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Eberly College of Science Mathematics Department

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March 1st, 2013 (12:20pm - 01:30pm)
Seminar: CCMA Luncheon Seminar
Title: Commuting Projections on Graphs and Two-Level Methods
Speaker: Ludmil Zikatanov, Penn State Math
Location: MB114

Motivated by the increasing importance of large-scale networks typically modeled by graphs we study properties associated with the popular graph Laplacian. We exploit its mixed formulation based on its natural factorization as product of two operators. The goal is to construct a coarse version of the mixed graph Laplacian operator with the purpose to construct two-level, and by recursion, a multilevel hierarchy of graphs and associated operators. In many situations in practice having a coarse (i.e., reduced dimension) model that maintains some inherent features of the original large-scale graph and respective graph Laplacian offers potential to develop efficient algorithms to analyze the underlined network modeled by this large-scale graph. One possible application of such a hierarchy is to develop multilevel methods that have the potential to be of optimal complexity. One result that we will report is the construction of a projection from the edge-space onto a properly constructed coarse edge-space associated with the edges of the coarse graph. This projection commutes with the discrete divergence operator, and the pair of coarse edge-space and coarse vertex-space offer the potential to construct two-level, and by recursion, multilevel methods for the mixed formulation of the graph Laplacian. The performance of one two-level method with overlapping Schwarz smoothing and correction based on the constructed coarse spaces for solving such mixed graph Laplacian systems is illustrated on a two of graph examples. This is a joint work with Panayot Vassilevski.

March 4th, 2013 (03:35pm - 04:35pm)
Seminar: Center for Dynamics and Geometry Seminars
Title: SPRING BREAK
Speaker: NO SEMINAR
Location: MB114
March 7th, 2013 (11:15am - 12:05pm)
Seminar: Algebra and Number Theory Seminar
Title: No Talk Today
Speaker: Spring Break, PSU
Location: MB106
March 11th, 2013 (03:35pm - 04:35pm)
Seminar: Center for Dynamics and Geometry Seminars
Title: Ergodic properties of skew products in infinite measure
Speaker: Yuri Lima, Weizmann Institute of Science, Israel
Location: MB114

Let (Ω,μ) be a shift of finite type with a Markov probability, and (Y,ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a measure-preserving automorphism of (Y,ν). We study skew products of the form (ω; y) -> (σω,Φ_ω_0 (y)), where σ is the shift map on (Ω,μ). We prove that, when the skew product is conservative, it is ergodic if and only if the Φ_i's have no common non-trivial invariant set. In the second part we study the skew product when Ω={0,1}^Z, μ is a Bernoulli measure, and Φ_0, Φ_1 are IR-extensions of a same uniquely ergodic probability-preserving automorphism. We prove that, for a large class of roof functions, the skew product is rationally ergodic with return sequence asymptotic to √n, and its trajectories satisfy the central, functional central and local limit theorem. Joint work with Patricia Cirilo and Enrique Pujals.

March 12th, 2013 (09:30am - 11:00am)
Seminar: Hyperbolic and Mixed Type PDEs Seminar
Title: Lipschitz Semigroups and quotient metrics for Hunter-Saxton equation I
Speaker: Geng Chen, Penn State
Location: MB216
March 12th, 2013 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: On Toda field theories
Speaker: Zhaohu Nie, Utah State University
Location: MB106

The Liouville equation is the nonlinear second order partial differential equation $u_{xy} = -e^{2u}$. It is related to surfaces of constant curvature, and Liouville found the general solution. Toda field theories are generalizations of the Liouville equation using general simple Lie algebras. These are a typical class of integrable systems. In this talk, we will concentrate on two aspects of Toda field theories. The first one is about characteristic integrals, that is, some differential polynomials (with respect to one variable) in the unknown functions whose derivative with respect to the other variable is zero. These are conserved quantities for the system. The second is about the explicit solutions to the Toda field theories in the above spirit of Liouville. Leznov's general solutions will be presented in the most explicit form using iterated integrals.

March 12th, 2013 (02:30pm - 03:45pm)
Seminar: Logic Seminar
Title: Pi^1_1-conservative extensions of subsystems of second-order arithmetic.
Speaker: Keita Yokoyama, Tokyo Institute of Technology
Location: MB315

For the study of reverse mathematics, (Pi^1_1-)conservation results between two different systems are very important. Many important theorems are well-known, e.g., WKL_0 and RCA_0^+ are Pi^1_1-conservative extensions of RCA_0. Moreover, we can show the following: for given a Pi^1_2-theory T, there exists a maximal Pi^1_2-theory which is Pi^1_1-conservative over T. Then, what is the maximal extension of RCA_0, ACA_0, etc. in this sense? In this talk, I will give a survey on this topic including Henry Towsner's resent results.

March 12th, 2013 (03:30pm - 06:00pm)
Seminar: Working Seminar: Dynamics and its Working Tools
Title: Gaussian Dynamical systems,I
Speaker: Shilpak Banerjee, Penn State
Location: MB216
March 13th, 2013 (03:35pm - 04:35pm)
Seminar: Center for Dynamics and Geometry Seminars
Title: Contact (Non-)Squeezing and spectral invariants
Speaker: Peter Albers, Westfälische Wilhelms-Universität Münster
Location: MB114

Gromov's famous non-squeezing theorem in symplectic geometry shows the strong rigidity properties of symplectic transformations. In this talk I will first explain why a theorem by Eliashberg-Polterovich shows that the obvious generalization of Gromov's non-squeezing theorem to contact geometry is not only the wrong one but also that in contact geometry squeezing and non-squeezing coexist. Then I will discuss how spectral invariants provide an easy tool to study contact non-squeezing and allows to establish many new cases of contact non-squeezing. Finally we will discuss some dynamical application, namely existence of infinitely many iterated translated points of contactomorphisms. This is joint work with Will Merry.

March 14th, 2013 (10:00am - 10:50am)
Seminar: Hyperbolic and Mixed Type PDEs Seminar
Title: Lipschitz Semigroups and quotient metrics for Hunter-Saxton equation II
Speaker: Geng Chen, penn State
Location: MB216
March 14th, 2013 (11:15am - 12:05pm)
Seminar: Algebra and Number Theory Seminar
Title: To be announced
Speaker: Daniel Vallieres, Binghampton University
Location: MB106
March 14th, 2013 (02:30pm - 03:20pm)
Seminar: PMASS Colloquium
Title: Prime number theorem
Speaker: Andrey Gogolev, Binghamton University
Location: MB113

The primes have been boggling the minds of humans since the ancient times. We will discuss the statistical distribution of primes within the set of positive integers. This distribution is described by the prime number theorem. Interestingly, finer versions of the prime number theorem are conjectures that are related to the core of modern mathematics such as the Riemann hypothesis.

March 14th, 2013 (03:35pm - 04:25pm)
Seminar: Department of Mathematics Colloquium
Title: Numerical methods for interfaces in fluids and regularizing effects in difference equations
Speaker: Tom Beale, Duke University
Location: MB114

We will discuss a class computational methods for a moving interface in viscous fluid and related error analysis. We will begin with a description of the Navier-Stokes equations and the model of a moving interface which interacts with the fluid. (It could be a drastically simplified model of biological tissue on a small scale.) We will describe several numerical methods for such problems in which the fluid variables are calculated at grid points and the moving interface is represented separately. Accuracy is most difficult to maintain near the interface, but it has often been observed that the solution can be uniformly more accurate than would be expected from the truncation error in satisfying the equations. We will explain this gain with discrete, finite difference versions of regularity estimates for elliptic and parabolic differential equations, such as Laplace's equation and the heat equation, in maximum norm, which are ``almost sharp''. We will present a numerical method for the coupled motion of an elastic interface with Navier-Stokes flow, developed with Anita Layton, in which the velocity is decomposed into a Stokes velocity and a more regular part. The advantage is that the interaction of the interface with Stokes flow (dominated by viscosity) is much simpler to deal with than the full Navier-Stokes equations. Simple test problems and partial analysis indicate the method is second-order accurate. It can be modified to make the interface motion partially implicit, in order to allow larger time steps.

March 15th, 2013 (12:20pm - 01:30pm)
Seminar: CCMA Luncheon Seminar
Title: Electrostatics in inhomogeneous media
Speaker: Dr. Zhenli Xu, Shanghai Jiao Tong University
Location: MB114

In this talk, I will present electrostatic algorithms for inhomogeneous media developed in my group. The talk is composed of three parts. Firstly, we studied the image charge method of the Green's function problem of a dielectric sphere immersed in a medium described by the Poisson-Boltzmann equation, which has been used for multiscale reaction-field model of ionic fluids. Second, we discuss the harmonic interpolation algorithm for ionic interaction in spatially varying media and its application in simulating inhomogeneous electric double layers. Third, we present an algorithm for multiple dielectric objects used in studying like charge attraction. Monte Carlo simulation results based on these algorithms are present to demonstrate the performance of these algorithms, and to understand properties of colloid/electrolyte interfaces.

March 15th, 2013 (03:35pm - 04:25pm)
Seminar: Computational and Applied Mathematics Colloquium
Title: Computing with Singular and Nearly Singular Integrals
Speaker: Tom Beale, Mathematics, Duke University
Location: MB106

We will describe a relatively simple, direct approach to computing a singular or nearly singular integral, such as a harmonic function given by a single or double layer potential on a smooth, closed curve in 2D or a surface in 3D. Integral formulations are used especially for Stokes flow (viscosity-dominated fluid flow)and in electromagnetics. The present approach can be useful for moving interfaces since the representation of the interface requires less work than boundary elements. The value of the integral is found by a standard quadrature, with the singularity replaced by a regularized version. Correction terms are then added for the errors due to regularization and discretization. These corrections are found by local analysis near the singularity. The accurate evaluation of a layer potential at a point near the curve or surface on which it is defined is not routine, since the integral is nearly singular. For a surface in 3D, integrals are computed in overlapping coordinate grids on the surface. A quadrature technique of J. Wilson allows this to be done without explicit knowledge of the coordinate charts, thus making the approach more practical. For a boundary value problem, an integral equation can be solved for the density of the needed potential. For a fluid interface, e.g. in Stokes flow, the velocity or pressure can be written in terms of layer potentials. Recent work with W. Ying in 2D shows that the method can accurately handle boundaries which are close to each other, such as two drops merging, and work in 3D is in progress. Collaborators include M.-C. Lai, A. Layton, S. Tlupova, J. Wilson, and W. Ying.

March 18th, 2013 (03:35pm - 04:35pm)
Seminar: Center for Dynamics and Geometry Seminars
Title: The associated sphere bundle to a negatively curved bundle.
Speaker: Andrey Gogolev, SUNY- Binghamton
Location: MB114

Consider a smooth fiber bundle over a compact simply connected manifold. We say that such bundle is negatively curved if the fibers can be equipped with continuously varying metrics of negative curvature. Conjecturally negatively curved bundles are trivial. We use a classical dynamical systems technique to verify a related result: the (fiberwise) associated sphere bundle to a negatively curved bundle is trivial. This work is joint with Tom Farrell.

March 19th, 2013 (11:15am - 12:05pm)
Seminar: Combinatorics/Partitions Seminar
Title: On the Parity of the Number of Parts in Distinct-Part Partitions
Speaker: James Sellers, PSU
Location: MB106

We will consider relatively recent work by Knopfmacher and Robbins related to the function s(n) which counts the *number* of parts in distinct-part partitions of n. Interestingly enough, the function s(n) has a strong tendency to be even (although it is not clear, a priori, that this should be so). We will shed light on why this phenomenon occurs and then close by proving a number of Ramanujan-like congruences modulo 2 satisfied by s(n) using elementary means.

March 19th, 2013 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: Richardson varieties and Poisson geometry
Speaker: Samuel Evens, University of Notre Dame
Location: MB106

I will discuss a Poisson structure on the flag variety of a complex semisimple Lie group with the property that its symplectic leaves are Richardson varieties, i.e., intersections between Schubert cells and opposite Schubert cells. This structure arises by taking a certain limit of the Bruhat-Poisson structure under a complex torus action. This talk is based on joint work with Arlo Caine.

March 19th, 2013 (02:30pm - 03:45pm)
Seminar: Logic Seminar
Title: Degree-invariant functions and uniform Borel reductions
Speaker: Jan Reimann, Pennsylvania State University
Location: MB315

A. Kechris asked whether Turing equivalence is a universal countable Borel equivalence relation. If this is true, it would imply that Martin's Conjecture on degree-invariant functions is false. We show that Turing equivalence cannot be universal by means of uniform reductions. This is joint work with A. Montalban and T. Slaman.

March 19th, 2013 (03:30pm - 06:00pm)
Seminar: Working Seminar: Dynamics and its Working Tools
Title: Gaussian Dynamical systems,II
Speaker: Shilpak Banerjee, Penn State
Location: MB216
March 21st, 2013 (10:00am - 11:30am)
Seminar: Computational and Applied Mathematics Colloquium
Title: Effects of Inhomogeneities—Nanoscale to Mesoscale—on the Durability of Li-Ion Batteries
Speaker: Stephen Harris, General Motors and Lawrence Berkeley
Location: MB315

We review work from our laboratory that suggests to us that most Li-ion battery failure can be ascribed to the presence of nano- and microscale inhomogeneities that interact at the mesoscale, as is the case with almost every material; and that these inhomogeneities act by hindering Li transport. (Li does not get to the right place at the right time.) For this purpose, we define inhomogeneities as regions with sharply varying properties—which includes interfaces—whether present by “accident” or design. We have used digital image correlation, X-Ray tomography, FIB-SEM serial sectioning, and isotope tracer techniques with TOF-SIMS to observe and quantify these inhomogeneities. We propose new research approaches to make more durable, high energy density lithium ion batteries.

March 21st, 2013 (11:15am - 12:05pm)
Seminar: Algebra and Number Theory Seminar
Title: A^1-ruled affine varieties.
Speaker: Tatiana Bandman, Bar-Ilan University
Location: MB106

I will speak about smooth affine complex surfaces with negative log-Kodaira dimension. Such a surface is always equipped with an A-fibration, i.e. with a morphism onto a curve, such that the general fiber of the morphism is isomorphic to an affine line. These fibrations may be used both for classification of the surfaces with negative log-Kodaira dimension and for building interesting examples in arithmetic dynamics.

March 21st, 2013 (02:30pm - 03:20pm)
Seminar: PMASS Colloquium
Title: Bezout's theorem
Speaker: Karl Schwede, Penn State
Location: MB113

How many times can a line intersect a parabola? You probably said the answer is zero, one, or two. How many times can two ellipses intersect? What if you count imaginary or complex intersection points? Do parallel lines ever intersect? What if we change the rules of the game and consider parallel lines in a twisted pac-man-like world? Bezout's theorem tells us exactly how many times two implicitly defined polynomial curves intersect, provided you count complex intersections, intersections at infinity (where parallel lines meet up), and double/triple/multiple roots correctly.

March 21st, 2013 (03:35pm - 04:25pm)
Seminar: Department of Mathematics Colloquium
Title: Evolutionarily stable strategies for dispersal in heterogeneous environments
Speaker: Yuan Lou, Ohio State University
Location: MB114

From habitat degradation and climate change to spatial spread of invasive species, dispersal plays a central role in determining how organisms cope with a changing environment. How should organisms disperse “optimally” in heterogeneous environments? In this talk I will discuss some recent development of game theoretic approach on the evolution of dispersal via Lotka-Volterra competition models.

March 22nd, 2013 (12:20pm - 01:30pm)
Seminar: CCMA Luncheon Seminar
Title: Multiscale problems in biology
Speaker: Chuan Xue, Ohio State University
Location: MB114

In this talk, I will give examples of multiscale problems arising from biology, including chemotaxis of cell populations and segregation of cell cytoskeleton. I will also show how multiscale methods and hybrid modelling methods can be used to explain phenomena observed at different scales.

March 22nd, 2013 (02:20pm - 03:20pm)
Seminar: Seminar on Probability and its Application
Title: Two parameter stochastic differential equations I
Speaker: Brian Nowakowski, PSU
Location: MB106

We survey two results by Yeh and Protter on the existence and uniqueness of solutions for stochastic differential equations where the integrator is a two parameter semimartingale and the integrand is a previsible process. This is an extension of the classical SDE, and it will be connected to my recently defended PhD dissertation.

March 22nd, 2013 (03:35pm - 04:25pm)
Seminar: Computational and Applied Mathematics Colloquium
Title: Multiscale models of chemotaxis
Speaker: Chuan Xue, Ohio State University Mathematics
Location: MB106

Chemotaxis is the directed movement of cells in response to extracellular chemical signals. Chemotaxis of cell populations have been modelled using both continuum models which involves systems of PDEs, and cell-based models that describe movement of each cell as a stochastic process with internal dynamics. In this talk, I will review current theory of both approaches and discuss recent developments on derivation of continuum models from cell-based models in the context of bacterial chemotaxis.

March 25th, 2013 (03:30pm - 05:00pm)
Seminar: Probability and Financial Mathematics Seminar
Title: All you wanted to know credit risk but were afraid to ask
Speaker: Nick Costanzino, ScotiaBank
Location: MB106

Credit risk manifests itself in different guises in almost all areas of Banking, including the CVA/DVA of a trade, the capital required for a loan portfolio, and the valuation of credit derivatives.  In all these areas, a unifying quantity is the probability of default. Default probability is a notoriously difficult quantity to estimate given the frequency of realized defaults. In this talk I will give a brief overview of the most popular mathematical models and calibration tools for quantifying the probability of default, including structural models such a Merton and Black-Cox as well as reduced form models such as Jarrow-Turnbull. Finally, I will introduce a new model and some related recent results on incorporating stochastic recovery rates in computing joint probabilities (joint work with Albert Cohen at MSU).

March 26th, 2013 (09:30am - 11:00am)
Seminar: Hyperbolic and Mixed Type PDEs Seminar
Title: Lipschitz metrics for a class of nonlinear wave equations I
Speaker: Geng Chen, Penn state
Location: MB216
March 26th, 2013 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: The hypersurface-deformation algebroid
Speaker: Martin Bojowald, Penn State (Physics Department)
Location: MB106

The symmetries and dynamics of space-time in general relativity follow from the algebroid of small deformations of spatial hypersurfaces. Quantum gravity modifies the algebroid in a characteristic way. It is not known how these modified algebroids could be integrated, but several intriguing effects, such as signature change at high density, have been derived.

March 26th, 2013 (02:30pm - 03:45pm)
Seminar: Logic Seminar
Title: Generic cuts in models of arithmetic, reprise.
Speaker: Tin Lok (Lawrence) Wong, Ghent University, Belgium
Location: MB315

Generic cuts are a particularly nice family of cuts (i.e., initial segments) in nonstandard models of arithmetic. They first appeared in Richard Kaye's 2008 paper, and are known to possess a number of pleasant model theoretic properties. However, it has been unclear what the key to genericity is. In the talk, I will describe generic cuts from various perspectives, e.g., the functions under which they are closed, the ways in which they are moved by automorphisms, and the types they realize. All these help one better picture generic cuts.

March 26th, 2013 (03:30pm - 06:00pm)
Seminar: Working Seminar: Dynamics and its Working Tools
Title: Gaussian Dynamical systems,III
Speaker: Shilpak Banerjee, Penn State
Location: MB216
March 26th, 2013 (03:30pm - 04:30pm)
Seminar: CCMA PDEs and Numerical Methods Seminar Series
Title: Designing geometric multigrid methods on semi-structured triangular grids
Speaker: Carmen Rodrigo, Department of Applied Mathematics, University of Saragossa, Spain
Location: MB114

Multigrid methods appear as the most suitable to handle the algebraic systems arising from discretized PDEs on unstructured grids. We will present geometric multigrid methods which are designed for semi-structured grids. Semi-structured grids are composed of regularly refined patches of initial unstructured meshes and, as we show, a geometric multigrid method can be done very efficiently on such grids. Since the good performance of the method depends on the particular choice of the components of the algorithm for an individual problem, the local Fourier analysis (LFA) is often used to predict the convergence rates of the multigrid method and to design suitable components. We use LFA on a patch of semi-structured grid to design an optimal and efficient block-wise multigrid algorithm.

March 28th, 2013 (10:00am - 10:50am)
Seminar: Hyperbolic and Mixed Type PDEs Seminar
Title: Lipschitz metrics for a class of nonlinear wave equations II
Speaker: Geng Chen, Penn state
Location: MB216
March 28th, 2013 (11:15am - 12:05pm)
Seminar: Algebra and Number Theory Seminar
Title: Questions with sums of three squares
Speaker: Robert Vaughan, Penn State
Location: MB106

We discuss some questions of recent interest in which sums of three squares have appeared

March 28th, 2013 (02:30pm - 03:30pm)
Seminar: Noncommutative Geometry Seminar
Title: Amenability and Ramsey theory
Speaker: John Roe, Penn State
Location: MB106

I'll summarize the machinery behind a recent attempt by Justin Tatch Moore, a logician at Cornell, to prove the amenability of Thompson's group. While this attempt was unsuccessful, the mechanism by which Moore relates amenability to Ramsey theory - roughly, the branch of logic that proves theorems of the form "large multicolored structures have fairly large monochromatic substructures" - is interesting in itself and yields some surprising results about amenable groups.

March 28th, 2013 (03:35pm - 04:25pm)
Seminar: Department of Mathematics Colloquium
Title: Composite fermions: magical beauty of emergence
Speaker: Jainendra Jain, Penn State University
Location: MB114

It is surprising that we can make any progress at all in understanding the behavior of a collection of a large number of interacting particles, given that even the problem of three interacting particles cannot be solved exactly. The trick is to guess emergent principles guided by experiment. I will talk about how new particles emerge in the context of the fascinating phenomenon known as the quantum Hall effect (1985 and 1998 Nobel Prizes), and how they help solve a problem that at first seemed utterly impossible.

March 28th, 2013 (04:40pm - 05:30pm)
Seminar: Computational and Applied Mathematics Colloquium
Title: Local Cell problems for numerical homogenization
Speaker: xingye yue, soochow university
Location: MB106

Local cell problems play a key role in numerical homogenization. It is the bridge between the microscopic scale and the macroscopic scale. We start from some model problems to show how to set up the local cell problems and then consider some multiscale multi-physics systems such as piezo-elasticity and thermo-elasticity.

March 29th, 2013 (12:20pm - 01:30pm)
Seminar: CCMA Luncheon Seminar
Title: Introduction to A Posteriori Error Estimation
Speaker: Zhiqiang Cai, Purdue University, Mathematics Department
Location: MB114
Abstract: http://www.math.psu.edu/nistor/SEMINARS/CaiPizza.pdf

See http://www.math.psu.edu/nistor/SEMINARS/CaiPizza.pdf

March 29th, 2013 (02:20pm - 03:20pm)
Seminar: Seminar on Probability and its Application
Title: Two parameter stochastic differential equations II
Speaker: Brian Nowakowski, PSU
Location: MB106

This is a continuation of last week's talk.

March 29th, 2013 (03:35pm - 04:25pm)
Seminar: Computational and Applied Mathematics Colloquium
Title: Efficient, Reliable and Robust A Posteriori Error Estimators
Speaker: Zhiqiang Cai, Purdue University Mathematics
Location: MB106
Abstract: http://www.math.psu.edu/nistor/SEMINARS/Cai.pdf

see http://www.math.psu.edu/nistor/SEMINARS/Cai.pdf