The UL Lafayette Mathematics Colloquium is an ongoing seminar series that features speakers from other universities and from our department. The topics cover all areas of mathematics and statistics. We try to schedule an interesting mix of topics ranging from very applied to more abstract in nature. These lectures are open to all UL Lafayette students, faculty and community members for the purpose of fostering continued discussion and networking in the various areas of mathematics. Please contact Paul L. Salceanu with questions or suggestions about the colloquium series.
Our colloquia are normally held on Thursday at 3:30 p.m. in room 208 of Maxim Doucet Hall. Refreshments are served at 3:15 in room 208. To accommodate outside speakers, the colloquium is occasionally held on a different day of the week, e.g., Tuesday at 3:30, instead of Thursday.
Remember, our colloquium is open to the public and everyone who is interested is encouraged to attend.
Spring 2017 Schedule
26 January 2017
Nonstandard Finite Difference Methods for Dynamical Systems in Biology
University of Texas at Arlington
A brief overview of the nonstandard finite difference methods is presented. Next, using the nonstandard discretization approach, a positive and elementary stable numerical method is developed for productive-destructive systems. Finally, a nonstandard finite-difference method for general autonomous dynamical systems is constructed. The proposed numerical methods preserve the positivity of solutions and the local behavior of the corresponding dynamical systems near equilibria; and are also computationally efficient and easy to implement. Applications to select problems in biology are given to demonstrate the performance of the new methods.
31 January 2017 (TUESDAY 3:45)
Variable Selection for discrete spatial data using Penalized Quasi-likelihood estimating equations
Department of Statistics and Probability
Michigan State University
To study real world applications of discrete data on a geographical domain we still face fundamental issues such as not being able to express the likelihood of correlated multivariate data. We circumvent this by assuming a parametric structure on the moments of a multivariate random variable and use a quasi-likelihood approach. In this talk, I propose a method that is able to select relevant variables and estimate their corresponding coefficients simultaneously. Under increasing domain asymptotics after introducing a misspecified working correlation matrix that satisfies a certain mixing condition we show that this estimator possess the” oracle” property as first suggested by (Fan and Li, 2001) for the non-convex SCAD penalty. Several simulation results and a real data example are provided to illustrate the performance of our proposed estimator.
2 February 2017
Estimation and Inference in High Dimensional Error‐in‐Variables Models and an Application to Microbiome Data
Biostatistics and Computational Biology
National Institute of Environmental Health Sciences
Research Triangle Park, North Carolina
We discuss three closely related problems in high dimensional error in variables regression, 1.Additive measurement error in covariates, 2.Missing at random covariates and 3.Precision matrix recovery. We propose a two stage methodology that performs estimation post variable selection in such high dimensional measurement error models. We show that our method provides optimal rates of convergence with only a sub‐block of the bias correction matrix, while also providing a higher computational efficiency in comparison to available methods. We then apply the proposed method to human microbiome data, where we classify observations to geographical locations based on corresponding microbial compositions. Lastly, we provide methods for constructing confidence intervals on target parameters in these high dimensional models, our approach is based on the construction of moment conditions that have an additional orthogonality property with respect to nuisance parameters. All theoretical results are also supported by simulations.
3 February 2017 (FRIDAY 11:00)
Symmetric Gini Covariance and Correlation
Department of Mathematics
The University of Mississippi
The most commonly used measure of dependence is the Pearson correlation. This measure is based on the covariance between two variables, which is optimal for the linear relationship between bivariate normal variables. However, the Pearson correlation performs poorly for variables with heavily-tailed or asymmetric distributions, and may be seriously impacted even by a single outlier. As a robust alternative, the Spearman correlation is defined as the covariance between the cumulative distribution functions (or ranks) of two variables, but it may loss the efficiency. Complementing these two measures, the traditional Gini correlations are based on the covariance between one variable and the rank of the other, and hence well balance in efficiency and robustness. The Gini correlations, however, are not symmetric due to different roles of two variables. This asymmetry violates the axioms of correlation measurement. We have proposed a symmetric Gini-type covariance and correlation based on the joint rank function, which takes more dependence information than the marginal rank in the traditional Gini correlations. The properties of the symmetric Gini correlation are fully explored. Theoretical results on efficiency and robustness are obtained. Numerical studies demonstrate that the proposed correlation have satisfactory performance under a variety of situations. The proposed symmetric Gini correlation provides an attractive option for measuring correlation.
14 February 2017 (TUESDAY 3:30)
Degenerate Diffusion in Phase Separations
New Mexico State University
The Cahn-Hilliard equation is a widely used phenomenological diffuse-interface model for the simulations of phase separation and microstructure evolution in binary systems. We consider a popular form of the equation with a smooth double-well potential, and with phase-dependent diffusion mobilities. The latter is a feature of many materials systems and makes both theoretical analysis and accurate numerical simulations challenging. In this talk, we discuss three aspects of the equation:
1. theoretical predictions on the coarsening dynamics based on asymptotic analysis;
2. numerical simulations that confirm the theoretical predictions;
3. the existence of weak solutions that potentially accommodate the features predicted by asymptotic analysis and exhibited in numerical simulations.
15 February 2017 (WEDNESDAY 2:30)
Special presentation: The Role of the Department Head
Mississippi State University
This is a special presentation by Prof Oppenheimer who is interviewing for the position of Department Head. Prof Oppenheimer will introduce himself and discuss his vision for the Department; all faculty and graduate students are encouraged to attend. The talk will last about 50 minutes. Please do your best to attend! After the talk, faculty will have an opportunity for an extended question-and-answer period with Dr Oppenheimer.
16 February 2017 (usual time)
On contraction of large perturbations of shock waves
University of Texas at Austin
Although mathematical understanding on hyperbolic conservation laws has made huge contributions across many fields of science, there remain many important unsolved questions. In particular, a global well-posedness of entropy solutions to the system of conservation laws in a class of large initial datas is completely open even in one space dimension. Recently, we have obtained a contraction (up to shift) of entropy shock waves to the hyperbolic systems in a class of large perturbations satisfying strong trace property. Moreover, concerning viscous systems, we have verified the contraction of large perturbations of viscous shock waves to the isentropic Navier-Stokes system with degenerate viscosity. Since the contraction of viscous shocks is uniformly in time and independent of viscosity coefficient, based on inviscid limit, we have the contraction (thus, uniqueness) of entropy shocks to the isentropic Euler in a class of large perturbation without any local regularity such as strong trace property. In this talk, I will present this kind of contraction property for entropy inviscid shocks and viscous shocks.
17 February 2017 (FRIDAY 11:00)
Solitary water waves
Courant Institute of Mathematical Sciences
New York University
The water wave equations describe the motion of a fluid (water) bounded above by a free surface. This free surface is subject to constant (atmospheric) pressure, while gravity acts as an external force. Traveling waves which are localized (solitary) and have small amplitude can be described by models such as the Korteweg–de Vries equation. To investigate their large-amplitude cousins, however, it is necessary to work with the full (Euler) equations. In this talk we will use continuation arguments to construct curves of large-amplitude solitary waves. We will also discuss whether the free surface of such a wave is necessarily a graph, and show that the wave speed exceeds the critical value appearing in the Korteweg-de Vries approximation.
20 February 2017 (MONDAY 2:30)
Computational Modeling of Multiphase Complex Fluids with Applications
University of North Carolina at Chapel Hill
Complex fluids are ubiquitous in nature and in synthesized materials, such as biofilms, synthetic and biological polymeric solutions. Modeling and simulation of complex fluids has been listed as one of the 21st century mathematical challenges by DARPA, which is therefore of great mathematical and scientific significance. In this talk, I will firstly explain our research motivations by introducing several complex fluids examples, and traditional modeling techniques. Integrating the phase field approach, we then derive hydrodynamic theories for modeling multiphase complex fluid flows. Secondly, I will discuss a general technique for developing second order, linear, unconditionally energy stable numerical schemes solving hydrodynamic models. The numerical strategy is rather general that it can be applied for a host of complex fluids models. All numerical schemes developed are implemented in C2FD, a GPU-based software package developed by our group for high-performance computing/simulations. Finally, I will present several applications in cell biology, materials science and soft matter physics. 3D numerical simulations will be given. The modeling, numerical analysis and high-performance simulation tools are systematic and applicable to a large class of fluid flow problems in science and engineering.
23 February 2017
Efficient numerical schemes for the Vlasov-Maxwell system in plasma applications
Michigan State University
Understanding complex behaviors of plasmas plays an increasingly important role in modern science and engineering. A fundamental model in plasma physics is the Vlasov-Maxwell system, which is a nonlinear kinetic transport model describing the dynamics of charged particles due to the self-consistent electromagnetic forces. As predictive simulation tools in studying such a complex system, efficient, reliable and accurate transport schemes are of fundamental significance. The main numerical challenges lie in the high dimensionality, nonlinear coupling, and inherent multi-scale nature of the system. In this talk, I will present several numerical methodologies to address these challenges. In this first part, I introduce a sparse grid discontinuous Galerkin (DG) method for solving the Vlasov equation, which is able to not only break the curse of dimensionality via a novel sparse approximation space, leading to remarkable computational savings, but also retain attractive properties of DG methods. In the second part, an asymptotic preserving Maxwell's solver is developed. The scheme is shown to be able to recover the correct asymptotic limit known as the Darwin limit and hence address the scale separation issue arising from plasma simulations. Theoretical and numerical results will be presented to demonstrate the efficiency and efficacy of the proposed schemes.
24 February 2017 (FRIDAY 11:00)
Bounding average quantities in dynamical systems using semidefinite programming
University of Michigan
I will discuss the task of proving bounds on average quantities in dissipative dynamical systems, including time averages in finite-dimensional systems and spatiotemporal averages in PDE systems. In the finite-dimensional case, I will describe computer-assisted methods for computing bounds by constructing nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proven by constructing Lyapunov functions. Nonnegativity of these polynomials is enforced by requiring them to be representable as sums of squares, a condition that can be checked computationally using the convex optimization technique of semidefinite programming. The methods will be illustrated using the Lorenz equations, for which they produce novel bounds on various average quantities. I will then discuss work in progress on extending these computer-assisted methods to dissipative PDEs, using the Kuramoto-Sivashinky equation as an example.
2 March 2017
A multi-scale model for vector-borne diseases
Arizona State University
There is recent interest in mathematical models which connect the epidemiological aspects of infectious diseases to the within-host dynamics of the pathogen and immune response. Multi-scale modeling of infections allows for assessing how immune-pathogen dynamics affect spread of the disease in the population. Here, I consider a within-host model for immune-pathogen dynamics nested in an age-since-infection structured PDE system for vector-borne epidemics. First, we study pathogen-host coevolution by analytically establishing evolutionary stable strategies for parasite and host, and by utilizing computational methods to simulate the evolution in various settings. We find that vector inoculum size can contribute to virulence of vector-borne diseases in distinct ways. Next, we develop a robust methodology for identifiability and estimation of parameters with multi-scale data, along with sensitivity analysis. The nested multi-scale model is fit to combined within-host and epidemiological data for Rift Valley Fever. An ultimate goal is to accurately model how control measures, such as vaccination and drug treatment, affect both scales of infection.
6 March 2017 (MONDAY 2:45)
From Picard groups to Picard categories
University of Sheffield
The Picard group of a commutative ring is a classical invariant that appears in a number of guises in algebraic geometry and number theory. This group can be enhanced to a more sophisticated invariant called a Picard category, and these appear naturally in the algebraic contexts previously mentioned as well as category theory and algebraic topology. While not a new structure, little work has been done on the algebra of Picard categories. I will introduce the basic notions, give examples and applications of Picard categories, and discuss how the algebra of Picard categories relates to that of both abelian groups and spectra via homological algebra.
8 March 2017 (WEDNESDAY 3:00)
Toric topology, polyhedral products and applications
Polyhedral products are the central objects in the emerging field of toric topology, which stands at the crossroads of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra. In this talk I will give an introduction to these combinatorial constructions in topology, and give a few applications, including calculations of monodromy representations.
10 March 2017 (FRIDAY 11:15)
Configuration space integrals and integer-valued cohomology classes in spaces of knots and links
University of Massachusetts
Configuration space integrals are a generalization of the Gauss linking integral which produce invariants of both knots and links. They can be used to construct all Vassiliev invariants, as well as nontrivial, real-valued “Vassiliev classes” in the cohomology of spaces of knots and links. I will review these ideas and then explain how configuration space integrals can be reinterpreted topologically to recover an integer lattice among the real-valued Vassiliev classes. This work also provides constructions of mod-p classes which need not be mod-p reductions of classes in this integer lattice.
28 March 2017 (TUESDAY 3:30)
Asymptotically Well-posed Boundary Conditions for Partitioned Fluid-Structure Algorithms
Rensselaer Polytechnic Institute
A new partitioned algorithm is described for solving fluid-structure interaction (FSI) problems coupling incompressible flows with elastic structures undergoing finite deformations. The new algorithm, referred to as the Added-Mass Partitioned (AMP) scheme, overcomes the added-mass instability that has for decades plagued partitioned FSI simulations of incompressible flows coupled to light structures. Within a Finite-Difference framework, the AMP scheme achieves fully second-order accuracy and remains stable, without sub-time-step iterations, even for very light structures when added-mass effects are strong. The stability and accuracy of the AMP scheme is validated through mode analysis and numerical experiments. Aiming to extend the AMP scheme to a Finite-Element framework, we also develop an accurate and efficient Finite-Element Method for solving the Incompressible Navier-Stokes Equations with high-order accuracy up-to the boundary.
24 April 2017 (MONDAY 3:30)
Global Hopf bifurcation for differential-algebraic equations with state dependent delay
University of Texas at Dallas
We discuss the type of differential equations with state-dependent delays and the associated global Hopf bifurcation problems. In particular, we develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the $S^1$-equivariant degree. We apply the global Hopf bifurcation theory to a model of genetic regulatory dynamics with degenerate threshold type state-dependent delays, for a description of the global continuation of the periodic oscillations.