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Math Colloquia: Spring 2012

Friday January 27

No talk today

Friday February 3

No talk today

Friday February 10

Title Asymptotic Analysis of Intracellular Cargo Transport by Multiple Molecular Motors
Speaker Avanti Athreya
Johns Hopkins University

We describe a system of stochastic differential equations (SDEs) which model the interaction between processive molecular motors and their biomolecular cargo within a cell. We show that the classical experimental environment fits within a parameter regime which is qualitatively distinct from conditions one expects to find in living cells. Through an asymptotic analysis of our system of SDEs, we are able make analytical predictions for motor-cargo transport in two parameter regimes that have thus far eluded direct experimental observation: 1) highly viscous in vivo transport and 2) dynamics when multiple identical motors are attached to the cargo and microtubule.

Friday February 17

Title Computational methods for the study of rare events
Speaker Maria Cameron
University of Maryland, College Park

Small scale processes in physics and chemistry such as conformal changes in molecules are often modeled using the stochastic differential equations with a small noise. I will show that the problem of finding the most likely transition paths and transition rates can be reduces to numerical solution of a Hamilton-Jacobi type PDE. I will discuss two cases. (1) The gradient case where a system is evolving according to the overdamped Langevin dynamics. I will show some applications to the problems coming from chemical physics. (2) A more general and more complex nongradient case with isotropic diffusion. I will show how one can compute the most likely transition paths and transition rates between two attractors.

Friday February 24

Title Industrial Applications of Image Registration
Speaker Nathan Cahill
Rochester Institute of Technology

Image registration is the process of determining geometric transformations that can be applied to one or more images in a way that aligns them in a common reference frame. The ability to register images is useful in many applications: stitching consumer photographs to generate a panoramic field of view, averaging satellite images to reduce noise, and comparing 3-D contrast-enhanced magnetic resonance images of the breast to diagnose cancer are a few such applications. In this talk, we discuss how image registration can generally be posed as an optimization problem, and we focus on how the choice of objective function, transformation model, and optimization procedure influence the ability to effectively align images in the various application settings.

Friday March 2

Title Pushing the limits of sequential quadratic programming methods
Speaker Daniel Robinson
Johns Hopkins University

Sequential quadratic programming methods form a class of very powerful active-set algorithms for finding local solutions of nonlinear non-convex optimization problems. For instance, they have been used successfully for solving optimal control problems since the problem formulations take advantage of the strengths of these methods. The basic form of sequential quadratic programming methods was introduced and popularized in the 1970’s by Han and Powell. After a flurry of research, these methods have received little attention until recently. This is likely due to three factors including (i) the difficulty of incorporating exact second derivative information in an efficient and effective manner; (ii) the expense associated with solving each subproblem and, therefore, making it difficult to solve very-largescale problems; and (iii) the surge of interior-point methods in the nonlinear programming setting that have scaled-up very well. In this talk I will present recent advances in sequential quadratic programming methods that (i) have increased the size of problems that sequential quadratic programming methods may solve; (ii) allow for effective use of second-derivative information (when available); and (iii) allow for the use of certain active-set strategies that do scale up.

Friday March 9

Title Quantum Knots and Quantum Braids
Speaker Samuel J. Lomonaco

In this talk, we show how to reconstruct knot theory in such a way that it s intimately related to quantum physics. In particular, we give a blueprint for creating a quantum system that has the dynamic behavior of a closed knotted piece of rope moving in 3-space. Within this framework, knot invariants become physically measurable quantum observables, knot moves become unitary transformations, with knot dynamics determined by Schroedinger’s equation. The same approach can also be applied to the theory of braids. Toward the end of the talk, we briefly look at possible applications to superfluid vortices and to topological quantum computing in optical lattices.

Biosketch: Professor Lomonaco received his PhD in Mathematics from Princeton University. He has been a full professor of Computer Science and Electrical Engineering at the University of Maryland Baltimore County (UMBC) since 1985, serving as Founding Chair of the CS Department 1985 to 1991.

Representative Awards, Accomplishments, and Honors:

  1. Professor Lomonaco is a member of the European Academy of Sciences.
  2. He was a Visiting Key Research Scientist at the Mathematical Sciences Research Institute (MSRI) at the University of California at Berkley in 2004.
  3. He was a Senior LaGrange Fellow at the Institute for Scientific Exchange in Torino, Italy in 2005.
  4. For contributions made to the development of the programming language Ada, he received an award from the United States Under Secretary of Defense for Research and Engineering, Dr. Richard DeLauer.
  5. He was the first to introduce Quantum Information Science to the American Mathematical Society (AMS) by organizing and giving a two day AMS Short Course on Quantum Computation at the Annual Meeting of the AMS in Washington, DC in January 2000.
  6. He has published four books on Quantum Computation and Information Science.
  7. He serves as associate editor of the Journal of Knot Theory
  8. He has currently accepted an invitation to be a guest editor of the Journal of Quantum Information Processing for a Special Issue on Topological Quantum Computation.

Friday March 16

Title Entire Solutions to Equivariant Elliptic Systems with Variational Structure
Speaker Nicholas D. Alikakos
University of Athens

We discuss the system

Laplace (u)- grad W(u)=0 , u: R^n to R^m

for W with several global minima, and symmetric under a finite reflection group G. The solutions we study derive their interest from Geometry (geometric evolution, minimal surfaces) and Phase Transitions. About half of the talk will be dedicated to these relationships.

Friday March 23

Spring Break

Friday March 30

Title IPC

Infinite Possibilities Conference

Friday April 6

Title Old and new problems on the discretization of the Stokes equation
Speaker Francisco Javier Sayas
University of Delaware

In this talk I will discuss on some recent activity on the Stokes problem. I will first comment on the use of Darcy Boundary Conditions to simulate contact of viscous flow with porous media flow and what are the conditions that the separate discretizations of the Stokes and Darcy equations have to satisfy in order to build a stable numerical scheme. In passing I will comment on a theoretical problem on the existence of a discrete lifting of the normal trace for the most popular mixed finite element spaces. I will then move on to discuss how exactly divergence free elements (or elements satisfying a certain gap condition) are apparently necessary to tackle simple-minded coupling of finite and boundary elements for Stokes flow. Time permitting, I will comment on new (theoretical and practical) ideas on discretization of time domain Stokes flow.

Friday April 13

Title TBA
Speaker Danielle C. Tarraf
Johns Hopkins University


Friday April 20

Title ICA and IVA: Theory, Connections, and Applications
Speaker Tulay Adali

Data-driven methods are based on a simple generative model and hence can minimize the assumptions on the nature of data. They have emerged as promising alternatives to the traditional model-based approaches in many applications where the underlying dynamics are hard to characterize. Independent component analysis (ICA), in particular, has been a popular data-driven approach and an active area of research. Starting from a simple linear mixing model and imposing the constraint of statistical independence on the underlying components, ICA can recover the linearly mixed components subject to only a scaling and permutation ambiguity. It has been successfully applied to numerous data analysis problems in areas as diverse as biomedicine, communications, finance, geophysics, and remote sensing. This talk reviews the fundamentals and properties of ICA, in particular, for the most widely used approach to achieve ICA, mutual information minimization, introduces the generalization of ICA for analysis of multiple datasets, independent vector analysis (IVA), and discusses the connections between ICA and IVA. Two key problems for achieving a successful ICA decomposition, matrix optimization and density matching, are discussed in detail, along with new powerful approaches for addressing these two key signal processing problems. Successful application examples in medical image analysis are presented, and important practical issues are emphasized.

Friday April 27

Title Brownian motion, the diffusion equation, and colliding particle models
Speaker Jason Swanson
University of Central Florida

The well-known diffusion equation describes how the density of a diffusing substance, such as a cloud of dust particles in the air, evolves over time. The changes in the density are the result of each individual particle performing a Brownian motion, which is a continuous analogue of a random walk. Despite the fact that each particle is moving randomly, the diffusion equation itself describes a deterministic evolution. In this talk, I will begin by illustrating how the diffusion equation can be derived from the stochastic model as a first-order limiting approximation, as the number of particles tends to infinity. I will then discuss the second-order approximation, which is not deterministic. Rather, it is described by a stochastic partial differential equation. Such models have applications in studying smaller molecular systems, where the number of particles is not large enough to justify using the classical diffusion equation.

In the above models, the individual particles behave independently, and without interaction. In many applications, however, such models are insufficient. We frequently wish to describe systems of particles that include some kind of interaction potential. I will discuss the case in which particles interact through elastic collisions. Preliminary results are proven only in one dimension. In this case, the ensemble behavior of the system is exactly the same as the previous models, although the individual particles no longer perform Brownian motions. Instead, their motions will depend on the number of particles in the system. I will discuss the deterministic, first-order limiting approximation of their motion, as well as the second-order stochastic approximation.

Friday May 4

Title Insurance Under Extreme Risks: A Ruin Model with Stable Claims
Speaker Tuncay Alparslan
American University

Determining the probability of exceedance of a deterministic barrier by a random walk with negative drift is a popular problem in several areas of applied probability. In the context of actuarial mathematics, this probability has a very intuitive meaning and is known as the ruin probability. In this talk, ruin problem will be introduced in the context of an insurance company facing extreme risks. Claims will be modeled using stable distributions, a very important class of probability laws used commonly in extreme event modeling. Asymptotic estimates for the ruin probability based on a particular integral representation of stationary stable processes will be given, and connections between the asymptotic behavior of the ruin probability and the dependence structure of the underlying claim process will be explored.