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January 25, Joint Probability/Applied Math/Math Bio (FRIDAY 3:05pm, LCB219)
Speaker: Scott McKinley, University of Florida, Mathematics
Title: Fluctuating hydrodynamics of microparticles in biological fluids: modeling, simulation and analysis
Abstract: Recent progress in advanced microscopy reveals that foreign microparticles in biological fluids exhibit anomalous diffusive behavior. Intrinsic to particle trajectories are time and length scale correlations that challenge conventional probabilistic models. One approach to dealing with these correlations stems from the Landau-Lifshitz formulation of thermally fluctuating fluids, and several models are emerging for simulating the dynamics of immersed interacting particles in a viscous environment. The stochastic PDEs associated with these models pose numerous computational and analytical challenges though. In this talk, after spending some time developing the mathematical model (and providing the motivation for a current collaboration with Christel Hohenegger), I will describe analytical work with Jonathan Mattingly and Natesh Pillai in which we establish geometric ergodicity of a bead-spring pair with stochastic Stokes forcing. The method employs control theoretic arguments, Lyapunov functions and hypoelliptic diffusion theory to prove exponential convergence via a Harris chain argument.

February 8, Joint Applied Math/Math Bio (FRIDAY 4pm, LCB 225)
Speaker: Sébastien Motsch, University of Maryland, CSCAMM
Title: Kinetic and macroscopic models for complex systems
Abstract: In complex systems such as a flock of birds or a school of fish, we observe at large scales the formation of complex structures. To model these phenomena, we can either use "microscopic models" describing the motion of each individual or "macroscopic models" (PDEs) describing the evolution of the density of individuals. In this talk, we discuss how we can "link" the two approaches using kinetic theory.
One of the main difficulty to study those systems is the lack of conserved quantities (e.g. momentum, energy). To overcome this difficulty, we introduce of new type of "collisional invariant" that allows us to derive the macroscopic limit of a large class of "microscopic models". Based on this method, we develop accurate numerical schemes for both kinetic and macroscopic models. We observe numerically new types of solutions that remain to be understood analytically.

February 11, Joint Math Bio/Applied Math (3:05pm, LCB 219)
Speaker: Kimberly Fessel, Rensselaer Polytechnic Institute, Mathematics
Title: Analyzing Nonlinear Waves in the Cochlea with Asymptotic and Numerical Techniques
Abstract: Mammalian hearing relies on several complicated processes for successful sound detection including mechanotransduction of traveling waves within the cochlea. Linear math models can be used to predict basic cochlear mechanics; however, many significant nonlinearities exist in the traveling wave and have not been explained. Experimentalists now point to outer hair cell activity as the primary source of these nonlinearities. In this work a mathematical model of cochlear biomechanics is developed to describe the inner ear's nonlinear behavior. Coupled equations of motion for the fluid pressure and the basilar membrane displacement are utilized in conjunction with a formulated model for outer hair cell electromotility. Asymptotic methods are used to reduce the complexity of the problem, and a hybrid analytic-numeric method is used to approximately solve for the nonlinear waves.

February 15, Joint Math Bio/Applied Math (FRIDAY 4-5pm, LCB 225)
Speaker: Jun Allard, University of California at Davis, Mathematics
Title: Mechanical modulation of immune receptors at cell-cell interfaces
Abstract: Receptors on the surface of cells initiate and regulate cell signaling processes and have been extensively studied because they form an important class of immune receptors, e.g., T-cell receptors, which bind to ligands that are anchored to other cells or surfaces, but remain poorly understood. The T-cell receptor-ligand complex spans 15 nanometers, while its phosphatase (a surface molecule that interacts with the receptor) spans 40 nanometers. This has been proposed to lead to size-based segregation that triggers signaling, but it is unclear whether the mechanochemistry supports such small-scale segregation. We present a nanometer-scale quantitative model that couples membrane elasticity with compressional resistance and lateral mobility of phosphatase. We find robust supradiffusive segregation of phosphatase from a single receptor-ligand complex. The model predicts a time-dependent tension on the complex leading to a nonlinear relationship between stressed and unstressed bond lifetimes, which could enhance the receptor's ability to discriminate between similar ligands. This provides a mechanical source of ligand sensitivity, in contrast to biochemical sources of sensitivity that have been proposed previously.

March 4
Speaker: Graeme W. Milton, University of Utah, Mathematics Department
Title: Non-linear metamaterials from rods and hinges
Abstract: A complete characterization is given of the possible macroscopic deformations of periodic nonlinear affine unimode metamaterials constructed from rigid bars and pivots. The materials are affine in the sense that their macroscopic deformations can only be affine deformations: on a local level the deformation may vary from cell to cell. Unimode means that macroscopically the material can only deform along a one dimensional trajectory in the six dimensional space of invariants describing the deformation (excluding translations and rotations). We show by explicit construction that any continuous trajectory is realizable to an arbitrarily high degree of approximation provided at all points along the trajectory the geometry does not collapse to a lower dimensional one. In particular, we present two and three dimensional dilational materials having an arbitrarily large flexibility window. These are perfect auxetic materials for which a dilation is the only easy mode of deformation. They are free to dilate to arbitrarily large strain with zero bulk modulus.

March 8 (FRIDAY 4pm, LCB 215)
Speaker: Alexander V. Mamonov, University of Texas at Austin, ICES
Title: A model reduction approach to numerical inversion for a parabolic partial differential equation.
Abstract: We propose a novel numerical inversion algorithm for parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The inversion algorithm considers a layered medium. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss-Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated computations of the time domain solutions of the forward problem. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments.

April 1
Speaker: Vladyslav Babenko, Dnepropetrovsk National University
Title: Optimal recovery of operators on various classes of functions
Abstract: In this talk we will discus problems of optimal recovery of operators in various settings and their connections with certain extremal problems of Analysis and Discrete Geometry. Review of known results will be given and new results will be presented. In particular we will present some results about optimal recovery of solutions of certain partial differential equations.

April 8 (Student talk)
Speaker: Lance Finn Helsten, University of Utah, Mathematics
Title: Global Navigation Satellite System (GNSS) Error Correction
Abstract: Global Navigation Satellite System (GNSS) errors result from a variety of sources in satellite ephemeris, clock drift, radio signal propagation, and relativistic effects. In this presentation I describe the basics of GNSS operation, and then examine each error source and the associated correction which allows for precise location measurements with errors less than two meters.

April 15
Speaker: Stewart Ethier, University of Utah. Mathematics
Title: An optimization problem in game theory
Abstract: The game of baccarat chemin de fer (briefly, baccarat) played a key role in the development of game theory. Bertrand's 1889 analysis of whether Player should draw or stand on a two-card total of 5 was the starting point of Borel's investigation of strategic games. In 1924 Borel described Bertrand's study as ``extremely incomplete'' but did not himself contribute to baccarat. In 1928 von Neumann, after proving the minimax theorem, remarked that he would analyze baccarat in a subsequent paper. But a solution of the game would have to wait until the dawn of the computer age. In 1957 Kemeny and Snell, assuming that cards are dealt with replacement from a single deck and that each of Player and Banker sees the total of his own two-card hand but not its composition, found the unique solution of the resulting $2\times 2^{88}$ matrix game. In practice, cards are dealt without replacement from a sabot, or shoe, containing six 52-card decks. In 1975 Downton and Lockwood, allowing a $d$-deck shoe dealt without replacement and assuming that Banker sees the composition of his own two-card hand while Player sees only his own total, found the unique solution of the resulting $2\times 2^{484}$ matrix game for $d=1,2,\ldots,8$.
Our aim here is to solve the game without simplifying assumptions. We allow a $d$-deck shoe dealt without replacement and allow each of Player and Banker to see the composition of his own two-card hand, making baccarat a $2^5\times 2^{484}$ matrix game. We find optimal Player and Banker strategies and determine the value of the game, doing so for $d=1,2,\ldots,12$. Optimality proofs are computer-assisted, with all computations done in infinite precision. Moreover, we conjecture optimal Player and Banker strategies and the value of the game for every integer $d\ge13$.
Joint work with Carlos Gámez, Universidad de El Salvador.