# Colloquia Archive

## A short look at the long history of the lemniscate of Bernoulli

Wednesday, March 6, 2013 @ 3:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Joel Langer, Professor, Case Western University

Abstract: TBA

## Vortex Filament Interactions and Hamiltonian PDEs

Wednesday, December 5, 2012 @ 3:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Walter Craig, Canada Research Chair, Department of Mathematics and Statistics, McMaster University

Abstract: Analysts intersted in PDEs and in Hamiltonian dynamical systems have developed techniques for the phase space analysis of many model nonlinear Hamiltonian evolution equations. In this talk I will describe some applications of these ideas to problems in fluid dynamics, and in particular a version of an infinite dimensional KAM theory. The main application concerns the interaction of two near-parallel vortex filaments in three dimensions. As well, I will speculate about further applications of phase space analysis of Hamiltonian PDEs to other nonlinear systems of fluid dynamics, in the form of nonlinear evolution problems of physical significance.

About the speaker: Dr. Craig earned the PhD at New York University's Courant Institute, as a student of Louis Nirenberg. He is an expert in the analysis of partial differential equations, with more than 75 papers, and with an emphasis on free-surface fluid dynamics.

## Elastic Capsules in Viscous Flow

Wednesday, November 14, 2012 @ 3:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Michael Siegel, Professor, Department of Mathematics, New Jersey Institute of Technology

Abstract: Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modeled but there are few analytical results. In this talk, complex variable techniques are used to derive semi-analytic solutions for the steady-state response and time-dependent evolution of elastic capsules in 2D Stokes flow. The analysis is complemented by spectrally accurate numerical simulations of the time-dependent evolution. We provide compelling evidence of the formation of finite-time cusp singularities, of which there are few rigorous examples in interfacial Stokes flow, with none involving elastic interfaces. We discuss the relevance of our findings to 3D capsules. This is joint work with Michael Booty and Michael Higley.

About the speaker: Dr. Siegel earned the PhD at the Courant Institute at New York University under the supervision of Russell Caflisch. He held positions at Ohio State University and California Institute of Technology before joining the faculty of NJIT. He has numerous publications in analysis and computing in fluid dynamics.

## Analysis Seminar

Friday, October 26, 2012 @ 12:00 PM
Korman Center, Room 245

Two-dimensional singular integral operators via poly-Bergman spaces, and Toeplitz operators with peudodifferential symbols.

http://www.math.drexel.edu/~tolya/analysis.html

## Stability and Restrictions of Time-Delayed Dynamical Networks

Wednesday, October 24, 2012 @ 3:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Benjamin Webb, Postdoctoral Associate, Laboratory of Statistical Physics, Rockefeller University

Abstract: Networks are found throughout nature. They are studied in numerous areas of science and are used in the design and implementation of technology. Because of finite processing speeds and transmission of signals over distances, the dynamics of such networks are inherently time-delayed. In this talk we consider the global stability of dynamical networks with distributed and non-distributed delays. In the case of non-distributed delays we show that the problem simplifies to considering networks without time-delays. By extending this technique we further describe how one can reduce the dimension of a network (dynamical system) to gain improved estimates of the network's (system's) global stability. This approach of "restricting" a network is illustrated by applications to various classes of Cohen-Grossberg neural networks.

About the speaker: Dr. Webb earned the Ph.D. at Georgia Tech, and recently started a postdoctoral position at Rockefeller University. His research is in the area of dynamical systems.

## Analysis Seminar

Friday, October 19, 2012 @ 12:00 PM
Korman Center, Room 245

Construction of a Sturm-Liouville vessel using Gelfand-Levitan theory. Solution of the Korteweg-de Vries equation on a half-line.

http://www.math.drexel.edu/~tolya/analysis.html

## Analysis Seminar

Friday, October 12, 2012 @ 12:00 PM
Korman Center, Room 245

Fixed-point theorems for noncommutative functions.

http://www.math.drexel.edu/~tolya/analysis.html

## Time-Stepping Methods for Stochastic Differential Equations

Wednesday, October 10, 2012 @ 3:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Jonathan Goodman, Professor, Department of Mathematics, Courant Institute, New York University

Abstract: We discuss computational methods for solving stochastic differential equations. Monte Carlo computations typically average some quantity of interest over many approximate sample paths of the SDE. We review the convergence theory for SDE time stepping methods in the weak and pathwise senses. We offer an alternative to these that studies the accuracy of the joint distribution of approximate paths at the discrete time steps. The convergence analysis in this sense centers on short time asymptotic approximations to the advection diffusion equation that governs the transition probability density for paths over a time step. The expansion in Gaussians and Hermite polynomials is not new, but we need new L^1 error estimates. This is joint work with Peter Glynn, Jose Antonio Peres, and Sangmin Lee.

About the speaker: Dr. Goodman earned the Ph.D. at Stanford University, and is a Professor at New York University's Courant Institute of Mathematical Sciences. He is an expert in the applied analysis of partial differential equations, and has been involved with Courant's program in Mathematical Finance for more than 15 years.

## Analysis Seminar

Friday, October 5, 2012 @ 12:00 PM
Korman Center, Room 245

2012 Fall Term Analysis Seminar

http://www.math.drexel.edu/~tolya/analysis.html

## Analysis Seminar

Friday, October 5, 2012 @ 12:00 PM
Korman Center, Room 245

Norm-constrained determinantal representations of multivariable polynomials.

http://www.math.drexel.edu/~tolya/analysis.html

## Analysis seminar: "$k$-Schur Functions and a New $k$-Pieri Rule"

Friday, June 8, 2012 @ 2:00 PM
Korman 245

Speaker: Avi Dalal, Graduate student, Department of Mathematics - Drexel University

Abstract: http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/Avi.pdf

Analysis

## Analysis seminar: "Gibbs Measures and the Young Bouquet: Work of A. Borodin and G. Olshanski (cont.)"

Friday, June 1, 2012 @ 2:00 PM
Korman 245

Speaker: Robert Boyer, Professor, Department of Mathematics - Drexel University

Abstract: There is a very close relationship between the characters of the infinite symmetric group and the infinite dimensional unitary group $U(\infty)$ which is not captured by attempts to extend the classical Schur-Weyl duality. In a recent paper, Borodin and Olshanski introduce a new object the Young bouquet" to explain this connection. I will also relate their work to my \href{http://www.mathjournals.org/jot/1992-028-002/1992-028-002-005.pdf}{1992 paper on infinite dimensional classical groups}.

Analysis

## Modeling Whisking In Air And Against Objects

Thursday, May 24, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: David Golomb, Professor, Department of Physiology and Neurobiology, Ben Gurion University

Abstract: A mechanistic description of the generation of whisker movements and its modulation by contact with objects is essential for understanding the control of whisking and vibrissal active touch. First, we study whisking in air and explore how facial-motoneuron spikes are translated, via an intrinsic muscle, to whisker movements. This is achieved by constructing, simulating, and analyzing a computational, biomechanical model of the motor plant. Our model predicts that contraction of a single intrinsic muscle results in movement of its two attached whiskers with different amplitudes; the relative amplitudes depend on the resting angles and on the attachment location of the intrinsic muscle on the anterior whisker. Second, we explore the effect of contact with object by developing and exploring models of bending whiskers. Preliminary results show that the real whisker bends more in response to contact than the model whisker.

About the speaker: Dr. Golomb works in theoretical and computational neuroscience. Before joining the faculty of Ben Gurion University, he held visiting positions at Cornell University and the National Institutes of Health. He is currently on sabbatical at the Howard Hughes Medical Institute research campus at Janelia Farm.

## Analysis seminar: "Proximity analysis in approximation methods for manifold-valued data"

Friday, May 18, 2012 @ 2:00 PM
Korman 245

Speaker: Thomas Yu, Associate Professor, Department of Mathematics - Drexel University

Abstract: http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/Thomas.pdf

Analysis

## Should Penalized Least Squares Regression Be Interpreted As Maximum A Posteriori Estimation?

Thursday, May 17, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Remi Gribonval, Directeur de Recherche, Institut de Recherche en Informatique et Systèmes Aléatoires

About the speaker: Dr. Gribonval earned the PhD from Université Paris Dauphine. He has numerous publications on the mathematics of signal processing, machine learning and pattern recognition. In 2011 he was awarded the Blaise Pascal Prize.

## Analysis seminar: "Gibbs Measures and the Young Bouquet: Work of A. Borodin and G. Olshanski"

Friday, May 11, 2012 @ 2:00 PM
Korman 245

Speaker: Robert Boyer, Professor, Department of Mathematics - Drexel University

Abstract: There is a very close relationship between the characters of the infinite symmetric group and the infinite dimensional unitary group $U(\infty)$ which is not captured by attempts to extend the classical Schur-Weyl duality. In a recent paper, Borodin and Olshanski introduce a new object the Young bouquet" to explain this connection. I will also relate their work to my \href{http://www.mathjournals.org/jot/1992-028-002/1992-028-002-005.pdf}{1992 paper on infinite dimensional classical groups}.

Analysis seminar

## Curves of Constant Torsion and Pseudospherical Surfaces

Thursday, May 10, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Ivan Sterling, Professor, Department of Mathematics and Computer Science, St. Mary's College of Maryland

Abstract: We'll start with new developments on a very old and simple problem: Classify the spherical curves of constant torsion. We will review the connection between curves of constant torsion and surfaces with constant negative Gaussian curvature ("pseudospherical surfaces"). We will survey the 200-year effort to classify pseudospherical surfaces, including some recent developments. An attempt will be made to show lots of graphics.

About the speaker: Dr. Sterling earned the Ph.D. from University of California at Berkeley. He held positions at U.C. Santa Cruz and Max Planck Institute (Bonn) before joining the faculty at St. Mary's College of Maryland. He has numerous publications in the theory of integrable systems and differential geometry.

## Analysis seminar: "On a solution of the Sturm-Liouville and the Korteweg-de-Vries equations with periodic and almost periodic parameters using theory of vessels"

Friday, April 27, 2012 @ 2:00 PM
Korman 245

Speaker: Andrey Melnikov, Assistant Teaching Professor, Department of Mathematics - Drexel University

Abstract: Using theory of vessels there will be presented a solution of the Sturm-Liouville equation with a spectral parameter \lambda: - y''(x) + q(x) y(x) = \lambda^2 y(x). The construction and existence of a vessel, solving this equation is closely connected to self-adjoint operators, related to this problem with different boundary conditions. We will state the main theorem of vessel existence in the case of periodic and almost periodic q(x). Using the notion of evolutionary vessel, we will construct a solution for the KdV equation q'_t = -3/2 qq'_x +1/4 q'''_{xxx} with a given periodic/ almost periodic initial value q(x,0). Bibliography: 1 (previous work) L. D. Fadeev. The inverse problem in the quantum theory of scattering. Journal of Mathematical Physics, 4(1):72--104, 1963 B. M. Levitan, I. M. Gelfand. On the determination of a differential equation from its spectral function (Russian). Izvestiya Akad. Nauk SSSR. Ser. Mat., 15, 1951. Levitan B. M. Sargsjan I.S. Introduction to spectral theory: selfadjoint ordinary differential operators, volume~39. Translations of mathematical Monographs, Providence, R.I., 1975. 2. (related to new results): A. Melnikov. Finite dimensional Sturm Liouville vessels and their tau functions. IEOT 74(4): 455--490, 2011. A. Melnikov. On a theory of vessels and the inverse scattering. http://arxiv.org/abs/1103.2392 A. Melnikov. Solution of the KdV equation using evolutionary vessels. http://arxiv.org/abs/1110.3495 A. Melnikov. On a solution of the Sturm-Liouville and the Korteweg-de-Vries equations with periodic and almost periodic parameters using theory of vessels, in preparation

Analysis seminar

## Numerical Methods for Interfaces in Fluids and Regularizing Effects in Difference Equations

Thursday, April 26, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: J. Thomas Beale, Professor, Department of Mathematics, Duke University

Abstract: We will discuss computational methods for a moving interface in viscous fluid and related error analysis. We will begin with a general description of numerical methods for partial differential equations with discontinuities at interfaces, in which the unknown is calculated at grid points and the (possibly moving) interface is represented separately. We will then describe work with Anita Layton in which we have designed a method for computing the coupled motion of an interface, made of elastic material, with fluid governed by the Navier-Stokes equations. The distinctive aspect of this approach is that we decompose the velocity at each time into a part determined by the Stokes equations (describing viscosity-dominated flow) and the interfacial force, and a ''regular'' remainder which can be calculated in a conventional way. Simple test problems indicate that this method is second-order accurate (error O(h^2) where h is the grid spacing), even though the truncation error near the interface is first order. This gain in accuracy has often been observed with methods of this type. For time-independent problems, related to the Laplacian, this observation can be explained by discrete elliptic estimates in maximum norm. For the time-dependent problem, maximum norm estimates for discrete versions of the linear diffusion equation show a regularizing effect, depending on the time discretization, and partially explain the numerical results for fluid flow with interfaces. We have developed versions of these methods which are partially implicit in the interface motion, in order to allow larger time steps. These use a strategy like that in work of Hou, Lowengrub and Shelley.

About the speaker: Professor Beale earned his PhD at Stanford University under the direction of Ralph Phillips. He held positions at the Courant Institute of New York University and Tulane University before joining the faculty of Duke University. The products of his research program have included fundamental contributions to the analytical theory and numerical analysis of free-surface fluid flows, as well as the celebrated Beale-Kato-Majda theorem.

## Analysis seminar: "Determinantal Representations and the Hermite Matrix" (after T. Netzer, D. Plaumann, and A. Thom)

Friday, April 20, 2012 @ 2:00 PM
Korman 245

Speaker: Hugo J. Woerdeman, Professor, Department of Mathematics - Drexel University

Abstract: The talk is based on the paper in arXiv: http://arxiv.org/abs/1108.4380

Analysis seminar

## Physical Reasoning in Mathematics

Thursday, April 19, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Mark Levi, Professor, Department of Mathematics, Pennsylvania State University

Abstract: Physics often provides mathematics not only with a problem, but also with the idea of a solution. Some calculus problems can be solved more quickly without calculus, by using physics instead. Quite a few theorems which may seem somewhat mysterious become completely obvious when given a proper physical incarnation. This is the case for some “elementary” theorems (the Pythagorean Theorem, Pappus' theorems, some trig identities, Euler's formula V-E+F=2, and more) and for some less elementary ones: Noether's theorem on conserved quantities, the preservation of Poincare's integral invariants, the Gauss-Bonnet theorem, the Riemann Mapping Theorem, Green's theorem, Moser's theorem on Jacobians, the uniformization theorem, and more (no familiarity with any of these is assumed). I will describe a miscellaneous sampling of problems according to the audience's preferences.

About the speaker: Professor Levi received his PhD from the Courant Institute, under the supervision of Jurgen Moser. He is an expert in the theory of dynamical systems, with more than 50 publications in scientific journals. His book, "The mathematical mechanic: using physical reasoning to solve problems," was published by Princeton University Press in 2009.

## Analysis seminar: "Approximation of Scattered Manifold-Valued Data"

Friday, April 13, 2012 @ 2:00 PM
Korman 245

Speaker: Thomas Yu, Associate Professor, Department of Mathematics - Drexel University

Abstract: Differential geometers have a trick called partition-of-unity which allows them to think globally and act locally' when they want to craft certain objects (tensor fields, embeddings etc.) on a manifold. Approximation theorists have a similar trick, except that they usually work on an Euclidean domain and the kind of partitions-of-unity they created are more sophisticated and specialized which give them approximation power. Motivated by an application in dimensionality reduction of dynamical system (which I will briefly discuss if time allows), I address the problem of approximating a function with an Euclidean domain and the range being a (fixed) manifold. Now, none of the tricks mentioned above apply directly. But with a few twists based on Karcher's Riemannian center of mass, the moving least square method, and a technique we call "proximity analysis", we found a natural solution to the problem with provable approximation and smoothness properties. When the manifold of interest is a symmetric space (e.g. sphere, Grassmannians, matrix Lie groups, etc.), the approximation method is also computationally efficient.

Analysis

## Migraine with Aura and Cortical Spreading Depression

Thursday, April 12, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Robert Miura, Distinguished Professor, Departments of Mathematical Sciences and of Biomedical Engineering, New Jersey Institute of Technology, and Graduate Faculty, Department of Biological Sciences, Rutgers-Newark

Abstract: Migraine with aura affects about 20% of the people who suffer from migraine (a severe headache affecting people around the world). The triggers for this disease are mainly undiagnosed and treatment is generally ad hoc from patient to patient. Migraine with aura has been linked to waves of cortical spreading depression (CSD) in the visual cortex of the brain. To devise rational treatments of migraine with aura, we need to learn much more about the brain and about CSD. We can learn a lot about the brain by studying extreme phenomena, such as CSD. CSD was discovered over 65 years ago by A.A.P. Leão, a Brazilian physiologist during his PhD research on epilepsy at the Harvard Medical School. CSD is characterized by nonlinear chemical waves that propagate at very slow speeds, on the order of mm/min, in the cortex of different brain structures in various experimental animals, and occurs in humans. CSD waves generate massive changes in extracellular ion concentrations. To date, we do not have a good explanation of how CSD occurs, although a number of mechanisms have been hypothesized to be important for CSD wave propagation. In this talk, I will review some of the characteristics of migraine with aura and CSD wave propagation, and describe some of the mechanisms that are believed to be important for CSD, including ion diffusion, membrane ionic currents, osmotic effects, the spatial buffer mechanism, neurotransmitter substances, gap junctions, metabolic pumps, and synaptic connections. The emphasis will be on developing dynamical system models and continuum models of CSD, consisting of coupled nonlinear diffusion equations. (Work done in collaboration with H. Huang, York University, Toronto, Canada and W. Yao, Fudan University, Shanghai, China.)

About the speaker: Professor Miura's current research interests are in mathematical modeling in neuroscience. He is a Fellow of the Society for Industrial and Applied Mathematics, of the American Association for the Advancement of Science, and of the Royal Society of Canada. He has received a Steele Prize of the American Mathematical Society for his fundamental contributions to the theory of solitons and integrable systems.

## Symbolic Dynamics and DNA

Thursday, April 5, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: David Koslicki, Postdoc, Department of Mathematics, Drexel University

Abstract: Symbolic dynamics provides a wealth of analytical tools for studying infinite strings of symbols that arise from continuous-time dynamical systems. Surprisingly, few of these techniques have been adapted for use with finite strings of symbols. In this talk, I will detail the adaptation of two such tools: topological entropy and pressure. Fortuitously, these tools are well adapted to studying DNA sequences. Results will be given on the efficacy of applying such techniques to a variety of genomic analysis problems (including intron/exon classification and gene density estimation).

## Analysis seminar: "A Tour of Free Noncommutative Convexity"

Friday, March 23, 2012 @ 2:00 PM
Korman 245

Speaker: Victor Vinnikov, Professor, Department of Mathematics - Ben-Gurion University

Abstract: There emerged over the course of the last decade a general paradigm for passing from the usual setting of $d$ commuting variables to the corresponding free noncommutative (in short, nc) setting. On the algebraic level, we replace the algebra of polynomials and its field of fractions --- the field of rational functions --- by the free algebra (the algebra of nc polynomials) and its universal skew field of fractions --- the free skew field (the skew field of nc rational functions). On the geometric level, we replace the vector space ${\mathbb R}^d$, and subsets thereof, by the nc space over ${\mathbb R}^d$ --- the disjoint union of square matrices over ${\mathbb R}^d$ of all sizes or equivalently of $d$-tuples of square matrices over ${\mathbb R}$ of all sizes, and nc subsets theoreof (subsets that are closed under the formation of direct sums). And on the analytic level, we replace differentiable or analytic functions on an open set in ${\mathbb R}^d$ by nc functions on an o! pen nc set in the nc space over ${\mathbb R}^d$. In this talk, I will discuss nc convex polynomials, rational, and entire functions, and (if time permits) nc convex semialgebraic sets. One reason for doing so, besides the general paradigm mentioned above, is that this leads to a beautiful and rigid theory. Another reason is that many optimization problems appearing in systems and control are {\em dimension-independent} --- the natural variables are matrices (rather than just collections of scalars) and the problem involves rational expressions in these matrix variables (rather than arbitrary expressions in the matrix entries) which {\em have therefore the same form independent of the matrix sizes}. Therefore applying such powerful recent methods in convex optimization as semidefinite programming often lead naturally to the study of nc convexity and nc positivity, with the polynomials and rational functions in commuting variables replaced by nc polynomials and nc rational functions.

Analysis

## Analysis seminar: "Schumaker's conjecture: do Bernstein operators induce P-matrices?"

Friday, March 9, 2012 @ 2:00 PM
Korman 245

Speaker: Simon Foucart, Assistant Professor, Department of Mathematics - Drexel University

Abstract:In the univariate case, the Berstein operator induce totally positive matrices. This is no longer true in the bivariate case, but it was conjectured by Schumaker that all principal minors are positive. The conjecture is still open to this day. I will try to spark the audience's interest in the simple-looking problem by showing that the conjecture is true up to degree 17 and by indicating possible avenues to prove or disprove it in general.

Analysis

## Soap Films in Electric Fields

Thursday, March 8, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: John Pelesko, Professor, Department of Mathematics, University of Delaware

Abstract: In 1968, in the context of investigating fundamental questions in electrohydrodynamics, G.I. Taylor studied the electrostatic deflection of elastic membranes. Utilizing soap film as the membrane material and applying a fixed high voltage potential difference between two supported circular membranes, Taylor showed experimentally that at a critical voltage the two membranes snap together and touch. That is, the equilibrium state where the membranes remained separate that existed at smaller voltages either became unstable or failed to exist. This instability is familiar to researchers in the MEMS (microelectromechanical systems) and NEMS (nanoelectromechanical systems) fields where it is known as the “pull-in” instability. In fact, in an interesting historical coincidence H.C. Nathanson and his coworkers studied this instability in the context of a primitive MEMS device at roughly the same time as Taylor was conducting his studies. Nathanson is responsible for the “pull-in” nomenclature and the analysis of a mass-spring model of this effect. Taylor, in conjunction with R.C. Ackerberg developed and numerically analyzed a more accurate membrane based model of electrostatic deflection. Recently, a rigorous analysis of this model was completed. Surprisingly, even this simple model of electrostatic deflection contains a rich solution set exhibiting a bifurcation diagram with infinitely many folds. In this talk, we provide an overview of recent results on the interaction of soap films with electrostatic fields. We discuss a re-creation of the Taylor experiment, some new experimental results and discuss the relevance of this research to MEMS and NEMS systems.

About the speaker: Professor Pelesko earned his PhD at New Jersey Institute of Technology, and held positions at Caltech and Georgia Tech before joining the faculty of University of Delaware. He has published many papers in the area of industrial and applied mathematics, as well as the book, "Modeling MEMS and NEMS," which was co-authored with David Bernstein.

## Analysis seminar: "Aligned semigroups of endomorphisms of $B(H)$"

Friday, March 2, 2012 @ 2:00 PM
Korman 245

Speaker: Daniel Markiewicz, Lecturer, Department of Mathematics - Ben-Gurion University of the Negev

Abstract: We will present a general survey of the classification problem for continuous one-parameter semigroups of endomorphisms of B(H), or E$_0$-semigroups for short. In the last part of the talk I will discuss the new class of \emph{aligned} E$_0$-semigroups, which exhibits several new interesting properties. For example, they are \emph{prime} in the sense that they cannot be decomposed as non-trivial tensor products (up to the appropriate equivalence condition). This is joint work with Christopher Jankowski and Robert T. Powers.

Analysis seminar

## Electromagnetic Cloaking At All Frequencies

Thursday, March 1, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Michael Vogelius, Professor, Department of Mathematics, Rutgers University

Abstract: I will provide an overview of some recent work on so-called techniques of "cloaking by mapping". The goal is to surround part of space with a (specially constructed) "cloak" in such a way that the cloak, and any object inside the cloak, is invisible (or nearly invisible) to electromagnetic inspection. I will focus on a particular approximate cloaking scheme, and rigorous estimates for the degree of near-invisibility it provides. The estimates have in order of increasing difficulty been carried out for the Conductivity Problem, the fixed frequency Helmholtz Problem, and the full, time-domain, Wave Problem. Time permitting I shall try to describe some of the problems and the solutions in all of these three settings. This work has been joint with R.V. Kohn, H-M. Nguyen, D. Onofrei and M. Weinstein.

About the speaker: Professor Vogelius is chairman of the Department of Mathematics at Rutgers University. He is a leading expert in the analysis of partial differential equations, with more than 75 scientific publications. Prior to joining Rutgers, Professor Vogelius held positions at University of Maryland, Stanford University, Ecole Polytechnique Federale de Lausanne, and the University of Copenhagen.

## Analysis semnar: "Principal minor assignment problem (after K. Griffin and M. J. Tsatsomeros)"

Friday, February 24, 2012 @ 2:00 PM
Korman 245

Speaker: Hugo Woerdeman, Professor, Department of Mathematics - Drexel University

Abstract: The talk is based on the paper by K. Griffin and M . J. Tsatsomeros, Principal minors. II. The principal minor assignment problem. Linear Algebra Appl. 419 (2006), no. 1, 125–171.

Analysis seminar

## From Analysis of Algorithms to Analytic Combinatorics

Thursday, February 23, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Robert Sedgewick, William O. Baker Professor, Department of Computer Science, Princeton University

Abstract: Analytic Combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. Primarily due to the efforts of Philippe Flajolet and his many research collaborators, the theory has emerged over recent decades as essential both for the scientific analysis of algorithms in computer science and for the study of scientific models in many other disciplines, including probability theory, statistical physics, computational biology and information theory. This talk surveys thirty years of joint work with Flajolet that was inspired by learning the analysis of algorithms from Knuth and that culminated in the publication of two books: "An Introduction to the Analysis of Algorithms" and "Analytic Combinatorics".

About the speaker: Prof. Sedgewick's research interests include mathematical techniques for the analysis of algorithms, design of data structures and algorithms, and program visualization. He has published widely in these areas and is the author of several books, including a well-known series of textbooks on algorithms that have sold over one-half million copies. The 4th edition of "Algorithms" (completely rewritten, with K. Wayne) was published in 2011. His other recently published books are "An Introduction to Programming in Java: An Interdisciplinary Approach" (with K. Wayne) in 2007 and "Analytic Combinatorics" (with P. Flajolet) in 2009.

## Analysis seminar: "Completely positive maps and E-semigroups"

Friday, February 17, 2012 @ 2:00 PM
Korman 245

Speaker: Chris Jankowski, Lecturer, Department of Mathematics - University of Pennsylvania

Abstract: Order-preserving (in the sense of positivity of operators) linear maps between C*-algebras arise naturally in functional analysis. We give an overview of a special subclass called the completely positive maps, which carry an additional positivity condition that makes them particularly useful and convenient to work with. Many basic maps, such as positive linear functionals and adjoint-preserving homomorphisms, are completely positive. We turn our attention to a classical result of Wigner that gives the form of all one-parameter semigroups of automorphisms of B(H) (the set of bounded operators acting on a Hilbert space H). If we replace automorphism with endomorphism, the picture changes drastically, and the classification theory for such semigroups (E-semigroups) is far from understood. We explore how to construct, and classify, E-semigroups using completely positive maps acting on the n x n complex matrices.

Analysis

Thursday, February 16, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Walter Strauss, L. Herbert Ballou University Professor, Department of Mathematics and Division of Applied Mathematics, Brown University

Abstract: Consider a classical 2D gravity wave (studied by Euler, Poisson, Cauchy, Airy, Stokes, Levi-Civita,..., as well as many modern mathematicians) with an arbitrary vorticity function. Let the wave travel at a constant speed over a flat bed. Using local and global bifurcation theory and topological degree, one can prove that there exist many such waves of large amplitude. I will outline the existence proof, joint with Adrian Constantin, and also exhibit some recent computations, joint with Joy Ko, of these waves using numerical continuation. The computations illustrate certain relationships between the amplitude, energy and mass flux of the waves. If the vorticity is sufficiently large, the first stagnation point of the wave occurs not at the crest (as with the much-studied irrotational flows) but on the bed directly below the crest or else in the interior of the fluid. The vorticity also affects the pressure beneath the fluid.

About the speaker: Dr. Strauss is one of the world's leading experts on the theory of partial differential equations. He has authored or coauthored over 100 scientific publications, which include fundamental contributions to the theory of nonlinear waves.

## Analysis seminar: "Stable symmetric polynomials and the Schur-Agler class"

Friday, February 10, 2012 @ 2:00 PM
Korman 245

Speaker: Hugo Woerdeman, Professor, Department of Mathematics - Drexel University

Abstract: The talk is based on the paper by G. Knese "Stable symmetric polynomials and the Schur-Agler class". http://arxiv.org/pdf/1008.4560.pdf

Analysis

## Hybrid Inverse Problems

Thursday, February 9, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Guillaume Bal, Professor, Department of Applied Physics & Applied Mathematics, Columbia University

Abstract: Hybrid inverse problems, also called coupled-physics inverse problems, aim to leverage the physical coupling between a high-contrast, low-resolution, modality and a low-contrast, high-resolution, modality to obtain a high-contrast, high-resolution, imaging procedure. From a mathematical standpoint, several such procedures involve the reconstruction of parameters from internal functionals of said parameters and solutions to partial differential equations. This talk will review several theoretical analyses and numerical results obtained recently in the field of hybrid inverse problems. Applications include the biomedical imaging modalities called Quantitative Photo-Acoustic Tomography, Transient and Magnetic Resonance Elastography, and Ultrasound Modulated Tomography.

About the speaker: Dr. Bal is an expert in the theory of partial differential equations and inverse problems, with more than 100 scientific publications. He received the Ph.D. from University of Paris VI, and held positions at University of Chicago and Stanford University before joining the faculty of Columbia University. Dr. Bal received the 2011 Calderon Prize from the Inverse Problems International Association.

## Analysis seminar: "Examples of solutions of the KdV equation using evolutionary vessels"

Friday, February 3, 2012 @ 2:00 PM
Korman 245

Speaker: Andrey Melnikov, Assistant Professor, Department of Mathematics - Drexel University

Abstract: In this talk I am going to present some classes of solutions of the KdV equation q'_t = q'_x q + q'''_{xxx} constructed from evolutionary Sturm-Liouville vessels.This will include: 1. Rational solutions, Solitons, polynomial-trigonometric, exponential-polynomial solutions arising from finite dimensional realizations of vessels, 2. Classical solutions based on the inverse scattering of the SL equation (Fadeev) -y''_{xx}+q (x,t) y = \lambda y, 3. Periodic solutions arising from realizations of vessels on $H=\ell^2$, 4. An approach to construction of almost-periodic solutions. These solutions are known to exist in general (Lax), but if we take the initial condition $q(x,0)=sin(x)+sin(\sqrt{2}x)$, no one knows whether a solution of KdV exists with this initial concrete value. These approach is general in the sense that many more (completely integrable) PDEs can be solved in this way, and evolutionary vessels are essentially generalize the Zakarov-Sabbath scheme. Some material can be found in 1. V. E. Zakarov; A. B. Sabbath. Integration of the nonlinear equations of mathematical physics by the method of the inverse scattering problem (Russian). Fun. Anal., 8(3):43{53, 1974. Translation in Funct. Anal. Appl., 1974, 8:3, 226{235. 2. "Solution of the KdV equation using evolutionary vessels" http://arxiv.org/abs/1110.3495, 3. "On a theory of vessels and the inverse scattering" http://arxiv.org/abs/1103.2392, 3. P.D. Lax. Periodic solutions of the KdV equation. Comm. Pure Appl. Math., 28:141-188, 1975. 4. P.D. Lax. Almost periodic solutions of the KdV equation. SIAM Rev., 18(3):351-375, 1976. 5. L.D. Fadeev. The inverse problem in the quantum theory of scattering. Journal of Mathematical Physics, 4(1):72{104, 1963.

Analysis

## Analysis seminar "Rational Inner Functions on the Disk and on Polydisks II"

Friday, January 27, 2012 @ 2:00 PM
Korman 245

Speaker: David Scheinker, Visiting Assistant Professor, Department of Mathematics - Drexel University

Abstract: http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/Scheinker2.pdf

Analysis seminar

## Analysis seminar:"Rational Inner Functions on the Disk and on Polydisks"

Friday, January 20, 2012 @ 2:00 PM
Korman 245

Speaker: David Scheinker, Visiting Assistant Professor, Department of Mathematics - Drexel University

Abstract: Fix a rational inner function $f$ on $D$ with degree $N$. If one chooses any $N+1$ distinct points $x_1$,...,$x_{N+1}$ on $D$, then the Nevanlinna-Pick problem on $D$ with data $x_1$,...,$x_{N+1}$ and $f(x_1)$,...,$f(x_{N+1})$ has a unique solution. Furthermore, essentially every Nevanlinna-Pick problem on $D$ with a unique solution arrises this way. In this talk, we give some examples of Nevanlinna-Pick problems on $D^n$ with $n>1$ demonstrating the ways in which this behavior of rational inner functions on $D$ fails to extend to $D^n$. We then introduce some definitions and theorems demonstrating the ways in which this behavior extends to 1 dimensional algebraic varieties passing through $D^2$.

Analysis seminar

## Perpetuities

Thursday, January 19, 2012 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Pawel Hitczenko, Professor, Department of Mathematics, Drexel University

Abstract: Perpetuities are random variables that appear quite frequently in various aspects of applied mathematics and other fields of science. They have been studied systematically since the influential paper by Kesten in 1973. In this talk I will introduce and review the main 'classical' results and will also discuss recent progress that has been made in the general theory.

About the speaker: Dr. Hitczenko is an expert in the theory of probability. He has more than 50 scientific publications, many of which focus on applications of probability to such topics as combinatorics and analysis of algorithms.

## Analysis seminar: "Transfer function realization of rational inner functions"

Friday, January 13, 2012 @ 2:00 PM
Korman 245

Speaker: Selcuk Koyuncu, Ph.D. student, Department of Mathematics - Drexel University

Abstract: My talk will be based on paper "Synthesis of two-dimensional lossless m-ports with prescribed scattering matrix" by A. Kummert

Analysis seminar

## Analysis seminar: "Good old Loewner's theorem: a proof by A. Koranyi and B. Sz.-Nagy (cont.)"

Friday, December 2, 2011 @ 1:00 PM
Korman 245

Speaker: Dmitry Kaliuzhnyi-Verbovetskyi, Associate Professor, Department of Mathematics - Drexel University

Analysis seminar

## Analysis seminar: "Good old Loewner's theorem: a proof by A. Koranyi and B. Sz.-Nagy"

Friday, November 18, 2011 @ 1:00 PM
Korman 245

Speaker: Dmitry Kaliuzhnyi-Verbovetskyi, Associate Professor, Department of Mathematics - Drexel University

Abstract: The classical Loewner theorem (C. Loewner, 1934) gives a necessary and sufficient condition for a function to be monotone on Hermitian matrices with the spectrum in a given interval. In my talk, I will present a proof given by A. Koranyi and B. Sz.-Nagy in 1958, which uses reproducing kernel Hilbert space techniques.

Analysis seminar

## Pieri RULES!!

Thursday, November 17, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Jennifer Morse, Professor, Department of Mathematics, Drexel University

Abstract: We will talk about several long-standing open problems involving the search for a description of non-negative numbers that arise in geometry, representation theory, and physics as connected to Schubert Calculus, Gromov-Witten invariants, and Macdonald polynomials. We will see how combinatorics was used to solve related problems and promises to shed light on these open problems.

About the speaker: Dr. Morse joined the faculty of Drexel University in 2006. She previously held a position at University of Miami, and earned her PhD from University of California at San Diego. She is known for the development of the theory of the k-Schur polynomials.

## Analysis seminar: "On the dimension of multivariate spline spaces, especially on Alfeld splits"

Friday, November 11, 2011 @ 1:00 PM
Korman 245

Speaker: Simon Foucart, Assistant Professor, Department of Mathematics - Drexel University

Abstract: In this talk, I will discuss recent advances on a major problem in multivariate spline theory, namely the determination of the dimension of spline spaces. I will introduce a computational method that finds an explicit formula for the dimension given a prescribed simplicial partition. It is based on concepts from Algebraic Geometry, and the key results will also be explained using Bernstein--Bezier techniques. I will illustrate the computational method on specific examples, especially the Alfled split of a tetrahedron.

Analysis seminar

## Dynamics on Random Graphs

Thursday, November 10, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Amitabha Bose, Professor, Department of Mathematical Sciences, New Jersey Institute of Technology

Abstract: Systems involving interacting nodes are ubiquitous arising in among other places biological, social and computer networks. Motivated by an example arising during development of neuronal networks, I will discuss the problem of finding periodic solutions on random graphs. In particular, we are interested in exploring what structural features of the graph are most likely to sustain periodic activity. The answer turns out to depend in surprising and non-intuitive ways on the type of dynamics that exist at each node and also the rules for interaction between nodes. No prior knowledge of neuronal networks or random graphs is needed to follow this talk.

About the speaker: Dr. Bose's research is in the area of the theory of dynamical systems, with applications to neuroscience. Before joining the faculty at NJIT, he held an appointment at Boston University. His research continually receives support from the National Science Foundation, and he has also held a Fulbright-Nehru fellowship.

## Analysis seminar: "Three variable stable polynomials"

Friday, November 4, 2011 @ 1:00 PM
Korman 245

Speaker: Hugo Woerdeman, Professor, Department of Mathematics - Drexel University

Analysis seminar

## Chimera States in Heterogeneous Kuramoto Networks

Thursday, November 3, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Carlo Laing, Senior Lecturer, Institute of Information and Mathematical Sciences, Massey University

Abstract: Chimera states occur when a network of identical oscillators splits into two groups, one consisting of synchronous oscillators, the other of partially-synchronous oscillators. We show how to analyze such states when the oscillators are not identical, using the recent ansatz of Ott and Antonsen to derive non-local differential equations governing the network dynamics in the continuum limit. The same techniques can be used to study transient fronts which connect regions of high synchrony with regions of asynchrony.

About the speaker: Dr. Laing earned the PhD from University of Cambridge. Before joining the faculty of Massey University, he held positions at University College London and University of Pittsburgh, among others. In 2008, he was awarded the JH Michell Medal by ANZIAM.

## Analysis seminar: "On the Distribution of the Number of Summands in a Random Partition with Respect to a Boltzmann Weighting"

Friday, October 28, 2011 @ 1:00 PM
Korman 245

Speaker: Daniel Parry, PhD student, Department of Mathematics - Drexel University

Abstract: http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/Parry.pdf

Analysis seminar

## Analysis seminar: "Zero product of two-level Toeplitz operators"

Friday, October 21, 2011 @ 1:00 PM
Korman 245

Speaker: Selcuk Koyuncu, PhD student, Department of Mathematics - Drexel University

Abstract: http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/Selcuk.pdf

Analysis seminar

## Exponential Screening and Optimal Rates of Sparse Estimation

Thursday, October 20, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Philippe Rigollet, Assistant Professor, Department of Operations Research and Financial Engineering - Princeton University

Abstract: We consider a general, not necessarily linear, regression problem with Gaussian noise and study an aggregation problem that consists in finding a linear combination of approximating functions, which is at the same time sparse and has small mean squared error (MSE). We introduce a new estimation procedure, called Exponential Screening (ES) that shows remarkable adaptation properties: it adapts to the linear combination that optimally balances MSE and sparsity, whether the latter is measured in terms of the number of non-zero entries in the combination or in terms of the global weight of the combination. The power of this adaptation result is illustrated by showing that ES solves optimally and simultaneously all the problems of aggregation in Gaussian regression considered previously. Tight minimax lower bounds establish optimal rates of sparse estimation and that the ES procedure is optimal. A numerical implementation of ES that results in a stochastic greedy algorithm is discussed and compared to state-of-the-art procedures for sparse estimation. Finally several extensions of the principle of sparsity pattern aggregation to other estimation problems that are structurally sparse will be presented.

About the speaker: Dr. Rigollet earned his PhD from University of Paris VI. He has held visiting positions at Grenoble and Georgia Tech before joining the Department of Operations Research and Financial Engineering at Princeton University.

## Analysis seminar: "The Bessmertnyi class: old and new results (cont.)"

Friday, October 14, 2011 @ 1:00 PM
Korman 245

Speaker: Dmitry Kaliuzhnyi-Verbovetskyi, Associate Professor, Department of Mathematics - Drexel University

Analysis seminar

## PDE/Applied Math Seminar: "The Analogy of Phantom Traffic Jams and Detonation Waves"

Tuesday, October 11, 2011 @ 3:00 PM
Korman 247

"The Analogy of Phantom Traffic Jams and Detonation Waves" Benjamin Seibold, Temple University.

http://www.math.drexel.edu/~jdoug/seminar

## Analysis seminar: "The Bessmertnyi class: old and new results"

Friday, October 7, 2011 @ 1:00 PM
Korman 245

Speaker: Dmitry Kaliuzhnyi-Verbovetskyi, Associate Professor, Department of Mathematics - Drexel University

Abstract: In the early 1980s, M. F. Bessmertnyi introduced a class of matrix-valued rational functions of $d$ variables which admit a so-called finite-dimensional long resolvent representation. The motivation came from electrical engineering, namely, this is the class of impedance functions of passive electrical $2n$ ports where impedances of elements (resistances, capacitances, inductances) are considered as independent variables. Bessmertnyi gave several necessary conditions for a function to be in this class, however no good characterization of the class in intrinsic terms (as opposed to existence of a representation' terms) was given. Later on, in my 2004 paper I extended the definition of Bessmertnyi's class to not necessarily rational, operator-valued functions, which admit an infinite-dimensional long resolvent representation, and gave three characterizations of the extended class, one of which was in intrinsic terms. Also, the connection to the Schur--Agler class of analytic functions on the unit polydisk was established. In my current project with J. A. Ball, characterizations of the original Bessmertnyi's class have been obtained, including an intrinsic one.

Analysis seminar

## Analysis seminar: "A new solution of the KdV equation using evolutionary vessels".

Friday, September 30, 2011 @ 1:00 PM
Korman 245

Speaker: Andrey Melnikov, Assistant Teaching Professor, Department of Mathematics - Drexel University

Abstract: We are going to see some progress in the theory of (semi-time-varying) vessels. Models of vessels on curves with an unbounded operator were created recently and enabled to create new families of potentials $q(x)$ appearing in Sturm-Liouville equation $y''_{xx} - q(x) y = s^2 y$ (possessing "inverse scattering"). Among them let us mention $L^1(\mathbb R), \int_0^\pi c(\theta) e^{R \sin(\theta x)} d\theta$. New notion of an evolutionary vessel will be presented. As a rule such a vessel generates solutions for some nonlinear (polynomial) PDEs. An example of an evolutionary vessel constructing solutions of the KdV equation will be discussed in details. Solitons corresponds to finite-dimensional realizations of evolutionary KdV vessels. I will need a lot of help from the audience. So questions/remarks/potential problems are the main reasons behind this talk.

Analysis seminar

## Water Wave Theory and Some of Its Applications

Thursday, September 29, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Jerry Bona, Department of Mathematics, Statistics, and Computer Science - University of Illinois at Chicago

Abstract: After some historical commentary, we investigate how model equations for water waves arise in practice. The theory forthcoming from this exercise is then brought to bear upon problems of importance in oceanography. The discussion will include issues arising in tsunami propagation, rogue wave generation and beach protection strategy, as time permits.

No prior knowledge of water-wave theory will be assumed.

About the speaker: Dr. Bona is one of the world's leading experts in the theory of partial differential equations and has made fundamental contributions to the mathematical theory of oceanography. He earned his PhD from Harvard University under the supervision of Garrett Birkhoff. He has held positions at University of Chicago, Pennsylvania State University, and University of Texas at Austin, in addition to numerous visiting positions throughout the world.

## PDE/Applied Math Seminar: "Well-posedness issues in degenerate dispersive equations"

Tuesday, September 27, 2011 @ 3:00 PM
Korman 247

"Well-posedness issues in degenerate dispersive equations" Doug Wright, Drexel University.

http://www.math.drexel.edu/~jdoug/seminar

## Existence and Uniqueness Theory for the Incompressible Euler Equations

Thursday, June 2, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Elaine Cozzi, Department of Mathematics - Drexel University

Abstract: In the 1960s, Yudovich proved existence of a unique solution to the two-dimensional incompressible Euler equations when the vorticity, or curl of the velocity, is bounded. In the last few decades, mathematicians have built upon Yudovich's result by proving existence and uniqueness of solutions with vorticity in various function spaces, such as classical Sobolev spaces and more modern Besov and Triebel-Lizorkin spaces. In this talk, we present some of these results, as well as open problems in this area of research. We also consider extensions of these results to the three-dimensional case under the assumption that the velocity is an axisymmetric vector field.

About the speaker: Dr. Cozzi is a Visiting Assistant Professor in the Department of Mathematics. She earned her PhD from University of Texas at Austin, and has been a postdoctoral scholar at Carnegie Mellon University.

## Test functions, kernel functions, and matrix-valued Schur-Agler class

Friday, May 27, 2011 @ 1:00 PM
Korman 245

Speaker: Joseph A. Ball, Professor, Department of Mathematics - Virginia Tech

Abstract: The classical Schur class consists of holomorphic functions on the unit disk with values equal to Hilbert-space contraction operators. Elements of this space have a number of equivalent characterizations: contractive multipliers on Hardy spaces over the unit disk, positivity of the associated de Branges-Rovnyak kernel function, realization as the transfer function of a dissipative (or even conservative) discrete-time input/state/output linear system. There has been much interest of late in extensions of these ideas to more general settings. One such setting is the test-function approach, explored by Agler-McCarthy and Dritschel-Marcantognini-McCullough, where one defines a generalized Schur class as the intersection of the contractive multiplier algebras over the collection of kernels for which each function in a preassigned collection of test functions is a contractive multiplier. We indicate extensions of this test-function approach to the case where the test functions, kernel functions, and Schur-class functions are allowed to be matrix- or operator-valued. As an application we consider the matrix-valued Schur class of a finitely-connected planar domain and indicate connections with the recent negative solution of the spectral set question for such domains (having at least three holes) due to Agler-Harland-Raphael and Dritschel-McCullough. This is joint work with Mois\'es Guerra-Huaman of Virginia Tech.

Analysis seminar

## Compressive Sensing and Banach Space Geometry

Thursday, May 26, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Simon Foucart, Assistant Professor, Department of Mathematics - Drexel University

Abstract: This talk will show how the applied field of Compressive Sensing offers particularly nice insights on deep results about the geometry of high-dimensional $\ell_1^N$-balls. After reviewing the main theoretical results in Compressive Sensing, we will specifically focus on three topics: the neighborliness of the images of $\ell_1^N$-balls under random projections, the Kashin decompositions of the $\ell_1^N$-space as two orthogonal almost-Euclidean subspaces, and the Gelfand widths of $\ell_1^N$-balls relative to $\ell_2^N$.

About the speaker: Dr. Foucart received his PhD from University of Cambridge, and has held visiting or postodoctoral positions at Vanderbilt University, and in Bonn and Paris. He joined the Department of Mathematics as an Assistant Professor in 2010.

## Littlewood-Richardson functions and our conjecture

Friday, May 20, 2011 @ 1:00 PM
Korman 245

Speaker: Lei Cao, graduate student, Department of Mathematics - Drexel University

Abstract: I will talk about Littlewood-Richardson functions and connections between Littlewood-Richardson functions and other combinatorics objects counted by Littlewood Richardson coefficients. Then I will formulate our conjecture in terms of Littlewood-Richardson functions.

Analysis seminar

## A matrix-valued generalization of Bochner's theorem

Friday, May 13, 2011 @ 1:00 PM
Korman 245

Speaker: David Kimsey, graduate student, Department of Mathematics - Drexel University

Abstract: http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/Kimsey.pdf

Analysis seminar

## Noncommutative calculus: A gentle introduction

Friday, May 6, 2011 @ 1:00 PM
Korman 111 (room F)

Speaker: Dmitry Kaliuzhnyi-Verbovetskyi, Assistant Professor, Department of Mathematics - Drexel University

Abstract: Noncommutative objects usually appear as elements of some algebraic structures, however they often can be evaluated on square matrices of all sizes. If such evaluations respect direct sums and similarities, then the objects are called noncommutative functions. We will introduce the difference-differential calculus for such noncommutative functions and discuss some applications and further research directions. The talk is based on a current joint work with Victor Vinnikov (Ben-Gurion University).

Analysis seminar

## The Cauchy Problem for Systems of Conservation Laws

Thursday, May 5, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Kris Jenssen, Associate Professor, Department of Mathematics - Penn State University

Abstract: Conservation laws provide partial differential equations governing the flow of conserved quantities. These equations are fundamental and ubiquitous in physical theories from continuum mechanics. Classical examples include gas flow and elasticity, but also models for traffic flow, dynamics of crowds, granular flow etc. are covered by this class of equations.

The talk will focus on inviscid equations where dissipative effects are ignored. We shall consider the Cauchy problem for systems of such equations in one and several space dimensions. To highlight the issues particular to this class of equations (loss of regularity, non-uniqueness, entropy conditions...) we discuss some recent results (both positive and negative).

About the speaker: Dr. Jenssen received a Ph.D. from Norwegian University of Science and Technology. He has been a faculty member at Indiana University and North Carolina State University, as well as his current institution, Penn State University. Dr. Jenssen has received a prestigious NSF CAREER award.

## Multivariable moment problems

Friday, April 29, 2011 @ 1:00 PM
Korman 245

Speaker: Hugo Woerdeman, Professor, Department of Mathematics - Drexel University

Analysis seminar

## A new class of potentials for Sturm Liouville ODE

Friday, April 22, 2011 @ 1:00 PM
Korman 245

Speaker: Andrey Melnikov, Assistant Teaching Professor, Department of Mathematics - Drexel University

Abstract: In this talk there will be presented a new class of potentials q(x) appearing in the classical Sturm Liouville differential equation: -y''+q(x)y = s y, s - complex number (spectral parameter). This class consists of functions q(x), whose integral behaves like 1/x at infinity, and is additionally integrable around zero after multiplication by x. There are developed explicit formulas for these potentials using theory of vessels. The class of parameters (=scattering data) consists of a function (or a spectral measure), defined on a finite interval and corresponds to the spectrum of the corresponding differential operator L y = -y'' + q(x) y. Generalizations to other classes of potentials (family of NLS equations, Airy equation, etc) will be discussed too.

Analysis seminar

## Synchronized Oscillations, Basal Ganglia, and Parkinson's Disease

Thursday, April 21, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Leonid Rubchinsky, Associate Professor, Department of Mathematical Sciences - Indiana University Purdue University Indianapolis

Abstract: The basal ganglia are a group of interconnected subcortical nuclei, which are involved in neural control of movement and are impacted in Parkinson’s disease. This disease is marked by an increase in synchronized oscillatory activity. Understanding the dynamical nature of this synchronization is essential for the understanding of the brain function and for the search of new treatment strategies. Moreover, this experimentally observed dynamics signifies the interest in the mathematical studies of intermittent complex dynamics of synchronous oscillations.

I will present the results of the study of the fine temporal structure of the synchronous activity in the experimental data and in the mathematical models of parkinsonian physiology. While the synchrony is essentially a non-instantaneous phenomenon, if some level of phase-locking between two signals is present, we can study how this phase-locking disappears and returns in time. I will describe this time-series analysis approach and will present the results for the data recorded in the human patients. This dynamics is further studied in the models of neural circuits (conductance-based models in the form of ordinary differential equations), where we can identify what properties are needed for generation of this kind of dynamics. I will discuss the implications of the experimental and modeling study for the function and dysfunction of the human brain.

No expertise in neurophysiology of Parkinson’s disease is expected by the speaker, who intends to emphasize the dynamical, rather than medical aspects.

About the speaker: Dr. Rubchinsky earned a Ph.D. in physics from the Institute for Applied Physics of the Russian Academy of Science. After performing postdoctoral research at University of California, Davis, he has been a faculty member in the Department of Mathematics at Indiana University Purdue University Indianapolis and the Stark Neurosciences Research Institute of the Indiana University School of Medicine.

## A sum of squares approximation of nonnegative polynomials

Friday, April 15, 2011 @ 1:00 PM
Korman 245

Speaker: Hugo Woerdeman, Professor, Department of Mathematics - Drexel University

Abstract:This talk is based on a paper of Lasserre, Jean B. A sum of squares approximation of nonnegative polynomials. SIAM Rev. 49 (2007), no. 4, 651–669

Analysis seminar

## Measuring Topology and Geometry by Distributed Sensing

Thursday, April 14, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Michael Robinson, Department of Mathematics - University of Pennsylvania

Abstract: How can one determine the shape of one's surroundings by listening to ambient signal sources? Traditional remote-sensing ideas place strong requirements on the topology and geometry of the sensorium in order to exploit linearity, often by using integral transform methods. I will explain some recent work that avoids making such assumptions, and instead recovers topological structure first. At the center of the approach is a variant of the Whitney embedding theorem, suitably extended to handle certain kinds of discontinuities inherent in the received signal profile. I will also exhibit some experimental apparatus and associated results that demonstrate the feasibility of this approach in practice.

About the speaker: Dr. Robinson received a Ph.D. in Applied Mathematics at Cornell University, and is currently a Postdoctoral Researcher in the Department of Mathematics at the University of Pennsylvania.

## Napoleon's Theorem and an Interesting Fact About Planar Rotations

Friday, April 8, 2011 @ 1:00 PM
Korman 245

Speaker: Sean Ballentine, Undergraduate Student, Department of Mathematics - Drexel University

Abstract: In proving Napoleon's Theorem, an old problem in Geometry, I came across an interesting fact about 30 degree planar rotations and the orientation of triangle vertices after a transformation on the vertex set. This brought about the question whether or not this angle was the only angle to have this property. I will find the set of angles satisfying this interesting property and show how this applies to Napoleon's theorem.

Analysis seminar

## Brother, can you spare a compacton?

Friday, April 8, 2011 @ 11:00 AM
Korman Center, Room 245

Drexel PDE/Applied Math Seminar

Speaker: Philip Rosenau, Professor, Department of Applied Mathematics - Tel-Aviv University

Abstract: Unlike certain personal or national tragedies which may extend indef- initely, patterns observed in nature are of finite extent. Yet, as a rule, the solitary patterns predicted by almost all existing mathematical models extend indefinitely with their tails being a by product of their analytical nature. Rather then viewing such tails as a manifestation of the inherent limitation of math to model physics in detail, we adopt the opposite view: the persistence of tails in a large variety of solitary patterns points to a missing mechanism capable to constrain the pattern. Clearly, to induce a compact pattern one has to escape the curse of analyticity. Differently stated, one has to supplement the existing models with a mechanism(s) which may beget a local singularity. When this is done the resulting local loss of solution’s uniqueness enables to connect a smooth part of the solution with the trivial ground state and thus to form an entity with a compact support: the com- pacton. We shall describe a variety of singularity inducing mechanisms that beget compact solutions of dispersive or dissipative uni and multi-dimensional phenom- ena. Compactified variants of the K-dV, Klein-Gordon and Schroedinger equations will be surveyed.

In Part two of the lecture we shall discuss the intriguing nature of these (weakly strong or strongly weak) solutions, the underlying singularities and their relation with a discrete antecedent where a sharp fronts are replaced with tails decaying at a doubly exponential rate.

This lecture is made possible through the generous funding of the Louis and Bessie Stein Family Fellowship.

PDE/Applied Math Seminar Schedule

## Schmidt's Game, Friendly Measures and Exceptional Sets on Fractals

Thursday, March 31, 2011 @ 1:00 PM
Korman Center, Room 245

Drexel Mathematics Colloquium

Speaker: Lior Fishman, Department of Mathematics - Brandeis University

Abstract: In this talk I shall describe new results regarding generic properties of certain sets on fractals. Questions regarding these (often exceptional) sets, arising from number theory, dynamics and Diophantine approximation theory, have been extensively studied in recent years utilizing Schmidt's game and properties of the class of friendly measures. In order to highlight the main ideas in many of these proofs, I shall reprove a slight modification of Schmidt's original result regarding badly approximable numbers, pointing out where generalizations have been made using modern ideas and techniques. I wish to emphasize that the talk is accessible to anyone interested, as I will try and make it as self contained as possible.

About the speaker: Dr. Fishman earned a Ph.D. from Ben Gurion University, and is currently a Lecturer in Mathematics at Brandeis University.

## Completely positive maps and noncommutative dynamics

Friday, March 11, 2011 @ 1:00 PM
Korman B-103

Speaker: Chris Jankowski, Postdoctoral Fellow, Department of Mathematics - Ben-Gurion University

Abstract: We present an overview of completely positive maps and their applications to the study of a class of semigroups of endomorphisms of $B(H)$ (the bounded linear operators on a Hilbert space $H$) called E$_0$-semigroups. We find that a special kind of completely positive map acting on the $n \times n$ complex matrices can be combined with linear functionals acting on $B(L^2(0, \infty))$ in order to induce E$_0$-semigroups. Through this construction, we obtain uncountably many non-equivalent E$_0$-semigroups, some of which have a very surprising property.

http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/index.htm

## Entrainment of a Thalamocortical Neuron to Periodic Sensorimotor Signals

Thursday, March 10, 2011 @ 1:00 PM
Korman Center, Room 245

Speaker: Dennis Yang, Postdoctoral Associate, Department of Mathematics - Drexel University

Abstract: We study a 3D conductance-based model of a single thalamocortical (TC) neuron in response to sensorimotor signals. In particular, we focus on the entrainment of the system to periodic signals that alternate between 'on' and 'off' states lasting for time T1 and T2, respectively. By exploiting invariant sets of the system and their associated invariant fiber bundles that foliate the phase space, we reduce the 3D Poincare map to the composition of two 2D maps and also simplify the two components of one of the 2D maps to a uniform shift and a uniform decay. Then based on these 2D maps, we analyze the bifurcation of the entrained limit cycles as the parameters T1 and T2 vary.

## Scattering theory of the Schrodinger operator on the line using the theory of vessels

Friday, March 4, 2011 @ 1:00 PM
Korman 245

Speaker: Andrey Melnikov, Assistant Teaching Professor, Department of Mathematics - Drexel University

Abstract: In this talk I will try to convince that vessels are good. In the classical theory starting from the Schrodinger (Strum-Liouville) equation $L(y) = - y'' + q(x) y = \lambda y$ and assuming that integral of $x |q(x)|$ is bounded from 0 to infinity, one study the spectrum of the operator L and show then how to reconstruct the potential from the spectrum (solving Gelfand-Levitan or Marchenko equation). In the theory of vessels one starts from the spectrum and constructs from this the corresponding vessel, which in turn uniquely determines the potential, corresponding to this spectrum. Advantage of this technique is that one can construct and study potentials with different from the classical one, spectrum. It is enough for the spectrum to be symmetric with respect to the real line and to be a "contour" in the complex plane(corresponding to the continuous spectrum). In the new setting Gelfand-Levitan equation is simply derived and moreover, tau function arises as a determinant of a I+trace class operator. The main feature of vessel construction is Krein space realizations of symmetric, identity at infinity functions. Other advantages: 1. Generalizable to a scattering theory of n-th order operator. 2. one can consider scattering when the basic waves are not just exponents, i.e. solutions of $- y'' = \lambda y$, but solutions of the Schrodinger equation for a fixed initial potential. Among the aims of this research i can mention 1. characterize potentials which have the continuous spectrum on the cut of imaginary axis and some discrete points. 2. generate new classes of potentials admitting scattering theory, different from the known ones in the literature. The project is new, I have just started and need a feedback.

http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/index.htm

## Analysis of consensus protocols

Friday, February 25, 2011 @ 1:00 PM
Korman Center, Room 245

Speaker: Georgi Medvedev, Associate Professor, Department of Mathematics - Drexel University

Abstract: Consensus protocols are simple dynamical systems, which are used in various control problems for dynamical networks. They provide a convenient setting for studying the role of the network topology in shaping the system dynamics. In this talk, I will show how the stability of a consensus protocol depends on the cycle subspace of the underlying graph.

http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/index.htm

## Piecewise to Wise: Multivariate Splines and Algebraic Geometry

Thursday, February 24, 2011 @ 1:00 PM
Korman Center, Room 245

Speaker: Patrick Clarke, Assistant Professor, Department of Mathematics - Drexel University

Abstract: We will discuss a current project with S. Foucart applying algebraic geometry to the theory of multivariate spline functions.

## Artin's Theorem (cont.)

Friday, February 18, 2011 @ 1:00 PM
Korman 245

Speaker: Andrey Melnikov, Assistant Teaching Professor, Department of Mathematics - Drexel University

http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/index.htm

## Artin's Theorem

Friday, February 11, 2011 @ 1:00 PM
Korman Center, Room 245

Speaker: Andrey Melnikov, Assistant Teaching Professor, Department of Mathematics - Drexel University

Abstract: We will sketch the proof, including main ideas of the following theorem of Artin: Let R be a real closed field and let r be a rational function in n independent variables over R. If substituting arbitrary elements of R into r we obtain a positive answer in the sense of real field, then r can be represented as a sum of squares of rational functions in the same independent n variables. We will define real closed field, consider its basic properties. Using Sylvester's theorem, we will prove the theorem of Artin. The book of V.V Prasolov "Polynomials" is used as the reference.

http://www.math.drexel.edu/~dmitryk/Analysis%20seminar/index.htm

## Decompositions for Matrices and Permutations

Thursday, February 3, 2011 @ 1:00 PM
Korman Center, Room 245

Speaker: Benjamin Pittman-Polletta, Assistant Teaching Professor, Department of Mathematics - Drexel University

Abstract: Lie theory is the home of many beautiful connections between geometry, combinatorics, and topology. In this talk, we'll explore connections between the combinatorics of the symmetric group Sn and the geometry and topology of the group of unitary matrices with determinant one, SU(n). In particular, we'll see that to each decomposition of the longest permutation in Sn, there corresponds a factorization of SU(n) into a product of two-spheres. This will lead us to flag manifolds and their cell decompositions. Interest and time permitting, we'll end up at some infinite-dimensional generalizations and open questions.

## From EPDiff to Diffeons: Finite Dimensional Control of Diffeomorphic Matching

Thursday, January 13, 2011 @ 1:00 PM
Korman Center, Room 245

Speaker: Laurent Younes, Johns Hopkins University

Abstract: We consider the framework, which corresponds to the so-called "large deformation diffeomorphic metric matching" family of algorithm, in which the problem of finding an optimal registration between two shapes, or images, is formulated as an optimal control problem, where the control specifies an Eulerian velocity associated to a time dependent diffeomorphism, with a cost represented by the norm of the velocity in a suitably chosen Hilbert space of vector fields. Because this Hilbert norm can also interpreted as the expression of a right-invariant Riemannian metric in the Lie algebra of the diffeomorphism group, this directly relates to a wellknown geodesic equation, often called EPDiff,that expresses momentum conservation. In this talk, we describe an approach in which additional contraints are placed on the Eulerian velocity that ensures that it belongs to a specific finite dimensional subspace of the originally considered Hilbert space. This space, which evolves with the motion, is generated by a finite number of well chosen time-dependent vector fields that we call diffeons. We will describe the resulting maximum principle and optimization algorithms for the resulting registration problems and other related algorithms, and provide some preliminary numerical experiments in two dimensions.

## Dynamical System Theory and the Two-Dimensional Navier-Stokes Equations

Thursday, December 2, 2010 @ 1:00 PM
Korman Center, Room 245

Gene Wayne, Boston University

Two- dimensional fluid flows exhibit a variety of coherent structures such as vortices and dipoles which can often serve as organizing centers for the flow. These coherent structures can sometimes be associated with the existence of special geometrical structures in the phase space of the equations and in these cases the evolution of these flows can often be studied with the aid of dynamical systems theory. The dynamical systems ideas also suggest new ways of numerically studying such coherent structures and I will describe recent results which generalize the classical point vortex model to systematically include the effects of viscosity and finite core size.

## Inequalities for Combinatorial Families of Symmetric Functions

Thursday, November 11, 2010 @ 1:00 PM
Korman Center, Room 245

Curtis Greene, Haverford College

We will explore and attempt to link two threads: the classical theory of inequalities, with its vast literature, and the combinatorial theory of symmetric functions, which has been greatly stimulated in recent decades by its links to algebra and representation theory. From a combinatorial point of view, symmetric functions fall naturally into "families", and many classical inequalities (e.g. Newton's inequalities, and the AGM inequality) find a natural setting within this framework. We will sketch the combinatorial foundations of this approach to symmetric function inequalities, and survey progress toward a comprehensive theory and classification.

## Transmission Eigenvalues and Inverse Scattering Theory

Thursday, October 28, 2010 @ 1:00 PM
Korman Center, Room 245

David Colton, University of Delaware

The transmission eigenvalue problem is a new class of non-selfadjoint eigenvalue problems that first appeared in inverse scattering theory. This problem can be viewed as the dual of the well known "cloaking problem" where now, for a given inhomogeneous medium, one seeks an incident wave for which the inhomogeneous medium is invisible, i.e. there is no scattered field. It can be shown that this can occur for at most a discrete set of values of the wave number and such values are called transmission eigenvalues. It has only recently been shown that for a non-absorbing medium real transmission eigenvalues exist and that these eigenvalues can be determined from a knowledge of the far field pattern of the scattered wave. Through the derivation of Faber-Krahn type inequalities for transmission eigenvalues one can obtain estimates for the index of refraction of the medium, thus opening up new possibilities for investigating the inverse scattering problem for both acoustic and electromagnetic waves. It can further be shown that for a spherically stratified medium the transmission eigenvalues uniquely determine the index of refraction up to a normalizing constant. This talk will provide a brief survey of the above results as well as the formulation of open problems whose solution is necessary for further progress.

## Surface Relaxation Below the Roughening Temperature: Steps, PDE, and Self-Similarity

Thursday, October 21, 2010 @ 1:00 PM
Korman Center, Room 245

Robert Kohn, New York University

Crystalline films are often grown or annealed below their roughening temperature. The microscopic physics involves the attachment and detachment of atoms at steps, and the diffusion of atoms across terraces. The macroscopic consequences of these atomic-scale mechanisms are still poorly understood. My talk will discuss recent progress with Hala Al Hajj Shehadeh and Jonathan Weare, concerning the evolution of a one-dimensional step-train separating two facets in the "attachment-detachment-limited" regime. I'll explain why the evolution is asymptotically self-similar, and why its continuum limit is associated with a certain fourth-order nonlinear PDE. The talk will be self-contained, requiring no prior background about crystal growth.

## Eigenvalues of Graphs

Thursday, October 7, 2010 @ 1:00 PM
Korman Center, Room 245

Sebastian Cioaba, University of Delaware

Graph theory is the study of networks. In many situations, the only way we can study key combinatorial parameters of graphs such as edge-distribution, connectivity or expansion, is by using their eigenvalues. In this talk, I will describe some connections between the structure of graphs and their eigenvalues. The talk is accessible to undergraduate students.

## Dynamics and Impact of Spike-Time Correlations

Thursday, September 23, 2010 @ 1:00 PM
Korman Center, Room 245

Andrea Barreiro, University of Washington

Correlations among neural spike times are found widely in the brain; they can be used to modulate or limit information in population coding, as well as opening the possibility for cooperative coding of sensory inputs across neural populations. In this talk we discuss recent work towards understanding how the structure and transfer of correlations is affected by both intrinsic neuron dynamics and network architecture.

## Banded Matrices with Banded Inverses

Thursday, April 15, 2010 @ 1:00 PM
Korman Center 245

Gilbert Strange - Massachusetts Institute of Technology

## Class RSI and Scattering Theory of the Sturm-Liouville Operator

Thursday, April 1, 2010 @ 1:00 PM
Korman Center, Room 245

Andrey Melnikov, Drexel University