PSU Mark
Eberly College of Science Mathematics Department

Abstracts of Invited Talks

 Jacob Rubinstain, Technion
Title: Optometry for mathematicians and mathematics for optometrists
Abstract:      The basic principles of classical optometry will be presented and cast in a mathematical language. Then a few novel ideas in vision correction will be discussed. In particular I shall present the math behind advanced spectacles design and wavefront technology for measuring visual aberrations and correcting them.

 

  Benedetto Piccoli, Rutgers University
Title: Flows on networks and the motion of intelligent groups
Abstract:      The talk will start revising recent results on fluid dynamic approach to network flows, with specific focus on vehicular traffic modeling. Starting from traffic, we then discuss motions of intelligent groups in general. The latter are meant as groups where each agent is not a passive particle but able to take decisions and act on the whole system. A new mathematical framework is introduced, based on time evolving measures. Both macroscopic and microscopic point of view can be integrated in such model.

 

  Steven Schiff, Penn State
Title: Understanding and Treating the Brain with the Mathematics of Control.
Abstract:      Since the 1950s, we have developed mature theories of modern control theory and computational neuroscience with almost no interaction between these disciplines. With the advent of computationally efficient nonlinear Kalman filtering techniques, along with improved neuroscience models which provide increasingly accurate reconstruction of dynamics in a variety of important normal and disease states in the brain, the prospects for a synergistic interaction between these fields are now strong. I show recent examples of the use of nonlinear control theory for the assimilation and control of single neuron dynamics, the modulation of oscillatory wave dynamics in brain cortex, and the use of optimized parameter model networks to assimilate complex network data called the ‘consensus set’. These findings lay the groundwork for a control framework for treating seizures and Parkinsons disease.

 

  James Keener, Penn State
Title: Mechanisms of length regulation of flagella in Salmonella
Abstract:      The construction of flagellar motors in bacteria is a carefully regulated genetic process. Among the structures that are built are the hook and the filament. The length of the hook is tightly controlled while the length of filaments is less so. However, if a filament is broken off it will regrow, while a broken hook will not regrow. The question that will be addressed in this talk is how Salmonella detects and regulates the length of these structures. This is related to the more general question of how physical properties (such as size or length) can be detected by chemical signals and what those mechanisms are. In this talk, I will present mathematical models for the regulation of hook and filament length and show how analysis of these models helps us understand these important physical processes.



  Terry L. Friesz, Penn State
Title: Dynamic Traffic Assignment: History, Recent Results and Unanswered Questions
Abstract:      Dynamic traffic assignment (DTA) is the positive (descriptive) modeling of time varying flows on vehicular networks consistent with established traffic flow. This presentation is concerned with a specific type of dynamic traffic assignment known as continuous time dynamic user equilibrium (DUE) for which unit travel cost, including early and late arrival penalties, is identical for those route and departure time choices selected by travelers between a given origin-destination pair.
It is shown that many of the historical formulations proposed to date may be viewed as special cases of an appropriately defined infinite dimensional variational inequality and abstract network loading problem, where the latter is based on either the so-called point-queue model or Lighthill-Whitham-Richards hydrodynamic theory.
We study the special case for which travel demand is constant during each given day of interest, although it evolves from day to day. This special case is the simplest plausible circumstance under which a discrete-time, day-to-day model of demand learning may be coupled to a continuous-time, within-day DUE model. In presenting such a dual-time scale theory, we employ demand evolution dynamics motivated by evolutionary game theory.
Numerical comparisons of alternative models are also provided. Limitations of the current state of knowledge are noted, and some interesting unanswered research problems are identified.

 

  Igor Aronson, Argonne National Laboratory
Title: Self-Assembled Magnetic Surface Microswimmers: Experiment and Multiscale Modeling
Abstract:      Self-propulsion of living microorganisms is a fascinating phenomenon attracting enormous attention in broad scientific community. The interest is driven by the need to create novel bio-inspired materials and systems. There exists a wide gap between man-made hard materials and living organisms: biological materials, unlike steel or plastics, are “alive”. They consume energy of the nutrients for self-assembly, adaption to their environment, and self-repair. A new type of micro-swimmers, magnetic snakes, is an excellent tool to model bio-inspired self-assembly and self-locomotion in a table-top experiment. The snakes self-assemble from a dispersion of magnetic microparticles suspended at a water-air interface and subjected to an alternating magnetic field. The snakes often exhibit “life-like behavior” reminiscent of their living counterparts. Formation and dynamics of the snakes is captured in the framework of multi-scale mathematical model coupling paradigm equation for the amplitude of surface waves, conservation law for the concentration of particles, and the Navier-Stokes equation for hydrodynamic flows. The results of continuum modeling are supplemented by first-principle molecular dynamics simulations of magnetic particles floating on a surface of fluid.



  Suncica Canic, University of Houston
Title: Mathematical Methods for Cardiology
Abstract:      Mathematical modeling, analysis and numerical simulation provide a powerful tool to study various aspects of cardiovascular treatment.
     This talk will address two examples: a mathematical study of fluid-structure interaction with a clinical application to 2D and 3D Doppler assessment of mitral regurgitation, and a novel multi-scale approach to modeling coronary stents as a 3D mesh of 1D curved rods (3D network of 1D hyperbolic conservation laws). An overview of the basic mathematical ideas underlying this research and several applications to cardiovascular treatment will be presented.
     This talk will be accessible to a wide scientific audience. The work reported in this talk has been performed together with medical collaborators Dr. W. Zoghbi, Dr. S. Little, Dr. C. Hartley and Dr. D. Paniagua of the Texas Medical Center in Houston and with mathematicians Prof. J. Tambaca (University of Zagreb, CRO), Prof. R. Glowinski, Prof. G. Guidoboni, post-doc A. Quaini, and graduate students M. Bukac, M. Kosor and T.B. Kim (UH).

 

  Miguel A. Herrero, Instituto de Matemática Interdisciplinar
Title: Mathematics and Social Dynamics: Modelling Criminality
Abstract:      Predicting social dynamics is one of the big scientific challenges of this century. From a modelling point of view, a given society can be considered as a playground for a number of interacting populations, whose joint evolution may result in the onset of emergent behaviours, sometimes termed as social phase transitions. Being able to forecast (and, when appropriate, to prevent) such transitions is a subject of considerable scientific and social interest.
     In recent years, mathematical methods have proved their strength in dealing with the dynamical aspects of social behaviours , a step forward with respect to the classical, static picture provided by statistical techniques. In this lecture . we shall focus on a particular model case, and will review some recent work describing the interaction of a number of social classes or subpopulations (criminal /cheaters, tax-payers, guards.) which compete for a number of resources that not all of them actually produce. We shall discuss on the dynamics of such interacting populations, and particular attention will be paid to the choice of optimal strategies aimed at keeping the cost of fighting criminals/cheaters as low as possible.
     The work to be presented has been done in collaboration with Juan C. Nuño ( Universidad Politécnica, Madrid, Spain) and Mario Primicerio (Università di Firenze, Florence, Italy).


Student talks:
NameAffiliationTitle
Martina Bukac U. Houston Numerical simulation of arterial blood flow under physiological conditions
Alvaro Kohn U. Complutense, Madrid Modelling vascular morphogenesis
Assieh SaadatpourPenn State Boolean dynamics for modeling biological networks
Yanping Ma Penn State Application of Population Dynamics to Study Heterotypic Cell Aggregations in the Near-wall Region of a Shear Flow
Brian Haines Penn State Effective viscosity of bacterial suspensions: a PDE model
R. Robyr U. Zurich Hamilton Jacobi equations with obstacles