Colloquia and Seminars

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**Wednesday August 23, 2017, 155 Fondren Science Building, 10am – 11am**

**Levitation and self-organization of liquid microdroplets over liquid surfaces and dry heated substrates**

Institute of Thermophysics and Novosibirsk State University

Host: V. Ajaev

Abstract: Levitating droplets of liquid condensate are known to organize themselves into ordered arrays over hot liquid-gas interfaces. Studies of such droplets are important for applications such as spray cooling, drug delivery by aerosol inhalation, and containerless synthesis of amorphous chemical compounds. Our recent experimental observations show similar behavior of droplets over a dry heated solid surface. Even though the life-time of the array is shorter in this case, its geometric characteristics are remarkably similar to the case of droplets levitating over liquid-gas interfaces. Mathematical models are developed that predict the mechanisms of both droplet levitation and inter-droplet interaction leading to pattern formation over dry surface; the model is shown to be in good agreement with the experimental data. Using the insights from the new experiments, we are able to resolve some long-standing controversies pertaining to the mechanism of levitation of droplets over liquid-gas interfaces. Finally, by studying transition of levitating droplets over the contact line region we are able to obtain velocity profiles for local gas flow there, using normal-mode stability theory.

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**Wednesday September 6, 2017, 126 Clements Hall, 3:30-4:30 (Refreshments at 3:15)
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**Mathematical Studies of Extraordinary Field Enhancement in Subwavelength Structures**

Dr. Junshan Lin, Auburn University

Host: A.J. Meir

Abstract: Since the discovery of the extraordinary optical transmission through nanohole arrays in metallic films by Ebbesen, a wealth of research has been sparked in the experimental and theoretical investigation of localized electromagnetic field enhancement in subwavelength nanostructures. This remarkable phenomenon can lead to potentially significant applications in near-field imaging, bio-sensing, etc. However, there has been a long debate on the interpretation of the enhancement effect since Ebbesen’s work. In addition, a quantitative analysis of the field enhancement in subwavelength structures is still widely open. In this talk, using two-dimensional slits as a prototype, I will present mathematical studies of the field enhancement in the subwavlength structures. Based upon the layer potential technique, asymptotic analysis and homogenization theory, the enhancement mechanisms for both the single slit and an array of slits are studied quantitatively.

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**Friday September 8, 2017, 126 Clements Hall, 3:30-4:30 (Refreshments at 3:15)**

**The Statistical Mechanics of Integrable Nonlinear Lattice Equations.
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Nick Ercolani, University of Arizona

Host: A. Aceves

Abstract: There are a number of, by now, classical studies of the statistical mechanics of nonlinear wave equations (for example by Lebowitz-Rose-Speer (1989) and McKean (1995) in the case of the Cubic Schrodinger Equation). We are currently looking at discrete analogues in the case of the Ablowitz-Ladik system with an eye to paralleling recent developments in random matrix theory that connect to the hyperbolic Brownian Carousel or Prufer Phase Stochastic Di erential Equations. For ease of exposition, in this talk we will focus on the simpler setting of the quasi-periodic Toda lattice and just try to motivate and describe the basic set-up. This is joint work with Diane Holcomb and Dylan Murphy.

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**Joint MathBio Monday Seminar/Mathematics Colloquium
Monday September 11, 2017, 0131 Dedman Life Sciences Building, 3:30-4:30 **

Dr. Jane Hawkins, University of North Carolina at Chapel Hill

Host: A.J. Meir

Abstract: The spread of HIV throughout a lymph node is a complex process and many aspects remain unknown; a noninvasive understanding can be gained through math modeling. We discuss a number of different approaches to analyzing the qualitative dynamics, focussing on some tools in symbolic dynamics such as cellular automata and Markov chains. We demonstrate the universality of some of these techniques as they apply to HIV, Ebola (EBV) and other viruses, and show how the differences in outcomes can emerge through the models. It turns out that some classical results in dynamical systems reflect or explain clinical findings and longterm differences quite well. For example, while EBV patients tend to recover or succumb quickly to the virus, HIV patients can live in a chronic state. Both topological and probabilistic dynamical tools are used. We touch on the more deterministic approaches to this problem as well, towards the end of the talk; namely the current role of ODEs and PDEs in modeling viral spread in an individual.

**Wednesday September 20, 2017, 126 Clements Hall, 3:30-4:30 (Refreshments at 3:15)
Unraveling tumor heterogeneity using single-cell RNAseq data: biology in a high-dimension sparse matrix**

Dr. Wei Lin

Host: Yunkai Zhou

Abstract: The recent advance of single-cell RNAseq technology makes it possible

to dissect heterogeneous tumor tissue samples and deconvolute the molecular signal into specific cell types. This technique generates high-dimensional gene expression profile data in a very sparse fashion. In a current study, by unsupervised clustering strategy and mapping the centroid of signal, we successfully identify multiple subpopulations of

cell types and characterize their roles in tumor progression.

**Thursday September 28, 2017, 126 Clements Hall, 3:45-4:45 (Refreshments at 3:30)
A positivity challenge in steady state reaction diffusion problems**

Dr. Ratnasingham Shivaji

University of North Carolina at Greensboro

Host: A.J. Meir

Abstract: We consider the semipositone problem of Poisson’s equation on bounded domains in R^N. Some results on the positivity of nonnegative solutions will be discussed.

**The colloquium by Luan Vu Thai has been postponed until: Thursday, Oct 19, 2017, New time: 3:45 (Refreshments 3:30)**

**Constructing efficient exponential integrators for large stiff systems**

Dr. Luan Vu Thai Department of Mathematics, Southern Methodist University

Host: A. Aceves

Abstract: Discretizing time-dependent PDEs by FDM, FEM, spectral methods or modeling flexible mechanical systems (highly nonlinear oscillating problems) often yields large stiff systems of ODEs. For solving such stiff systems, explicit (classical) integrators usually lack stability and are required to use tiny time steps. Thus standard implicit integrators such as Radau, DIRK, BDF, IMEX are commonly used but are usually costly as they require the solution of large (non)linear systems at every step. Moreover, for problems whose Jacobian possesses large imaginary eigenvalues, implicit methods usually introduce artificial damping if large stepsize is used. With the recent developments of numerical linear algebra in computing matrix functions, it turns out that exponential integrators can overcome these shortcomings. In recent years, they have received a lot of interests and it has been shown that the integrators are highly competitive: they are fully explicit and thus do not require the solution of large linear systems in each step. Moreover, they offer very high accuracy and do not suffer from the time-step restricted by the CFL (stability) condition inherent in the explicit methods.

In this talk, first I will briefly present the basic idea behind exponential integrators such as the exponential Runge-Kutta/Rosenbrock methods. Then I will focus on how to design customized exponential methods for the purpose of efficiency improvement. In particular, one can construct parallel exponential Rosenbrock methods which can be implemented on a multi-processor system or parallel computers. For problems with very stiff nonlinearities, the implicit-explicit exponential methods have been constructed. They can be used with any preconditioners and thus offer significant computational savings compared to the widely used standard IMEX integrators. Finally, I will demonstrate the efficiency of the new integrators in solving a number of PDEs (such as advection-diffusion-reaction equations and Schnakenberg systems), as well as their applications in computer graphics e.g. in computational modeling of elastodynamic systems.

Multi-transmission-line-beam interactive system

Speaker: Prof. Alexander Figotin, Math, UCI

Host: W. Cai

ABSTRACT: We construct a Lagrangian field formulation for a system consisting of an electron beam interacting with a slow-wave structure modeled by a possibly non-uniform multiple transmission line (MTL). In the case of a single line we recover the linear model of a traveling wave tube (TWT) due to J.R. Pierce. Since a properly chosen MTL can approximate a real waveguide structure with any desired accuracy, the proposed model can be used for design and optimization. Furthermore, the Lagrangian formulation provides for: (i) a clear identification of the source of amplification and its mathematical representation, (ii) exact expressions for the conserved energy and its flux distributions obtained from Noether's theorem. In the case of uniform MTLs we carry out an exhaustive analysis of eigenmodes and find sharp conditions on the parameters of the system to provide for amplifying regimes.

Host: W. Cai

ABSTRACT: We construct a Lagrangian field formulation for a system consisting of an electron beam interacting with a slow-wave structure modeled by a possibly non-uniform multiple transmission line (MTL). In the case of a single line we recover the linear model of a traveling wave tube (TWT) due to J.R. Pierce. Since a properly chosen MTL can approximate a real waveguide structure with any desired accuracy, the proposed model can be used for design and optimization. Furthermore, the Lagrangian formulation provides for: (i) a clear identification of the source of amplification and its mathematical representation, (ii) exact expressions for the conserved energy and its flux distributions obtained from Noether's theorem. In the case of uniform MTLs we carry out an exhaustive analysis of eigenmodes and find sharp conditions on the parameters of the system to provide for amplifying regimes.

Wed., Oct. 25, 2017, 126 Clements Hall, 3:30-4:30p (Refreshments at 3:15)

Opening Band Gaps in Two-dimensional Photonic Crystals

Host: A. Aceves

ABSTRACT: High-contrast photonic crystals have been studied intensely by both mathematicians and physicists for the past several decades. A desirable property of such media is the existence of band gaps, i.e. bands of frequencies at which no wave can propagate in the given medium. In this talk, we will identify an explicit lower bound on material contrast necessary to open band gaps in two-component, two-dimensional photonic crystals, along with the location and estimates on the size of these gaps. All of these properties are determined entirely by spectral quantities associated with the geometry of the crystalline structure, including Dirichlet and Neumann-Poincaré spectra.

Host: W. Cai

ABSTRACT: Many problems in control and optimization require the treatment of systems in which continuous dynamics and discrete events coexist. This talk presents a survey on some of our recent work on such systems. In the setup, the discrete event is given by a random process with a finite state space, and the continuous component is the solution of a stochastic differential equation. Seemingly similar to diffusions, the processes have a number of salient features distinctly different from diffusion processes. After providing motivational examples arising from wireless communications, identification, finance, singular perturbed Markovian systems, manufacturing, and consensus controls, we present necessary and sufficient conditions for the existence of unique invariant measure, stability, stabilization, and numerical solutions of control and game problems.

Wed., Nov. 1, 2017, 126 Clements Hall, 3:30-4:30p (Refreshments at 3:15p)

Application of Large Deviations to Genetic Evolution of E. coli Populations

Speaker: Prof. Brett Geiger, Southern Methodist University

Host: A. Aceves

ABSTRACT: Radical shifts in the genetic composition of large cell populations are rare events with quite low probabilities, which direct numerical simulations generally fail to evaluate accurately. In this talk, we develop an applicable large deviations framework for a class of Markov chains used to model genetic evolution of E. coli bacteria. We then apply this framework using realistic parameter sets in order to solve several difficult numerical and mathematical questions of high biological interest, such as computing the most likely evolutionary path linking two given population states in the fitness landscape and evaluating transition probabilities between successive genotype fixations.

Thu., Nov. 16, 2017, 126 Clements Hall, 4:00-5:00p (Refreshments at 3:45p)

Host: W. Cai

ABSTRACT: In this talk, we propose a spectral method for discretizing the Boltzmann collision operator that satisfies a discrete version of the H-theorem. The method is obtained by modifying the existing Fourier spectral method in order to match a classical form of the discrete velocity method. It preserves the positivity of the solution on the Fourier collocation points and as a result satisfies the H-theorem. The fast algorithms appeared previously in the literature can be readily applied to this method to speed up the computation.

Host: W. Cai