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Math Colloquia: Spring 2008

Monday January 28

Title The ‘LifeLine’ – Vascular Access for Hemodialysis
Speaker Paula Budu-Grajdeanu
Mathematical Biosciences Institute
Ohio State University

When the kidneys fail to perform their functions to full capacity, one cannot live long without some form of renal replacement therapy, such as hemodialysis. In order to perform hemodialysis, the patient must have suitable vascular access to allow adequate flow of blood to the hemodialysis circuit. Hemodialysis vascular access complications due to progressive neointimal hyperplasia formation (narrowing of the blood vessel) remains the most common cause of morbidity/hospitalization among dialysis patients worldwide. In this talk I will show how mathematical modeling can be used to understand the influence of oxidative stress and turbulent flow on the hyperplasia formation, to predict access stenosis and to suggest interventions aimed at specific growth factors that may be successful in prolonging the life of the vascular access, while reducing the high costs of vascular access maintenance.

Wednesday January 30

Title Spatial problems in mathematical ecology
Speaker Andrew Nevai
Mathematical Biosciences Institute
Ohio State University

In this talk, I will introduce two spatial problems in theoretical ecology together with their mathematical solutions.

The first part of the talk concerns competition between plants for sunlight. In it, I use a mechanistic Kolmogorov-type competition model to connect plant population vertical leaf profiles (or VLPs) to the asymptotic behavior of the resulting dynamical system. For different VLPs, conditions can be obtained for either competitive exclusion to occur or stable coexistence at one or more equilibrium points.

The second part of the talk concerns the spatial spread of infectious diseases. Here, I use a family of SI-type models to examine the ability of a disease, such as rabies, to invade or persist in a spatially heterogeneous habitat. I will discuss properties of the disease-free equilibrium and the behavior of the endemic equilibrium as the mobility of healthy individuals becomes very small relative to that of infecteds. The family of disease models consists variously of systems of difference equations (which I will emphasize), ODEs, and reaction-diffusion equations.

Monday February 04

Title A Mathematical Model for Wound Angiogenesis as a Function of Tissue Oxygen Tension
Speaker Richard Schugart
Ohio State University

Wound healing represents a well-orchestrated reparative response that is induced by injuries. Angiogenesis, the formation of blood vessels from existing vasculature, plays a central role in wound healing. In this talk, I will present a mathematical model that addresses the role of tissue oxygen tension in cutaneous wound healing. Key components of the developed model include capillary tips, capillary sprouts, fibroblasts, inflammatory cells, chemoattractants, oxygen, and the extracellular matrix. The model consists of a system of nonlinear partial differential equations describing the interactions in space and time of the above variables. The simulated results agree with the reported literature on the biology of wound healing. The proposed model represents a useful tool to analyze strategies for improved healing and can be used to generate novel hypotheses for experimental testing.

Wednesday February 06

Title Intracellular Signaling in Excitable Cells
Speaker Bradford Peercy
National Institute of Health

How excitable cells process external stimuli dictates adaptation to the environment. These processes have a profound affect on diverse, organism-level functions such as learning and memory and blood sugar level. In this talk I will describe results from modeling intracellular signaling in neurons, and I will briefly discuss recent projects on pancreatic beta cells.

Experiments have shown the existence of evanescent calcium waves traveling from stimulated dendrites to the cell body. Nuclear calcium elevation triggers upregulation of genes hypothesized as necessary for changing neural sensitivity. I model the generation and propagation of the calcium waves for various stimulus protocols and confirm several experimental results as well as predict important signal characteristics. I’ll briefly discuss work to incorporate this model with a biochemical model at the synapse and nucleus to explain late long term potentiation.

Friday February 08

Title Mathematical Models of T-cell Development and Function
Speaker Stanca M. Ciupe
Duke University Medical Center

The immune response to infectious agents involves the presence and maintenance of a large number of T cells with highly variable antigen receptors and functional diversity. In the first part of the talk, we will explain the mechanisms underlying the establishment and maintenance of T cell receptor diversity and T cell population homeostasis using mathematical models and statistical analysis of data from DiGeorge syndrome patients undergoing thymus transplantation. In the second part, we will present the functional properties of the T cells when stimulated by antigen, in our case the Hepatitis B virus. We analyze DDE models of noncytolytic immune response and the cytolytic T cell response, explaining how the relative contribution of these two processes can lead to viral clearance or persistence.

Wednesday February 13

Title Growth and Symmetry: Pattern Formation on Plants
Speaker Patrick Shipman
University of Maryland College Park

Tiling planforms dominated by diamonds (such as the diamond-shaped seeds on a sunflower head), hexagons, or ridges (such as those on saguaro cacti) are observed on many plants. We analyze PDE models for the formation of these patterns that incorporate the effects of growth and biophysical and biochemical mechanisms. The aim is to understand both the underlying symmetries and the information specific to the mechanisms. The patterns are compared to Voronoi tessellations, and we will start to draw a bigger picture of growth and symmetry in biological systems.

Friday February 15

Title Pricing American Contingent Claims by Stochastic Linear Programming
Speaker Mustafa C. Pinar, Fulbright Scholar
Department of Operations Research and Financial Engineering
Princeton University (On sabbatical leave from Bilkent University, Ankara, Turkey)

In this talk I consider pricing of American contingent claims (ACC) and also different types of contingent claims as special cases of ACC’s, in a multi-period, discrete time, discrete state space setting. Determining the buyer’s price for ACC’s requires solving an integer program unlike European contingent claims for which solving a linear program is sufficient. However, I show that a relaxation of the integer programming problem which is a linear program, can be used to get the same lower bound for the price of the ACC. Therefore, solving a linear program is essentially enough to compute the fair price to the buyer of the option. This is joint work with PhD candidate Ahmet Camci from Bilkent University.

Friday February 22

No talk today

Friday February 29

Title Examples of Non-local Interactions in Biomathematical Models: Some Properties from Analysis and Simulation
Speaker Jonathan Bell

Biology returns something of value to mathematics via new classes of mathematical problems to study. In this talk I will discuss two topical areas leading to nonlinear partial differential-integral equations, where the defining characteristic is non-local interactions between dynamic populations of one sort or another. My first topic concerns a neural field involving populations of nerve cells organized in layers. As one example question, I will discuss, for a two-layer model, dependence of the shape of traveling wave solutions on a network parameter and a cellular parameter. As a second topic, I will introduce “degeneracy” into population dynamics. Numerous biological interactions, such as interactions between T cell receptors with antigens, interactions between enzymes and substrates, or interactions between predators and prey are not strictly specific. This less specific, or “sloppy”, specificity is called “degeneracy”. I will investigate the concept through generalizing the Lotka-Volterra and Verhulst population models, and highlight some effects of the degeneracy. I will also briefly mention other problems and questions to explore.

Friday March 07

No talk today

Friday March 14

Title Engineering Applications of Noncommutative Harmonic Analysis
Speaker Gregory S. Chirikjian
Mechanical Engineering
Johns Hopkins University

Classical Fourier Analysis has many applications in engineering and the physical sciences ranging from the study of heat conduction to image processing and optics. The non-Abelian extension of Fourier analysis (Noncommutative Harmonic Analysis) is far less known in engineering and the sciences. This area of mathematics, which draws on group representation theory, geometry and analysis also has many applications. In this talk, it is shown how the Fourier transform of functions on the group of rigid-body motions is applied to solve problems in DNA statistical mechanics, optical communications, and robotics.

About the Author: Greg Chirikjian is a professor of Mechanical Engineering at JHU, where he has been since 1992. He is the author of 150 articles ranging in topics including robotics, polymer theory, image processing, applied mathematics. He is the primary author of the 2001 book “Engineering Applications of Noncommutative Harmonic Analysis”.

Friday March 21

Spring Break

Friday March 28

Title Complete families and inverse obstacle scattering
Speaker Giovanni Franco Crosta
Department of Environmental Sciences
Università degli Studi Milano – Bicocca

Families of functions which are linearly independent and complete (complete, for short) play a role in the approximate representation of the wave functions impinging on and scattered by an obstacle, both in acoustics and electromagnetics. For simplicity, the scattering by a star-shaped, nonpenetrable, perfectly conducting obstacle in two spatial dimensions will be addressed, where the involved wave functions are scalar.

The following results will be presented.

  • A review of the approximate forward propagation method (G. F. CROSTA, FIELDS Institute Communications 25 (2000) pp 287–301; MR2001c:78010, Zbl 0963.35193).
  • Its application to the experimental IPSWICH data-set, including the inversion of magnitude-only scattered data.
  • The ongoing attempt at justifying the performance of the above method without requiring the RAYLEIGH hypothesis to hold and by using a complete family of functions derived from the minimization of the boundary defect.

Friday April 04

Title Kernel Functions and Interior-Point Methods for Sufficient Linear Complementarity Problems
Speaker Goran Lesaja
Department of Mathematical Sciences
Georgia Southern University

In this talk we discuss the importance of barrier and kernel functions in the design and analysis of interior ~V point methods. Furthermore, we present a class of polynomial primal-dual interior-point algorithms for sufficient linear commplementarity problems based on a new class of kernel functions. This class is fairly general and includes the classical logarithmic function, the prototype self-regular function, and non-self-regular kernel functions as special cases. The obtained complexity bounds match the currently best known complexity bounds obtained for these methods.

Friday April 11

Title Convergence and Stability of the Inverse Scattering Series for Diffuse Waves
Speaker Shari Moskow
Department of Mathematics
Drexel University

The application of the inverse Born series for the inverse scattering problem for diffuse light has been very promising, (Markel et. al. J. Opt. Soc. Am. A, 2003) . The technique gives nonlinear corrections to the inverse Born approximation which do not require any further operator inversion, and formally the entire series yields the absorption coefficient. Here we examine the radius convergence of this functional series and the stability of the limit with respect to perturbations in the measured data. We find in particular that its stability is determined by the stability of the linearized inverse problem. We also examine the error due to regularization. The approach extends to scalar waves in the near-field.

Friday April 18

Title Quantum Information, Mutually Unbiased Bases and Finite Geometry: An Unexpected Connection
Speaker Arthur Pittenger
Department of Mathematics and Statistics

The emergent interdisciplinary field of quantum information theory motivates research in areas ranging from experimental physics to computational complexity. In this talk we examine a mathematical problem motivated by measurements of finite dimensional quantum systems and its unexpected link with finite geometry. Some open problems will be highlighted.

Friday April 25

Title Theoretical and Experimental Analysis of Chemotactic Systems in Biology
Speaker Pablo A Iglesias, Professor
Electrical & Computer Engineering, Biomedical Engineering and Applied Mathematics & Statistics
Johns Hopkins University

Many biological systems have the ability to sense the direction of external chemical sources and respond by polarizing and migrating toward chemoattractants or away from chemorepellants. This phenomenon, referred to as chemotaxis, is crucial for proper functioning of single cell organisms, such as bacteria and amoebae, as well as multi-cellular systems as complex as the immune and nervous systems. Chemotaxis also appears to be important in wound healing and tumor metastasis.

I will discuss our groups efforts at elucidating the mechanisms underlying chemotaxis. Using known biochemical data, we have developed mathematical models that can account for many of the observed chemotactic behavior of the model organism Dictyostelium. I will discuss experiments used to test these models. Finally, I will describe how information-theoretic methods can be used to evaluate the optimality of the gradient sensing mechanisms.

Friday May 02

Title Cardiac motion estimation from cine-MR images
Speaker George Biros, Assistant Professor
Mechanical Engineering and Applied Mechanics, Bioengineering, and Computer and Information science
University of Pennsylvania

Cine Magnetic Resonance Imaging is routinely used for diagnosing cardiovascular disorders. Clinical interpretation of MR images requires semi-automatic image processing by certified technicians. Image processing is necessary to recover global and localized measures of the motion of the myocardium but it has significant intra- and inter-rater variability.

Along with collaborators at the Hospital of the University of Pennsylvania, we are developing novel technologies for cardiac motion estimation. The key feature of our approach is an optimization formulation that is constrained by partial differential equations. The optimality conditions for this problem result in a nonlinear four-dimensional, boundary value problem. I will briefly review the related work in cardiac motion estimation, discuss the formulation, and algorithmic issues related to discretization and solution. I will conclude with a discussion on numerical results on synthetic and clinical datasets.

This joint work with Christos Davatzikos, Harold Litt, and Hari Sundar.

Biosketch: George Biros is an assistant professor in Mechanical Engineering and Applied Mechanics, Bioengineering, and Computer and Information Science at the University of Pennsylvania. He received his BS in Mechanical Engineering from Aristotle University Greece (1995), his MS in Biomedical Engineering from Carnegie Mellon (1996), and his PhD in Computational Science and Engineering, also from Carnegie Mellon (2000). He joined Penn in 2003 after a postdoctoral appointment at the Courant Institute of Mathematical Sciences. He is affiliated with the Computer Science Research Institute (CSRI) at the Sandia National Laboratories.