ways to formulate the problem so the uniqueness can be restored. This is joint work with Chi Hin Chan.

- April 8th, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**The deterministic random walk**Speaker:**Omri Sarig, The Weizmann Institute for Science**Location:**MB114The simple random walk on Z can be simulated by picking randomly uniformly in the unit interval, iterating T(x)=2x mod 1, and moving the "walker" one step to the right every time the orbit lands at [0,1/2) and one step to the left every time the orbit lands at [1/2,1). What happens if we replace the chaotic T(x)=2x mod 1 by the zero entropy R(x)=x+\alpha mod 1 for \alpha irrational? The resulting process, called the "deterministic random walk", is again a recurrent zero drift walk -- but one that visits zero much more often. This is a joint work with Avila, Dolgopyat, and Duriev

- April 9th, 2013 (09:00am - 11:00am)
**Seminar:**Ph.D. Oral Comprehensive Examination**Title:**"Numerical simulation of solar cells"**Speaker:**Yicong Ma, Adviser: Jinchao Xu, Penn State**Location:**319 HHD-East**Abstract:**http://Solar energy plays an important role nowadays and can make considerable contributions to solving some of the most urgent problems the world now faces. Its efficiency is usually measured by the so-called power conversion efficiency, which mainly depends on the nano-structure of unit solar cell. Instead of time-consuming and expensive experiments, numerical simulation plays a crucial role in designing unit solar cell to improve the power conversion efficiency with much lower costs. In the simulation of solar cells, we need to solve Maxwell equation with complex and jump coefficients, and the resulting linear systems are highly indefinite and challenging. In this talk, I will introduce the basic physical model of the optical process of the unit solar cell. We use finite element method to discrete the model problem and use efficient linear solvers based on Hiptmair-Xu preconditioner to solve the linear system from finite element discretization. Preliminary test results will be presented.

- April 9th, 2013 (11:15am - 12:05pm)
**Seminar:**Combinatorics/Partitions Seminar**Title:**Moments of Askey-Wilson polynomials**Speaker:**Jang Soo Kim, University of Minnesota**Location:**MB106The Askey-Wilson polynomials are the most general orthogonal polynomials among those classified by the Askey scheme. These are orthogonal polynomials in one variable with 5 parameters. In this talk I will talk about 3 combinatorial methods to study the nth moment of the Askey-Wilson polynomials. The first method is Viennot's theory of weighted Motzkin paths. The second method uses staircase tableaux introduced by Corteel and Williams. The third method is a modification of an idea of Ismail, Stanton, and Viennot on matchings and q-Hermite polynomials. Using the third method we express the nth moment as a fraction of two generating functions for certain matchings and obtain a new formula for the moment. This is joint work with Dennis Stanton.

- April 9th, 2013 (02:30pm - 03:30pm)
**Seminar:**GAP Seminar**Title:**Gamma factors and normalising distributions I**Speaker:**Pierre Clare, Penn State University**Location:**MB106We will present some classical features of L-functions and gamma-factors in relation with the representation theory of reductive groups. Our goal will be to describe some strong similarities observed between the convolution picture for gamma-factors of Braverman and Kazhdan and normalising distributions for C*-algebraic universal principal series.

- April 9th, 2013 (02:30pm - 03:45pm)
**Seminar:**Logic Seminar**Title:**The random graph and its properties.**Speaker:**John Pardo, Pennsylvania State University**Location:**MB315In graph theory, the probabilistic method of constructing countable graphs with randomly determined edges has been used to prove significant results where traditional methods have failed. Interestingly, these sorts of graphs have also appeared in other areas of mathematics using deterministic means, in particular in logic as the Fraisse limit of a certain class of graphs, and have been found to have some very fascinating properties. In my talk I will explain both the probabilistic and deterministic constructions as well as some of those fascinating properties, in the process showing that the random graph may not be as random as it first appears.

- April 9th, 2013 (03:30pm - 06:00pm)
**Seminar:**Working Seminar: Dynamics and its Working Tools**Title:**Transfer Operator method, I**Speaker:**Omri Sarig, Weizmann Institute of Science, Rehovot, Israel**Location:**MB216Suppose T is a non invertible expanding map preserving a measure m. The action of T on the points of the space induces an action on the space of "mass densities" f dm. This action is called the transfer operator, and it can be viewed as an operator on L^1 (the space of integrable signed densities f). As it turns out, the more chaotic the behavior of T, the better is the behavior of the transfer operator. This observation is the starting point for a collection of methods for analyzing the ergodic and stochastic properties of m by a studying the operator theoretic properties of the transfer operator. We will develop the basic theory and explore some of the applications, such as decay of correlations and (time permitting) the central limit theorem.

- April 10th, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**TBA**Speaker:**Bruce Kleiner, NYU**Location:**MB114- April 11th, 2013 (11:15am - 12:05pm)
**Seminar:**Algebra and Number Theory Seminar**Title:**The distribution of the variance of primes in arithmetic progressions**Speaker:**Daniel Fiorilli, University of Michigan**Location:**MB106Gallagher's refinement of the Barban-Davenport-Halberstam states that V(x;q), the variance of primes up to x in the arithmetic progressions modulo q, is at most x log q, on average over q in the range x/(log x)^A < q < x. It was then discovered by Montgomery that in this range V(x;q) is actually asymptotic to x log q (on average over q); his result was refined by a long list of authors including Hooley, Goldston and Vaughan, and Friedlander and Goldston. Tools used in these papers include the circle method and divisor switching techniques, and under GRH and a strong from of the Hardy-Littlewood Conjecture it is now known that V(x;q) is asymptotic to x log q in the range x^{1/2+o(1)} < q< x. While it is not clear that the asymptotic should hold for more moderate values of q, Keating and Rudnick have proven an estimate for the function field analogue of V(x;q) which suggests that this range could be extended to x^{o(1)} < q< x. In this talk we will show how one can use probabilistic techniques to give evidence that V(x;q) should be asymptotic to x log q in the even wider range (log log x)^{1+o(1)} < q < x, and that this range is best possible.

- April 11th, 2013 (01:25pm - 02:15pm)
**Seminar:**Algebra and Number Theory Seminar**Title:**The distribution of the variance of primes in arithmetic progressions**Speaker:**Daniel Fiorilli, University of Michigan**Location:**MB114**Abstract:**http://Gallagher's refinement of the Barban-Davenport-Halberstam states that V(x;q), the variance of primes up to x in the arithmetic progressions modulo q, is at most x log q, on average over q in the range x/(log x)^A < q < x. It was then discovered by Montgomery that in this range V(x;q) is actually asymptotic to x log q (on average over q); his result was refined by a long list of authors including Hooley, Goldston and Vaughan, and Friedlander and Goldston. Tools used in these papers include the circle method and divisor switching techniques, and under GRH and a strong from of the Hardy-Littlewood Conjecture it is now known that V(x;q) is asymptotic to x log q in the range x^{1/2+o(1)} < q< x. While it is not clear that the asymptotic should hold for more moderate values of q, Keating and Rudnick have proven an estimate for the function field analogue of V(x;q) which suggests that this range could be extended to x^{o(1)} < q< x. In this talk we will show how one can use probabilistic techniques to give evidence that V(x;q) should be asymptotic to x log q in the even wider range (log log x)^{1+o(1)} < q < x, and that this range is best possible.

- April 11th, 2013 (02:30pm - 03:30pm)
**Seminar:**Noncommutative Geometry Seminar**Title:**Gamma factors and normalising distributions II**Speaker:**Pierre Clare, Penn State University**Location:**MB106We will present some classical features of L-functions and gamma-factors in relation with the representation theory of reductive groups. Our goal will be to describe some strong similarities observed between the convolution picture for gamma-factors of Braverman and Kazhdan and normalising distributions for C*-algebraic universal principal series.

- April 12th, 2013 (12:20pm - 01:30pm)
**Seminar:**CCMA Luncheon Seminar**Title:**Some puzzles about Couette flow**Speaker:**Zhiwu Lin, Georgia Tech Mathematics**Location:**MB114Couette flows are shear flows with a linear velocity profile between two plates. It is one of the simplest butÂ important laminar flows. We will discuss two unresolved issues about them. First, Couette flows are known to be linearly stable for any Reynolds number, but become turbulent for large Reynolds numbers by both experimental and numerical results. We will discuss several attemps to explain the nature of transient turbulence near Couette. Second, starting from the work of Orr in 1907, the velocity of the linearized inviscid equation at Couette flows is known to decay in time. But the nonlinear inviscid damping is still open. The mechanism of such inviscid damping is important to understand the appearance of coherent structures in 2D turbulence.

- April 12th, 2013 (02:20pm - 03:20pm)
**Seminar:**Seminar on Probability and its Application**Title:**Almost sure central limit theorems**Speaker:**Lucia Tabacu, PSU**Location:**MB106We give an overview of the almost sure central limit theorems and show their connection to the classical central limit theorems. We also introduce a new result, the almost sure central limit theorem for linear rank statistics.

- April 12th, 2013 (03:35pm - 04:25pm)
**Seminar:**Computational and Applied Mathematics Colloquium**Title:**Nonlinear Landau damping**Speaker:**Zhiwu Lin, Georgia Tech Mathematics**Location:**MB106Consider electrostatic plasmas described by Vlasov-Poisson equation with a fixed ion background. In 1946, Landau discovered the linear decay of electric field near a stable homogeneous state. The nonlinear Landau damping was recently proved under analytic perturbations by Mouhot and Villani, but for general perturbations the problem is still largely open. With Chongchun Zeng, we construct nontrivial traveling waves (BGK waves) with any spatial period which are arbitrarily near any homogeneous state in H^s (s<3/2) Sobolev norm of the distribution function. Therefore, the nonlinear Landau damping is not true in H^s (s<3/2) spaces. We also showed that in small H^s (s>3/2) neighborhoods of linearly stable homogeneous states, there exist no nontrivial invariant structures. Our results suggest that for subcritical perturbations (s<3/2) the nonlinear trapping effect cannnot be ignored even in the limit of small amplitude; for supercritical perturbations (s>3/2), such trapping effect might have no influence on the long time dynamics and the nonlinear damping is hopeful. Similar results were also obtained for the problem of nonlinear inviscid damping of Couette flow, for which the linear decay was first observed by Orr in 1907.

- April 15th, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**Invisibility and retro-reflection in billiards**Speaker:**Alexandre Plakhov, University of Aveiro, Portugal and Institute for Information Transmission Problems, Russia**Location:**MB114We consider the problem of invisibility for bodies with mirror surface within the scope of geometrical optics. The problem amounts to studying billiards in the exterior of bounded regions. Examples of bodies invisible in 1, 2, and 3 directions and bodies invisible from 1 and 2 points are provided in the talk. It is proved that there do not exist bodies invisible in all directions. The question of maximum number of directions and/or points of invisibility of a body remains open. The duality between invisibility (unperturbed billiard trajectories outside a bounded domain) and periodic billiard trajectories inside the domain is also discussed. Further, we consider retro-reflecting bodies with mirror surface. A body is called a perfect retroreflector, if the direction of any beam of light incident on it is changed to the opposite. We provide several examples of asymptotically retro-reflecting sequences of bodies. On the other hand, it is not known if there exist perfect billiard retroreflectors.

- April 16th, 2013 (09:00am - 11:00am)
**Seminar:**Ph.D. Oral Comprehensive Examination**Title:**"Aggregation based AMG on weighted Graph Laplacian problems"**Speaker:**Fei Cao, Adviser: James Brannick, Penn State**Location:**23 McAllister BuildingIt has been known that graph laplacian problem plays an important role in large-scale computational applications such as semi-supervised machine learning, spectral clustering of images, genetic data and web pages, transportation network flows, and electrical resistor circuits. While algebraic multigrid method (AMG) is well-known as "black box" solver method. In this talk we propose to discuss variants of algebraic multigrid for solving (weighted) graph Laplacian problems defined on general graphs. The multigrid solvers we intend to develop are aggregation-based methods that will be designed to use the geometry from the problem at hand to select the coarse variables (whenever such information is available, e.g., when finite element, finite volume, and finite difference discretizations of 2d and 3d anisotropic diffusion problems are considered) and, then, use algebraic techniques to construct the corresponding interpolation operators.

- April 16th, 2013 (11:00am - 01:00pm)
**Seminar:**Ph.D. Thesis Defense**Title:**"Optimal Pricing Strategies in a Limit Order Book"**Speaker:**Giancarlo Facchi, Adviser: Alberto Bressan, Penn State**Location:**011 Ag Science Industries**Abstract:**http://We study a continuum model of the limit order book, viewed as a noncooperative game for n players. An external buyer asks for a random amount X > 0 of a given asset. This amount will be bought at the lowest available price, as long as the price does not exceed a given upper bound P . One or more sellers offer various quantities of the asset at different prices, competing to fulfill the incoming order, whose size is not known a priori. Depending on the probability distribution of X , we prove the existence (or non-existence) of a unique Nash equilibrium. In the positive case, the optimal pricing strategies of the various agents are explicitly determined.

- April 16th, 2013 (11:15am - 12:05pm)
**Seminar:**Combinatorics/Partitions Seminar**Title:**A periodic approach to plane partition congruences**Speaker:**Matt Mizuhara, PSU**Location:**MB106Ramanujan's celebrated congruences of the partition function p(n) have inspired a vast amount of results on various unrestricted and restricted partition functions. In 1989 Kwong characterized several cases of periodicity of rational polynomial functions modulo prime powers. These results yield an alternative approach to prove known congruences on restricted plane partitions as well as prove a new congruence.

- April 16th, 2013 (02:30pm - 03:30pm)
**Seminar:**GAP Seminar**Title:**Hermitian Variations of Hodge Structure of Calabi-Yau type**Speaker:**Radu Laza, Stony Brook University**Location:**MB106Except a few special cases (e.g. abelian varieties and K3 surfaces), the images of period maps for families of algebraic varieties satisfy non-trivial Griffiths' transversality relations. It is of interest to understand these images of period maps, especially for Calabi-Yau threefolds. In this talk, I will discuss the case when the images of period maps can be described algebraically. Specifically, I will show that if a horizontal subvariety Z of a period domain D is semi-algebraic and it is stabilized by a large discrete group, then Z is automatically a Hermitian symmetric domain with a totally geodesic embedding into the period domain D. I will then discuss the classification of the semi-algebraic cases for variations of Hodge structures of Calabi-Yau type, with a special emphasis on the classification over Q (which is partially based on earlier work of Zarhin). This is joint work with R. Friedman.

- April 16th, 2013 (02:30pm - 03:45pm)
**Seminar:**Logic Seminar**Title:**The logic of graph decompositions.**Speaker:**Stephen Flood, Pennsylvania State University**Location:**MB315The theory of simplicial graph decompositions studies the infinite graphs that can be built using a sequence of irreducible graphs which are attached together at complete subgraphs. We study the logical strength required to prove the area's "existence theorems," which say that certain classes of graphs admit such a decomposition. We will discuss the strength of these existence theorems from the perspective of reverse mathematics and computability theory. In addition, we will give upper and lower bounds on the possible ordinal lengths of prime decompositions.

- April 16th, 2013 (03:30pm - 06:00pm)
**Seminar:**Working Seminar: Dynamics and its Working Tools**Title:**Transfer Operator method, II**Speaker:**Omri Sarig, Weizmann Institute of Science, Rehovot, Israel**Location:**MB216Suppose T is a non invertible expanding map preserving a measure m. The action of T on the points of the space induces an action on the space of "mass densities" f dm. This action is called the transfer operator, and it can be viewed as an operator on L^1 (the space of integrable signed densities f). As it turns out, the more chaotic the behavior of T, the better is the behavior of the transfer operator. This observation is the starting point for a collection of methods for analyzing the ergodic and stochastic properties of m by a studying the operator theoretic properties of the transfer operator. We will develop the basic theory and explore some of the applications, such as decay of correlations and (time permitting) the central limit theorem.

- April 16th, 2013 (04:00pm - 05:15pm)
**Title:***Private***Location:**MB106- April 17th, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**Problems of optimal resistance in Newtonian aerodynamics**Speaker:**Alexandre Plakhov, University of Aveiro, Portugal and Institute for Information Transmission Problems, Russia**Location:**MB114A rigid body moves in a rarefied medium of resting particles and at the same time very slowly rotates (somersaults). Each particle of the medium is reflected elastically when hitting the body boundary (multiple reflections are possible). The resulting resistance force acting on the body is time-dependent; we consider the time-averaged value of resistance. The problem is: given a convex body, find a roughening of its surface that minimizes or maximizes its resistance. (The problem includes mathematical definition of roughening.) This problem is solved using the methods of billiard theory and optimal mass transportation. Surprisingly, the minimum and maximum depend only on the dimension of the ambient Euclidean space and do not depend on the original body. In particular, the resistance of a 3-dimensional convex body can be decreased by (approx.) 3.05% at most and can be increased at most twice by roughening.

- April 18th, 2013 (10:00am - 10:50am)
**Seminar:**Hyperbolic and Mixed Type PDEs Seminar**Title:**On total variation bounds for inviscid, isentropic flow**Speaker:**Kris Jenssen, Penn State**Location:**MB216Abstract: We will describe a recent result with Geng Chen ruling out certain types of variation bounds for isentropic Euler flow. We also comment on simple examples showing arbitrary magnification of variation.

- April 18th, 2013 (11:15am - 12:05pm)
**Seminar:**Algebra and Number Theory Seminar**Title:**Multiplicative and Inhomogeneous Diophantine Approximation**Speaker:**Sanju Velani, University of York**Location:**MB106**Abstract:**http://www.math.psu.edu/rvaughan/PennStateNT2013.pdf- April 18th, 2013 (02:30pm - 03:20pm)
**Seminar:**PMASS Colloquium**Title:**A short history of length**Speaker:**Joel C. Langer, Case Western Reserve University**Location:**MB113A handy old device called a waywiser - basically a wheel and axle mounted on a handle - may be used to measure the length of a path, straight or curved. If the wheel is one meter in circumference, the waywiser measures the length of the path in meters by counting revolutions of the wheel as it is walked from beginning to end of the path. It works well enough in practice - but does it also work in theory? In fact, the waywiser and the concept of arc length may be used to illustrate both successes of ancient geometers and some of the struggles faced by subsequent mathematicians and philosophers in coming to terms with innity, innite processes and associated computations. The story of arc length alternates between geometry and the theory of numbers, between the continuous and the discrete, over two thousand years.

- April 18th, 2013 (03:35pm - 04:25pm)
**Seminar:**Department of Mathematics Colloquium**Title:**Metric Diophantine approximation: the Lebesgue and Hausdorff theories**Speaker:**Sanju Velani, University of York**Location:**MB114There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well-approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We discuss these theorems and show that Lebesgue statement implies the general Hausdorff statement. The key is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup sets to Hausdorff measure-theoretic statements. In view of this, the Lebesgue theory of limsup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

- April 19th, 2013 (12:20pm - 01:30pm)
**Seminar:**CCMA Luncheon Seminar**Title:**About D'Alembert's paradox in mathematical fluid mechanics**Speaker:**David Gerard-Varet, Jussieu Paris, France**Location:**MB114I will give a mathematical presentation of the so-called D'Alembert's paradox (1755): "Bodies moving at constant speed should not experience any drag or lift". Explaining this strange statement will lead us to discuss the mathematical difficulties associated with the transition from viscous to inviscid fluid models. This prepares for the colloquium talk I will give on boundary layer equations.

- April 19th, 2013 (02:20pm - 03:20pm)
**Seminar:**Seminar on Probability and its Application**Title:**Talk shifted to Math-Bio Seminar Tu 2:30**Speaker:**David Koslicki, Oregon State University**Location:**MB106- April 19th, 2013 (03:35pm - 04:25pm)
**Seminar:**Computational and Applied Mathematics Colloquium**Title:**Mathematical theory of the boundary layer equations**Speaker:**David Gérard-Varet, Mathematics, Université Denis Diderot Paris 7**Location:**MB106We shall discuss the mathematical theory of the Prandtl equations, involved in the description of high Reynolds number flows near a solid wall. The Cauchy problem for these equations is tricky, and turns out to be very dependent on the choice of the functional spaces, due to underlying fluid instabilities. The talk (or at least the end of it) will be based on recent joint work with N. Masmoudi.

- April 22nd, 2013 (09:00am - 11:00am)
**Seminar:**Ph.D. Thesis Defense**Title:**"An asymptotic Mukai model of M6"**Speaker:**Evgeny Mayanskiy, Adviser: Yuri Zarhin, Penn State**Location:**132 EE EastI will talk about Mukai models of the moduli space of genus 6 curves

- April 22nd, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**Dynamical coherence and intrinsic ergodicity for partially hyperbolic diffeomorphisms isotopic to Anosov**Speaker:**Todd Fisher, Brigham Young University**Location:**MB114We discuss partially hyperbolic diffeomorphisms that are isotopic to a hyperbolic toral automorphism and contained in a connected component. If the splitting of the partially hyperbolic diffeomorphism satisfies certain dimensional constraints, then we show the diffeomorphism is dynamically coherent. We then prove that if the center direction is one dimensional, then the topological entropy is locally constant and there is a unique measure of maximal entropy. This is joint work with Rafael Potrie and Martin Sambarino.

- April 23rd, 2013 (09:30am - 11:00am)
**Seminar:**Hyperbolic and Mixed Type PDEs Seminar**Title:**Systems with moving boundaries**Speaker:**Giuseppe Coclite, University of Bari**Location:**MB216We consider a system of scalar balance laws in one space dimension coupled with a system of ordinary differential equations. The coupling acts through the (moving) boundary condition of the balance laws and the vector fields of the ordinary differential equations. We prove the existence of solutions for such systems passing to the limit in a vanishing viscosity approximation. The results were obtained in collaboration with Professor Mauro Garavello.

- April 23rd, 2013 (10:00am - 11:30am)
**Seminar:**Ph.D. Oral Comprehensive Examination**Title:**"Counting rational points near planar curves"**Speaker:**Ayla Gafni, Adviser: Bob Vaughan, Penn State**Location:**MB315**Abstract:**http://ucs.psu.edu/service/home/~/comps%20abstract.tex?auth=co&loc=en_US&id=52953&part=2- April 23rd, 2013 (11:15am - 01:15pm)
**Seminar:**Ph.D. Oral Comprehensive Examination**Title:**"Modular forms and de Rham cohomology"**Speaker:**Haining Wang, Adviser: Winnie Li, Penn State**Location:**MB106- April 23rd, 2013 (02:30pm - 03:30pm)
**Seminar:**GAP Seminar**Title:**Dynamical Quantum Algebras, Quantum Function Algebras, and their Representations**Speaker:**Bharath Narayanan, Penn State**Location:**MB106I will explain the representation theory for quantized function algebras and dynamical quantum algebraic structures on a simple Lie group. Dynamical Quantum Algebras are a class of important noncommutative algebras, originating in exactly solvable models in statistical mechanics and conformal field theory. They have also proven to be very useful for easily veriftying many identities of hypergeometric series. Quantized function algebras, commonly known as the quantum matrix algebras, are quantum versions of usual coordinate rings of Lie groups. They can also be viewed as degenerations of the dynamical algebras, by sending the spectral parameter to infinity, and have numerous applications in q-series, noncommutative geometry and mathematical physics. After an introduction to the basic characters of our investigation - the R-matrices, Yang Baxter equations, Hopf Algebroids and Dynamical Representations, we will explore important relationships between the different algebras and their irreducible representations, to discover a surprising result, known as “Self-Duality”. The examples of sl2 and sl3 will be presented in detail since more or less all of the important ideas can be illustrated explicitly in these 2 cases.

- April 23rd, 2013 (02:30pm - 03:45pm)
**Seminar:**Logic Seminar**Title:**Graph limits and random graphs**Speaker:**Jan Reimann, Pennsylvania State University**Location:**MB315When does a sequence of finite graphs converge? And what should the limit object be? With the theory of random graphs in mind, Lovász and collaborators over the past ten years have built an impressive theory of graph limits that brings together combinatorics, probability theory, and logic. The talk aims to give an overview of this theory, introducing the cut distance and graphons.

- April 23rd, 2013 (03:30pm - 06:00pm)
**Seminar:**Working Seminar: Dynamics and its Working Tools**Title:**Algebraic K-theory and its applications to dynamics, IV**Speaker:**Kurt Vinhage, Penn State**Location:**MB216- April 23rd, 2013 (03:30pm - 05:00pm)
**Seminar:**CCMA PDEs and Numerical Methods Seminar Series**Title:**Reduction to Bidiagonal Form—New Algorithms and Issues**Speaker:**Jesse Barlow, Department of Computer Science and Engineering The Pennsylvania State University**Location:**MB023Since Golub and Kahan’s 1965 landmark paper, bidiagonal reduction has been an important first step in computing the singular value decomposition. The algorithm also plays an important role in the solution of ill-posed least squares problems and the computation of matrix functions. Two algorithms for bidiagaonal reduction were presented in that original paper---a Householder based reduction usually used for dense matrices and a Lanczos based reduction that is traditionally used to extract a subset of singular values and vectors from a large, sparse or structured matrix. Variants to the Householder based reduction have been proposed by Lawson and Hansen, Chan, and Trefethen and Bau. The Lanczos-based reduction has always suffered from the same problem as the Lanczos algorithm for symmetric matrices—loss of orthogonality in the reduction resulting in simple singular values converging as clusters. Simon and Zha proposed a version of the algorithm that reorthogonalizes just one of the two sets of Lanczos vectors, the speaker along with Bosner and Drmač developed a hybrid algorithm that produces one set of Lanczos vectors using Householder transformations, the other using the Lanczos recurrence. In this talk, a framework based on an observation by Charles Sheffield and recent work by Paige, is used to understand both the Simon and Zha and Barlow, Bosner, and Drmač variants of the Golub-Kahan-Lanczos (GKL) bidiagonal reduction. It is shown that if good orthogonality is maintained in just one set of Lanczos vectors, the algorithm retains a number of desirable properties. These include good orthogonality in leading left and right singular vectors, and that the algorithm acts exactly as the GKL algorithm in exact arithmetic on a nearby matrix. At the end of the talk, issues involving generalization to a block GKL algorithm and implementation are discussed.

- April 23rd, 2013 (05:00pm - 06:30pm)
**Seminar:**SIAM Student Chapter Seminar**Title:**Energetic Variational Approaches: Some Simple Examples**Speaker:**Chun Liu, Pennsylvania State University**Location:**MB106- April 24th, 2013 (03:35pm - 04:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**The dynamics of Kuperberg flows**Speaker:**Steve Hurder, University of Illinois at Chicago**Location:**MB114In 1993 the Seifert Conjecture was resolved by Krystyna Kuperberg, who constructed, for any 3-manifold, a smooth flow with no periodic orbits. Kuperberg introduced a completely new type of ``aperiodic plug'' for flows, that was used to trap their periodic orbits. The dynamical properties of the flows constructed via her method have remained only partly understood. The work described in this talk, which is joint with Ana Rechtman, explores their properties in greater depth. We introduce the notion of a "zippered lamination", and show that there exists an invariant zippered lamination for a generic Kuperberg flow. The study of the dynamics of zippered laminations leads to a precise description of the topology and dynamical properties of the minimal set for a generic Kuperberg flow, including a type of chaotic behavior with non-trivial entropy.

- April 25th, 2013 (10:00am - 10:50am)
**Seminar:**Hyperbolic and Mixed Type PDEs Seminar**Title:**Structure of shock-free solutions for 1-D compressible Euler equations**Speaker:**Geng Chen, Penn State**Location:**MB216In this talk, we discuss a result with Robin Young for large shock-free solutions and shock formation on 1-D compressible Euler equations. A current work on general 1-D hyperbolic conservation laws will also be discussed.

- April 25th, 2013 (11:15am - 12:05pm)
**Seminar:**Algebra and Number Theory Seminar**Title:**Zeros of $\zeta$, of $\zeta'$, and of Siegel**Speaker:**Maksym Radziwill, Stanford University**Location:**MB106Motivated by applications to the class number problem and the non-existence of Siegel zeros, Farmer and Ki have recently conjectured a precise relationship between the vertical distribution of the zeros of the Riemann zeta-function and the horizontal distribution of the zeros of \zeta'(s). I will describe the ideas behind my proof of Farmer and Ki's conjecture, the connection between the distribution of the three sets of zeros (of $\zeta$, $\zeta'$ and of Siegel), and the relevance of each to number-theoretic problems.

- April 25th, 2013 (01:00pm - 03:00pm)
**Seminar:**Ph.D. Oral Comprehensive Examination**Title:**"Local Rigidity of Algebraic Higher-Rank Abelian Group Actions"**Speaker:**Kurt Vinhage, Adviser: Anatole Katok, Penn State**Location:**MB114Structural Stability of Anosov diffeomorphisms is a prized result in dynamical systems, and the stability of orbits in the flow case is the best that can be hoped for. Remarkably, in the case of higher-rank abelian group actions, one can expect something much stronger: First, the Anosov condition can be reduced to partial hyperbolicity. Second, orbit equivalence can be replaced by smooth conjugacy. A technique using algebraic K-theory was first introduced by Katok and Damjanovic to prove this in the case of Weyl Chamber flows in SL(n,R) and SL(n,C). Similar results will be presented in the case of quaternionic groups, and the scheme for completing the program for all simple noncompact Lie groups will be outlined.

- April 25th, 2013 (03:35pm - 04:25pm)
**Seminar:**Department of Mathematics Colloquium**Title:**Traces, fixed points, characters, loops**Speaker:**David Ben-Zvi, University of Texas at Austin**Location:**MB114The dimension of a vector space, the trace of a linear map and the character of a group representation can be seen as instances of an abstract notion of character. I will explain how loop spaces and fixed points provide nonlinear analogues of these notions in the brave new world of derived algebraic geometry, a strange blend of topology and geometry. Following ideas from topological field theory, we'll see how passing back to the linear world results in interesting trace and character formulas. This is joint work with David Nadler (Berkeley).

- April 26th, 2013 (12:20pm - 01:30pm)
**Seminar:**CCMA Luncheon Seminar**Title:**Struggle with Discrete Stability and Convergence**Speaker:**Leszek Demkowicz, ICES, University of Texas Austin**Location:**MB114I will review the history of analyzing stability and convergence of Galerkin methods including Finite Element methods. We will start with the main points of Babuska's Theorem and then review the history of the struggle from Ritz and Rayleigh, through Galerkin, Bubnow and Petrov, asymptotic stability result of Mikhlin, concept of optimal testing of Barret and Morton, stabilized methods including SUPG, bubble methods, DG methods and, finally, minimum residual methods including the DPG method.

- April 26th, 2013 (02:20pm - 03:20pm)
**Seminar:**Seminar on Probability and its Application**Title:**Poisson limit theorem for Gibbs-Markov systems**Speaker:**Xuan Zhang, PSU**Location:**MB106Sinai studied Poisson limit distribution behaviour for eigenvalues in the quantum kicked-rotator model. Inspired by Sinai's work, Pitskel investigated the Poisson limit distribution for return times of dynamical systems. He proved results for Markov chains with finite states and for hyperbolic toral automorphisms. Independently Hirata proved his result for subshifts of finite types with Gibbs measures and for Axiom A systems. Many results have been proved for diffenrent kinds of systems in recent years. In this talk, we will try to explain that Poisson limit distribution also exists for Gibbs-Markov systems. Certain Markov chains with countable states will be special cases. We basically follow Hirata's method with a few revisions.

- April 26th, 2013 (03:35pm - 04:25pm)
**Seminar:**Computational and Applied Mathematics Colloquium**Title:**Discontinuous Petrov Galerkin Method (DPG) with Optimal Test Functions**Speaker:**Leszek Demkowicz, ICES, UT Austin**Location:**MB106I will give a short tutorial on the DPG method proposed by Jay Gopalakrishnan and myself four years ago, emphasizing the main points and illustrating them with numerical examples. Here is a few of them: 1/ The DPG method is a minimum-residual method with the residual evaluated in a dual norm. 2/ The method can be interpreted as a Petrov-Galerkin method with optimal test functions (realizing the sup in the inf-sup condition). 3/ The optimal test functions are computed on the fly by inverting (approximately) the Riesz operator corresponding to the test space. 4/ With broken test spaces and localizable norms, the inversion is done elementwise, i.e. the optimal test functions are computed within the element routine. This is more expensive then for standard FE method but it is compatible with the standard FE technology. 5/ The main price paid for the localization is the presence of additional unknowns: traces and fluxes. Compared with standard conforming FE methods or hybridizable DG methods, the number of (non-local) unknowns doubles and it is of the same range as for DG methods. Contrary to DG methods based on numerical flux, in the DPG method, the flux enters as additional unknown. 6/ The method can be interpreted as a preconditioned least squares method. The stiffness matrix is hermitian and positive-definite. Its condition number is the same as for standard FEs. 7/ The formulation based on a first order system is very natural but not necessary. You can work with the second order equation if you wish. The key point is to break the test functions. 8/ There is nothing exotic about the ultra-weak variational formulation behind the DPG method. If the operator is well posed in the L^2 sense (the operator is L^2 bounded below), the ultra-weak variationalformulation is also well posed with the corresponding inf-sup constant being of the same order. 9/ With the use of optimal test functions, the issues of approximability and stability are fully separated. This is illustrated by using hp-adaptivity. 10/ The method is especially suited for singular perturbation problems e.g. convection-dominated diffusion, high wave number wave propagation, elasticity for thin-walled structures etc. For problems of this type, one can systematically design a test norm to accomplish robustness, i.e. a stability uniform in the perturbation parameter. 11/ If you have a hybrid FE code, converting it to a DPG code is very easy. 12/ The methodology extends to nonlinear problems. I will show examples for compressible NS equations.

- April 29th, 2013 (01:00pm - 03:00pm)
**Seminar:**Ph.D. Oral Comprehensive Examination**Title:**"Long-time Behavior of Stochastic Models"**Speaker:**Chao Tian, Adviser: Qiang Du, Penn State**Location:**104 OsmondIn this talk, I will present two examples to illustrate the rich long time dynamics of stochastic models. The first example is related to fluid-structure interaction (FSI) which appears in many scientific and engineering problems. The stochastic implicit-interface method (SIIM) is designed to account for the microscopic fluctuation in phase-field or level-set models of FSI. Formulated as a system of non-linear stochastic partial differential equations with a degenerate additive noise term, the SIIM is formally derived using the fluctuation-dissipation theorem. It is interesting to study its long time asymptotic behavior. The second example is a bi-stable mean-field model whose mathematical analysis, including the study of mean-field limits, existence of multiple equilibria, and fluctuation theory was initiated by D.A. Dawson et al. Using the large-deviations theory G. Papanicolaou et al. recently studied the dynamical phase-transitions in such systems due to stochastic fluctuation. We consider various generalizations of such mean-field models and study the phase transitions between different equilibria in various settings.

- April 29th, 2013 (03:30pm - 05:00pm)
**Seminar:**Probability and Financial Mathematics Seminar**Title:**Long-Run Bond Risk Premia**Speaker:**Olesya Grishchenko, Federal Reserve Board, Washington**Location:**MB315NOTE: if you are interested in being on the mailing list for this seminar, please contact one of the organizers (nistor at math dot psu dot edu or mazzucat at math dot psu dot edu) to request that you be included in the list. Abstract of the talk: In the paper that I will present, we rationalize the failure of the expectations hypothesis and the time-varying bond risk premia in the generalequilibrium model with the long-run consumption risks coupled with stochastic consumption volatility process coupled with recursive preferences and inflationuncertainty dynamics, which interact with both consumption growth and volatility. The last two effects are also present in the inflation modeling, and thusaffect both real and nominal side of the economy. This link paves a way of explaining long standing puzzle of the failure of expectations hypothesis, which wasearlier attempted to be explained only via a real economy variables. This is joint work with Hao Zhou.

- April 29th, 2013 (03:35pm - 05:35pm)
**Seminar:**Center for Dynamics and Geometry Seminars**Title:**Arnold Diffusion via Invariant Cylinders and Mather Variational Method**Speaker:**Vadim Kaloshin, University of Maryland TWO HOUR TALK**Location:**MB114The famous ergodic hypothesis claims that a typical Hamiltonian dynamics on a typical energy surface is ergodic. However, KAM theory disproves this. It establishes a persistent set of positive measure of invariant KAM tori. The (weaker) quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian dynamics on a typical energy surface has a dense orbit. This question is wide open. In early 60th Arnold constructed an example of instabilities for a nearly integrable Hamiltonian of dimension $n>2$ and conjectured that this is a generic phenomenon, nowadays, called Arnold diffusion. In the last two decades a variety of powerful techniques to attack this problem were developed. In particular, Mather discovered a large class of invariant sets and a delicate variational technique to shadow them. In a series of preprints: one joint with P. Bernard, K. Zhang and two with K. Zhang we prove Arnold's conjecture in dimension $n=3$.

- April 30th, 2013 (10:00am - 12:00pm)
**Seminar:**Ph.D. Oral Comprehensive Examination**Title:**"Modular forms for finite index subgroups of SL(2,Z)"**Speaker:**Will Chen, Adviser: Wen-Ching W. Li, Penn State**Location:**MB113Since its inception, the theory of modular forms has almost exclusively focused on modular forms for congruence subgroups, for which a beautiful and deep theory has been developed culminating in the celebrated proof of the modularity theorem and FLT. On the other hand, very little is known about the world of modular forms for noncongruence subgroups. A theorem of Belyi implies that every smooth projective curve defined over a number field can be realized as a modular curve, usually noncongruence. Thus, noncongruence modular curves are very general, and one might expect that as a result, relatively little can be said about them. On the other hand, their uniformization by the upper half plane is rather special, and hence one might hope that there maybe be at least certain classes of noncongruence modular curves for which a theory comparable to that of congruence curves may be established. In my talk, I'll compare the two worlds, highlighting their key differences, as well as some striking similarities. I will conclude the talk by presenting some directions for further investigation.

- April 30th, 2013 (01:30pm - 02:30pm)
**Seminar:**Probability and Financial Mathematics Seminar**Title:**Optimal Bidding in a Limit Order Book**Speaker:**Giancarlo Facchi, PSU, Mathematics**Location:**MB106**Abstract:**http://An external buyer asks for a random amount $X>0$ of a certain asset. This agent will buy the amount $X$ at the lowest available price, as long as this price does not exceed a given upper bound $P$. One or more sellers are competing to fulfill the incoming order, by offering various quantities of the same asset for sale at different "limit" prices. The collection of all these sell orders at different prices is the "Limit Order Book". Having observed the prices asked by his competitors, each seller must determine an optimal pricing strategy, maximizing his expected payoff. Clearly, when other sellers are present, asking a higher price for the asset reduces the probability of selling it. In our model we assume that the $i$-th seller owns an amount $\kappa_i$ of stocks. He can put all of it on sale at a given price, or offer different portions at different prices. If the selling prices are allowed to be any real number in $[0,P]$, then a general pricing strategy is described by a measure on $[0,P]$. We analyze in detail two different scenarios. Let $\psi(s) = {\rm Prob.}[X>s]$ denote the tail distribution function of $X$. \begin{itemize} \item If $(\ln\psi(s))^{\prime\prime} \geq 0~ \forall s$, then a unique Nash equilibrium exists and can be explicitly determined in the special case where every player has the same payoff function. We show that the all the optimal strategies (except at most one) consist of measures which are absolutely continuous with respect to the Lebesgue measure. \item If $(\ln\psi(s))^{\prime\prime} < 0~ \forall s$, a Nash equilibrium does not exist, and the competition between sellers does not settle near any equilibrium state. \end{itemize} We also consider a different model where there is a positive tick size, which means that the only admissible pricing strategies are purely atomic, supported on a finite set of prices. In this case, we can prove the existence of Nash equilibria also in the more general case of heterogeneous players. Moreover, as the tick size goes to 0, any weak limit of discrete Nash equilibria provides a Nash equilibrium for the continuum model.