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Eberly College of Science Mathematics Department

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December 1st, 2014 (12:20pm - 01:30pm)
Seminar: CCMA Luncheon Seminar
Title: Finite time singularities of inhomogeneous elasticity
Speaker: Tao Huang, Penn State University
Location: MB114

We study the formation of finite time singularities in the form of super norm blowup for a spatially hyperbolic system modeling inhomogeneous elasticity, which is related to the variational wave equations. The system possesses a unique $C^1$ solution before the emergence of vacuum in finite time, for given initial data that are smooth enough, bounded and uniformly away from vacuum. At the occurrence of blowup, the density becomes zero, while the momentum stays finite, however the velocity and the density of the energy are both infinite.

December 1st, 2014 (02:30pm - 03:30pm)
Seminar: Computational and Applied Mathematics Colloquium
Title: Uncertainty quantification and geophysical hazard mapping
Speaker: Elaine Spiller, Marquette University (Host: J Conway)
Location: MB106

PDE models of granular flows are invaluable tools for developing probabilistic hazards maps for volcanic landslides, but they are far from perfect. First, any probabilistic hazard map is conditioned on assumptions about the aleatoric uncertainty -- how mother nature rolls the dice -- and is hence tied to the choice of probability distributions describing various scenarios (e.g. initial and/or boundary conditions). Thus new data, differing expert opinion, or emergent scenarios may suggest that the original assumptions were invalid and thus the hazard map made under those assumptions is not terribly useful. Epistemic uncertainty -- uncertainty due to a lack of model refinement -- arises through assumptions made in physical models, numerical approximation, and imperfect statistical models. In the context of geophysical hazard mapping, we propose a surrogate-based methodology which efficiently assesses the impact of various uncertainties enabling a quick yet methodical comparison of the effects of uncertainty and error on computer model output.

December 1st, 2014 (03:35pm - 04:35pm)
Seminar: Dynamical systems seminar
Title: An avalanche principle for dynamical systems
Speaker: Manfred Denker, The Pennsylvania State University
Location: MB114

Consider a product transformation $S=S_1\times S_2\times...S_N$ of $N$ transformations. Given sets $U_i$ in the domain of $S_i$, one can define a new transformation: whenever a coordinate falls into $U_i$ a transformation $T$ is applied as many times as this happens until such a process stops. Otherwise apply $S$ at each time step. The talk will make this construction precise and discuss the case when $T=S$ and the case when $T$ and $S$ are product transformations on $[0,1]^N$ given by two rotations. The results are on basic properties of such transformations, as topological transitivity, stationary measures, ergodicity.

December 2nd, 2014 (10:00am - 10:50am)
Seminar: Hyperbolic and Mixed Type PDEs Seminar
Title: Uniqueness of conservative solutions to a variational wave equation
Speaker: Alberto Bressan, Penn State
Location: MB216

An interesting class of variational wave equations take the form u_tt - c(u)(c(u)u_x)_x = 0, where c(u) > 0 is the wave speed. It is well known that solutions remain uniformly Holder continuous, but their gradient can blow up in finite time. When this happens, multiple solutions can be constructed. Uniqueness can be achieved by further imposing that the total energy remains constant in time. The uniqueness proof relies on a refined analysis of characteristics, which in this case satisfy an ODE with Holder continuous right hand side. The talk will present the main ideas in the construction, and review some earlier results on uniqueness for ODEs with possibly discontinuous right hand side.

December 2nd, 2014 (11:15am - 12:05pm)
Seminar: Combinatorics/Partitions Seminar
Title: Infinitely Many Congruences Modulo 5 for 4-Colored Frobenius Partitions
Speaker: James Sellers, Penn State
Location: MB106

In his 1984 AMS Memoir, Andrews introduced the family of functions c\phi_k(n), which denotes the number of generalized Frobenius partitions of n into k colors. Recently, Baruah and Sarmah, Lin, Sellers, and Xia established several Ramanujan-like congruences for c\phi_4(n) relative to different moduli. In this paper, which is joint work with Michael D. Hirschhorn of UNSW, we employ classical results in q-series, the well-known theta functions of Ramanujan, and elementary generating function manipulations to prove a characterization of c\phi_4(10n+1) modulo 5 which leads to an infinite set of Ramanujan-like congruences modulo 5 satisfied by c\phi_4. This work greatly extends the recent work of Xia on c\phi_4 modulo 5.

December 2nd, 2014 (01:00pm - 01:50pm)
Seminar: Theoretical Biology Seminar
Title: Properties of networks with partially structured and partially random connectivity
Speaker: Yashar Ahmadian, Columbia University
(Host: Vladimir Itskov)
Location: MB106

Networks studied in many disciplines, including neuroscience, have connectivity that may be stochastic about some underlying mean connectivity represented by a nonnormal matrix. Furthermore the stochasticity may not be i.i.d. across elements of the connectivity matrix. More generally, the problem of understanding the behavior of stochastic matrices with nontrivial mean structure and correlations arises in many settings. We address this by characterizing large random $N\times N$ matrices of the form $A = M + LJR$, where $M$, $L$ and $R$ are arbitrary deterministic matrices and $J$ is a random matrix of zero-mean independent and identically distributed elements. $M$ can be nonnormal, and $L$ and $R$ allow correlations that have separable dependence on row and column indices. We first provide a general formula for the eigenvalue density of $A$. For $A$ nonnormal, the eigenvalues do not suffice to specify the dynamics induced by $A$, so we also provide general formulae for the transient evolution of the magnitude of activity and frequency power spectrum in an $N$-dimensional linear dynamical system with a coupling matrix given by $A$. These quantities can also be thought of as characterizing the stability and the magnitude of the linear response of a nonlinear network to small perturbations about a fixed point. We derive these formulae and work them out analytically for some examples of $M$, $L$ and $R$ motivated by neurobiological models. We also argue that the persistence as $N\rightarrow\infty$ of a finite number of randomly distributed outlying eigenvalues outside the support of the eigenvalue density of $A$, as previously observed, arises in regions of the complex plane $\Omega$ where there are nonzero singular values of $L^{-1} (z\one - M) R^{-1}$ (for $z\in\Omega$) that vanish as $N\rightarrow\infty$. When such singular values do not exist and $L$ and $R$ are equal to the identity, there is a correspondence in the normalized Frobenius norm (but not in the operator norm) between the support of the spectrum of $A$ for $J$ of norm $\sigma$ and the $\sigma$-pseudospectrum of $M$.

December 2nd, 2014 (02:30pm - 03:45pm)
Seminar: Logic Seminar
Title: Gelfand duality and Ramsey theory
Speaker: Willem Fouché, University of South Africa
Location: MB315

In this talk, I want to discuss recent developments arising from the paper [1] (1987). In this paper, Andreas Blass introduced the idea of, what he called, a Ramsey action of a group on a discrete space in the context of understanding the tension, in set theory, between the Axiom of Choice and the Prime Ideal Theorem for Boolean algebras. In this talk, I will discuss this very tension, with the benefit of hindsight, as to what has happened to his viewpoint as viewed from the world of the dynamical aspects of structural Ramsey theory. Some recent links with Gelfand duality of commutative C∗- algebras will be discussed.

December 2nd, 2014 (03:30pm - 06:00pm)
Seminar: Working Seminar: Dynamics and its Working Tools
Title: Free Products as Topological Groups in Dynamics, II
Speaker: Kurt Vinhage, Penn State
Location: MB216

In the study of partially hyperbolic homogeneous systems, the Lyapunov manifolds become cosets of unipotent subgroups and their free product appears as a natural object of study. At best, the natural topology of such a free product can be described as unpleasant, with key properties like local compactness and first countability failing. In these talks, we will see how dynamical and topological arguments can be used to tame these complexities, leading to a local rigidity result. In particular, we will see two powerful and classical theorems on topological groups appear (one of Montgomery-Zippin and another of Gleason-Palais), and prove one of them. Time-permitting, we will see how free products may also appear in a non-homogeneous setting.

December 3rd, 2014 (12:05pm - 01:20pm)
Seminar: Geometry Luncheon Seminar
Title: CANCELLED! Hilbert geometry and non-Euclidean geometry
Speaker: CANCELLED! Athanase Papadopoulos, U Strasbourg, visiting CUNY
Location: MB114

I will present some results in Hilbert geometry with analogues in spherical and hyperbolic geometry

December 3rd, 2014 (02:30pm - 03:20pm)
Seminar: Applied Algebra and Network Theory Seminar
Title: A recipe for post-classical probabilistic theories
Speaker: Alex Wilce, Susquehanna University
Location: MB106

I describe a simple recipe for building (generally, non-classical) probabilistic models having strong symmetry properties from group-theoretic data. Where it can be made functorial, this construction becomes a recipe for symmetric monoidal categories of such models. A version of quantum mechanics arises as a special case.

December 3rd, 2014 (03:35pm - 04:25pm)
Seminar: Teaching Mathematics Discussion Group Seminar
Title: What is "True" Understanding?
Speaker: Attendees, Penn State
Location: MB102

What exactly is mathematical understanding and when has a person achieved it? Although it may seem clear cut, further thought on the subject may convince you otherwise. This week, we read an article that attempts to tackle the ambiguity of "true" mathematical understanding and discuss our experiences (both as students and as teachers).
Sfard, Anna. "Reification as the Birth for Metaphor." For the Learning of Mathematics 14.1 (1994): 44-55. Web.

December 3rd, 2014 (03:35pm - 04:35pm)
Seminar: Geometry Working Seminar
Title: CANCELLED! Mapping class groups of surfaces
Speaker: CANCELLED! Athanase Papadopoulos, U Strasbourg, visiting CUNY
Location: MB114

I will survey several results on actions of mapping class groups of surfaces on several spaces

December 3rd, 2014 (03:35pm - 04:25pm)
Seminar: Logic Seminar
Title: Fourier properties of algorithmically random Brownian motion
Speaker: Willem Fouché, University of South Africa
Location: MB315

We discuss the Fourier dimensions of sets generated by an algorithmically random Brownian motion. We outline how these results open the way for studying the local time of such a Brownian motion.

December 3rd, 2014 (05:01pm - 06:01pm)
Seminar: Student Geometric Functional Analysis Seminar
Title: classification program for C*-algebras I
Speaker: Hung-Chang Liao
Location: MB216

I will talk about the classification program for C*-algebras. The ultimate goal of this program is to classify a large class of C*-algebra (namely simple nuclear ones) by K-theory. This will be a two-part talk: tomorrow I will discuss some history and basic definitions, and next week I will try to discuss the recent development of this program.

December 4th, 2014 (11:15am - 12:05pm)
Seminar: Algebra and Number Theory Seminar
Title: Uniform Dilations in High Dimensions
Speaker: Michael Kelly, University of Michigan
Location: MB106

It is a theorem of Glasner that given an infinite subset X of the torus R/Z and an epsilon greater than 0 there exists a positive integer n such that any interval of length epsilon in R/Z contains a point of the set nX (that is, nX is epsilon-dense in R/Z). The set nX is called a dilation of X by n. Alon and Peres have shown that the dilation factor n can be chosen to be a prime or n=f(m) for some integral polynomial f with degree(f)>0 and integer m. We will discuss various developments on these sorts of topics and I'll present joint work with Le Thai Hoang where we consider this phenomenon in higher dimensions.

December 4th, 2014 (02:30pm - 03:30pm)
Seminar: Noncommutative Geometry Seminar
Title: Affine Weyl groups and Langlands duality
Speaker: Roger Plymen, University of Manchester
Location: MB106
December 5th, 2014 (03:35pm - 04:35pm)
Seminar: Probability and Financial Mathematics Seminar
Title: A distribution for avalanche sizes in neural dynamics
Speaker: Anirban Das, PSU
Location: MB106

In 2002 Eurich, Herrmann and Ernst proposed a distribution for the size of avalanches in complete networks of neurons without leaking (Phys. Rev. E, 2002). The talk will present some new additional properties for this and similar distributions, in particular, a formula for the variance and a complete derivation of Wenbo Li's formula for expectations.

December 8th, 2014 (12:20pm - 01:30pm)
Seminar: CCMA Luncheon Seminar
Title: Solitary waves in lattices: an introduction
Speaker: Anna Vainchtein, University of Pittsburgh (Host: X Li)
Location: MB114

The interplay between discreteness and nonlinearity in many physical systems leads to the formation of solitary waves. For example, such waves were experimentally observed in granular materials, electrical transmission lines and optical fibers. Much of the interest in these nonlinear waves was triggered by the pioneering study by Fermi, Pasta and Ulam (1955). The subsequent work of Zabusky and Kruskal (1965) has revolutionized the nonlinear science by connecting the FPU problem to its continuum near-sonic limit described by the KdV equation. In integrable systems solitary waves, known as solitons, are now well understood, with one-dimensional Toda lattice being the most prominent example that has an exact solution covering a broad range of behaviors from delocalized low-energy waves in the KdV limit to highly localized high-energy waves. Most discrete systems, however, are non-integrable. In this case understanding the transition from the KdV limit to the strongly discrete waves has mostly relied on numerical and quasicontinuum approximations. In this introductory talk I will describe some of the important developments in this area.

December 8th, 2014 (02:30pm - 03:30pm)
Seminar: Computational and Applied Mathematics Colloquium
Title: Solitary waves in non-integrable lattices
Speaker: Anna Vainchtein, University of Pittsburgh (Host: X Li)
Location: MB106

The interplay between discreteness and nonlinearity in many physical systems leads to the formation of solitary waves. For example, such waves were experimentally observed in granular materials, electrical transmission lines and optical fibers. Much of the interest in these nonlinear waves was triggered by the pioneering study by Fermi, Pasta and Ulam (1955). The subsequent work of Zabusky and Kruskal (1965) has revolutionized the nonlinear science by connecting the FPU problem to its continuum near-sonic limit described by the KdV equation. In integrable systems solitary waves, known as solitons, are now well understood, with one-dimensional Toda lattice being the most prominent example that has an exact solution covering a broad range of behaviors from delocalized low-energy waves in the KdV limit to highly localized high-energy waves. Most discrete systems, however, are non-integrable. In this case understanding the transition from the KdV limit to the strongly discrete waves has mostly relied on numerical and quasicontinuum approximations. In this talk I will review some of these results and describe recent work with Lev Truskinovsky on a non-integrable FPU problem with piecewise quadratic potential. We construct an exact solitary wave solution that captures the entire crossover velocity range between the low-energy limit and strongly localized waves that involve only one particle moving at a time. The solution is expressed in the form of an infinite series. A truncation of the series involving progressively smaller characteristic wavelengths produces a nested set of approximate solutions. Even the simplest solution of this type that accounts only for the longest wave lengths provides a better overall approximation of solitary waves than some conventional quasicontinuum models. I will also discuss recent work with Aaron Hoffman, Yuli Starostvetsky and Doug Wright on solitary waves in a diatomic FPU lattices. At generic mass ratios, pulses propagating through such lattices radiate lattice waves traveling behind them, thus precluding formation of genuine solitary waves. However, numerical simulations suggest that under certain conditions there is a sequence of special 'anti-resonance' values of mass ratio at which there is no radiation and solitary waves do exist. Using multiscale asymptotic analysis, we find a Fredholm-type condition, made explicit for the diatomic Toda lattice, for mass ratios approximating such values.

December 9th, 2014 (01:00pm - 01:50pm)
Seminar: Theoretical Biology Seminar
Title: Constraints from data on the architecture of neural and synaptic representations
Speaker: John Collins, Penn State (Physics)
(Host: Carina Curto)
Location: MB114

I explore some ways in which key properties of behavioral and other data can constrain the architecture of the representations and functions in biological memory systems. First I examine working memory. Then I revisit the old issue of local v. distributed representations, and show that the concept of distributed representations needs to be split into several very different kinds.

December 9th, 2014 (02:30pm - 03:30pm)
Seminar: GAP Seminar
Title: The abstract Hodge--Dirac operator and its stable discretization
Speaker: Ari Stern, Washington University in St Louis
Location: MB106

Dirac operators play an important role in linking geometry and topology with the analysis of PDEs. I will discuss a relatively new approach to studying Hodge--Dirac-type operators (due to Axelsson, Keith, and McIntosh, Invent. Math., 2006), which uses an abstract formalism involving nilpotent operators on Hilbert spaces, but which nevertheless preserves many key properties, including the Hodge decomposition. I will then present some recent work (joint with P. Leopardi) on the stable discretization of these operators, with applications to certain problems in "discrete Clifford analysis" and numerical PDEs.

December 9th, 2014 (03:30pm - 06:00pm)
Seminar: Working Seminar: Dynamics and its Working Tools
Title: Dynamics near low-dimensional attractors of neural networks.
Speaker: Vladimir Itskov, Penn State
Location: MB216

After a brief introduction to generic rate models of neural networks I will talk about "continuous attractor networks" that are hypothesized to underlie some critical functions in the mammalian brain. These networks are perturbations of systems that have a low-dimensional submanifold of steady states in the continuum limit. I will discuss basic properties of such networks and also pose some open questions.

December 9th, 2014 (04:00pm - 05:00pm)
Seminar: Applied Analysis Seminar
Title: Fluctuations of the effective conductance in a random conductor
Speaker: Jim Nolen, Duke University
Location: MB106

I will talk about solutions to a linear, divergence-form elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably; this talk is about fluctuations of the solution around its mean behavior. Suppose an electric potential is imposed at the boundary of some heterogeneous conducting material. Some current will flow through the material. What is the net current? For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe recent results about fluctuations of this quantity. In particular, I'll explain a central limit theorem for the effective conductivity.

December 10th, 2014 (03:35pm - 04:25pm)
Seminar: Teaching Mathematics Discussion Group Seminar
Title: Metaphors for Learning
Speaker: Attendees, Penn State
Location: MB102

We often understand fundamental ideas like learning through metaphor. This week, we read an article that discusses two kinds of metaphors employed and argues that both are incomplete without the other.
Sfard, Anna. "On Two Metaphors for Learning and the Dangers of Choosing Just One." Educational Researcher 27.2 (1998): 4. Web.

December 10th, 2014 (03:35pm - 04:35pm)
Seminar: Geometry Working Seminar
Title: Around basics of h-principle and its applications to symplectic topology III
Speaker: Alena Erchenko, Penn State
Location: MB114

Тhis is the third talk on the subjet. Even though h-principle originated from Nash's embedding theorems and Gromov's book "Partial Differential Relations",the easiest applications are to symplectic topology.One need to realize that h-principle is not a number of ready-to-use tools but rather an philosophy of how to approach a certain class of problems. We follow books by Eliashberg and McDuff et al. Our goal is not to nearly cover the subject but to spark interest towards this very impressive set of ideas and techniques in (geometric) PDEs and PD inequalitiues.

December 10th, 2014 (05:01pm - 06:01pm)
Seminar: Student Geometric Functional Analysis Seminar
Title: classification program for C*-algebras II
Speaker: Hung-Chang Liao
Location: MB216

I will talk about the classification program for C*-algebras. The ultimate goal of this program is to classify a large class of C*-algebra (namely simple nuclear ones) by K-theory. This will be a two-part talk: I will discuss some history and basic definitions, and I will try to discuss the recent development of this program.

December 11th, 2014 (11:15am - 12:05pm)
Seminar: Algebra and Number Theory Seminar
Title: Integral points and orbits in the projective plane
Speaker: Aaron Levin, Michigan State University
Location: MB106

We will begin by discussing the problem of classifying the behavior of integral points on affine subsets of the projective plane. As an application of this study, we will examine the problem of classifying endomorphisms of the projective plane with an orbit containing a Zariski dense set of integral points (with respect to some plane curve). This is joint work with Yu Yasufuku.

December 11th, 2014 (03:30pm - 04:20pm)
Seminar: Department of Mathematics Colloquium
Title: Why am I interested in the Feynman transform of the operad governing commutative algebras?
Speaker: Vasily Dolgushev, Temple University (Host: Mathieu Stienon)
Location: MB114

Operads and their generalizations are ubiquitous in mathematics. One possible way to get an example of an operad is to consider the collection $\{\operatorname{Hom}(X^n, X)\}_{n \ge 1}$ of sets of maps for a fixed set $X$. Another example comes from the Fulton--MacPherson compactification of the configuration space of points. While for usual operads multiplications are encoded by planar trees, for \emph{modular operads}, introduced by Getzler and Kapranov, multiplications are encoded by certain graphs with some additional data. My talk is devoted to a modular operad which was introduced by Kapranov in his seminal paper on Rozansky--Witten invariants. I will explain how this modular operad is related to Vassiliev finite type invariants of framed knots and to computation of homotopy groups of spaces of long knots ``modulo immersions.''