Colloquia
Department of Mathematics and Statistics
Texas Tech University
In this talk, we’ll discuss my work since joining TTU including novel perspectives on two themes in geometry. The first is to study the connection between the geometry and the topology of a manifold. Here, geometry is about the shape and the measurement while topology is about properties that are preserved under continuous deformations. We’ll describe a new viewpoint leading to a resolution of a conjecture proposed by S. Nishikawa in 1986 and progress towards other fundamental problems. In addition, our work provides a framework for further investigations including the much-anticipated application of Ricci flows in dimension four parallel to the resolution of Poincare's conjecture in dimension three.
The second theme is the search for a hierarchy of stationary surfaces, which are critical points of a certain geometric functional. In Calculus, we use derivative tests to classify critical points and determine whether one is a minimum, a maximum, or a saddle. An abstract generalization is to study the index of an operator associated with the second variation. In this direction, we develop a novel mechanism, via partial differential equations when studying surfaces with boundaries, with several applications. Specifically, there is a partial resolution of a conjecture proposed by R. Schoen and A. Fraser and closely related to the Willmore conjecture. Another significant contribution is to introduce a new approach based on functional analysis to investigate a setup with constraints.
Dr. Hung Tran's Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
The Vlasov system is known as a fundamental model in plasma physics which describes the dynamics of dilute charged particles due to self-induced electrostatic forces. The main numerical challenges lie in the high dimensionality of the phase space, multi-scale feature, and the inherent conservation property of the solutions. In this talk, we introduce a conservative adaptive low-rank tensor method. The approach takes advantage of the fact that the differential operators in the Vlasov equation are tensor friendly, based on which we propose to build up the low-rank solution basis dynamically and adaptively by exploring the intrinsic low-rank structure of Vlasov dynamics. We further develop a novel low-rank scheme with local mass, momentum, and energy conservation by considering the corresponding macroscopic equations. The mass and momentum conservation are achieved by a conservative low-rank truncation, while the energy conservation is achieved by replacing the energy component of the Vlasov solution by the one obtained from a conservative scheme for the macroscopic energy equation. The algorithm is extended to high-dimensional problems with the hierarchical Tucker tensor decomposition of Vlasov solutions, overcoming the curse of dimensionality. An extensive set of nonlinear Vlasov examples are performed to demonstrate the effectiveness and conservation property of the proposed conservative low-rank tensor approach.
Dr. Wei Guo's Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
In the past few years, I am interested in recurrent diseases and multiple disease outcomes shown in pathogen-immune interaction models. The crucial role of the immune system is to eliminate foreign pathogens and cancerous cells, simultaneously, preserving self-tolerance. The failure of complete elimination leads to persistent antigens and chronic inflammation, which in turn actives immune tolerance and inhibits immune functions. Through applied bifurcation theory, my work reveals the causal mathematical mechanisms for "slow-fast" motions in recurrent diseases and identifies the key parameter determining multiple disease outcomes. The mathematical results lead to a better understanding of the underlining biological mechanisms, which will shed new light to improve immunotherapy.
Dr. Wenjing Zhang's Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
Simulation of phenomena in nature requires a faithful model, an accurate discretization method, an efficient solution algorithm, and software that is both efficient and adaptable. Use of simulation in real-world decision making requires an understanding of the uncertainties in the model data and their propagation through the simulation. Through the course of my career I’ve worked on all of those aspects of applied scientific computing. Here I concentrate on three problems: (1) quantifying the effect of parametric uncertainties in medical infrared thermography, (2) a method to measure model uncertainty in nonlinear dynamics, and (3) a family of preconditioners for implicit Runge-Kutta timesteppers applicable to a large family of hyperbolic and parabolic partial differential equations.
Dr. Katharine Long's Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
Progress continues in the unification and extension of discrete and continuous analysis since Stefan Hilger’s landmark paper in 1988. The general idea is to prove a result once for a dynamic equation where the unknown function’s domain is a time scale T, which is an arbitrary, nonempty, closed subset of the real numbers.
A few years ago, I changed the direction of my research, focusing on more applied problems instead of theoretical ones. Of particular interest are issues arising in mathematical biology. While infusing an introduction to the time scales theory, in this talk, we use dynamic equations on time scales to model intermittent androgen deprivation (IAD) therapy. Our approach is novel since time scales combine continuous and discrete time and have not been used to model IAD therapy. Traditionally, continuous ordinary differential equations are used to estimate prostate-specific antigen levels, which are used to determine the timing of a pause in IAD treatment. In this work, we use dynamic equations to estimate prostate-specific antigen levels and construct a time scale model to account for continuous and discrete time simultaneously. We discuss preliminary results on the development of specific models and present the next steps.
We conclude the talk with activities related to my service mission.
Dr. Raegan Higgins' Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
Linear algebra plays a central role in scientific computing. My research in numerical linear algebra is centered around preconditioning, which is a key to the performance of iterative solvers for systems of linear equations. In this talk, I will discuss a few recent research projects. The first is in preconditioning for implicit Runge-Kutta (IRK) methods. IRK methods have attractive stability properties, but until recently have not been used frequently due to challenges in solving the resulting linear systems. I will present some recent results, an analysis of the preconditioners via eigenvalues and field of values and preliminary results from a new optimized preconditioner.
In addition, I will briefly describe two other projects. One is using machine learning algorithms to improve preconditioners, and the other is in simulation and machine learning for classification of skeletal trauma.
Dr. Victoria Howle's Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
A classical result, at the interface between harmonic analysis and partial differential equations, asserts that the $L^p$-Dirichlet problem for the Laplacian is well posed in the upper half-space. In fact, the same is true for constant coefficient homogeneous second-order systems satisfying the Legendre-Hadamard
(strong) ellipticity condition. In my talk I will show that this well-posedness result may fail if the system in question is assumed to be only weakly elliptic (i.e., its characteristic matrix is merely invertible rather than strictly positive definite). In fact, the aforementioned failure is at a fundamental level, in the sense that there exist weakly elliptic systems for which the $L^p$-Dirichlet problem in the upper half-space is not even Fredholm solvable.
Figure skating is a beautiful sport combining elegance, precision, and athleticism. To understand some of the mechanics and complexity involved in this sport, we derive and analyze a three dimensional model of a figure skater at a continuous contact with the ice (i.e., no jumps). We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. We derive a surprising result that for a static (i.e., non-articulated) skater, the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. We also consider the case when the projection of the center of mass on skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior. We then extend the study to see how an articulated skater traces trajectories on ice by considering a two-dimensional skater (a Chaplygin's sleigh) with controlled moving mass. We derive a control procedure by approximating the trajectories using circular arcs. We show that there is a control procedure minimizing the 'relative kinetic energy' of the control mass which leads to well-posed equations for the control masses. We demonstrate examples of our system tracing actual compulsory figure skating trajectories. We also discuss further extensions of the model and applications to real-life figure skating.
Slideshow pdf available
We consider high-order discretizations of a Cauchy problem where the evolution operator comprises a hyperbolic part and a parabolic part with diffusion and stiff relaxation terms. Assuming that this problem admits non-trivial invariant domains, in the talk we discuss approximation techniques in time that preserve these invariant domains. Before going into the details, we are going to give an overview of the literature on the topic. Emphasis will be put on explicit and explicit Runge Kutta techniques using Butcher's formalism. Then we are going to describe techniques that make every implicit-explicit time stepping scheme invariant-domain preserving and mass conservative. The proposed methodology is agnostic to the space discretization and allows to optimize the time step restrictions induced by the hyperbolic sub-step.
This Departmental Colloquium is sponsored by the Applied Math seminar group and may be attended at 4 PM (UT-5) via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied
Professors Research Talk Title
Dr. Alvaro Pampano Some Geometric Variational Open Problems
Dr. Juntao Huang Structure-Preserving Machine Learning Moment Closures for the Radiative Transfer Equation
Dr. Suddhasattwa Das Learning Theory for Dynamical Systems
Dr. Travis Thompson Mathematics makes a difference: Modeling, Analysis, Computing, and Frontier of Brain Research
Dr. Hongwei Mei Stochastic Systems, Control and Optimization, and Optimal Stopping
Dr. Ignacio Tomas Energy Stability of Diagonally Implicit Runge-Kutta Methods
This Departmental Colloquium was sponsored by the TTU chapters of SIAM & AWM and is archived in the Mediasite Catalog.
Kinetic models are used to simulate the collective behavior of particle systems. They provide a mesoscopic description that forms a link between continuum fluid models, which are not valid in non-equilibrium settings, and molecular dynamics models, which are often too expensive for practical purposes. In this talk, I will introduce the basic formalism of kinetic theory and present some relevant applications. I will then discuss some of the challenges of solving kinetic equations numerically, including a more detailed look at some implicit solver strategies for multiscale problems.