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Mathematics Colloquium

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 Leonel Robert 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.

Fall 2019 Schedule

  • 10 October 2019 CANCELLED
  • 24 October 2019
    Accurate interval estimation of the intra-class correlation in a mixed effects model with a non-linear mean
    Thomas Mathew
    Department of Mathematics & Statistics
    University of Maryland Baltimore County
    For the assessment of agreement among repeated measurements obtained at different laboratories, or by different medical devices or physicians, the intra-class correlation coefficient (ICC) is a widely used index. The ICC can also be used as a measure of heterogeneity in meta-analysis. The present talk is on a meta-analysis application where the problem of interest is the dose-response relationship between different doses of several antipsychotic drugs, and the dopamine D2 receptor occupancy. The mean response is modeled using the Michaelis-Menten curve, which is a non-linear function of unknown parameters, where the dose levels appear as a covariate. An additive random effect and an additive error term are included so as to capture the between- and within-study variabilities. The model is investigated in Lako et al. (2013, Journal of Clinical Psychopharmacology), and considered by Demetrashvili and Van den Heuvel (2015, Biometrics). The latter authors investigated the extent to which the between-study variability could be dominating. This was assessed using the ICC, and the authors approximated the distribution of the estimated ICC using a beta distribution. The approximation was used to derive confidence limits for the ICC, and the authors refer to this as the beta approach. In the present talk, we shall consider likelihood based confidence limits for the ICC and a corresponding small sample asymptotic procedure so as to achieve accurate coverages in small sample scenarios. Simulation results and data analysis will be reported, and the proposed method will be compared with the beta approach. Interval estimation of the ICC under linear mixed effects models will be briefly discussed (time permitting).
    Joint work with Xiaoshu Feng and Kofi Adragni.
  • 31 October 2019
    Gabriele Morra
    UL Lafayette
  • 14 Novermber 2019
    Scott Hottovy
    United States Naval Academy
  • 21 November 2019
    Jason McCullough
    Iowa State University
  • 3 December 2019
    Jo Nelson
    Rice University

Spring 2019 Schedule

  • 24 January 2019
    How to build a surface of genus six
    Ben Knudsen
    Harvard University

    The study of the configuration spaces of graphs began in the context of motion planning problems in robotics. A surge of recent research has been reexamining them through the lenses of algebraic topology, geometric group theory, representation stability, physics, tropical geometry, and equivariant stable homotopy theory. This talk will be a low-tech and example-driven introduction to these spaces, which, along the way, will provide an explanation for its title.

  • 31 January 2019
    The Feit-Thompson Odd Order Theorem, 56 years later
    George Glauberman
    University of Chicago

    The Feit-Thompson Odd Order Theorem asserts that every finite group of odd order is solvable. I plan to discuss the background of the theorem, the proof, and some related open questions and recent developments.

  • 14 February 2019 (Oliver Hall auditorium (room 112))
    Using Mathematics to Fight Cancer
    Ami Radunskaya
    Pomona College

    What can mathematics tell us about the treatment of cancer? In this talk I will present some of work that I have done in the modeling of tumor growth and treatment over the last fifteen years. Cancer is a myriad of individual diseases, with the common feature that an individual's own cells have become malignant. Thus, the treatment of cancer poses great challenges, since an attack must be mounted against cells that are nearly identical to normal cells. Mathematical models that describe tumor growth in tissue, the immune response, and the administration of different therapies can suggest treatment strategies that optimize treatment efficacy and minimize negative side-effects. However, the inherent complexity of the immune system and the spatial heterogeneity of human tissue gives rise to mathematical models that pose unique challenges for the mathematician. In this talk I will give a few examples of how mathematicians can work with clinicians and immunologists to understand the development of the disease and to design effective treatments. I will use mathematical tools from dynamical systems, optimal control and network analysis.
    This talk is intended for a general math audience: no knowledge of biology will be assumed.

  • 21 February 2019
    An introduction to calculus of functors and its applications to knots and links
    Ismar Volić
    Wellesley College

    Calculus of functors is a theory which “approximates” functors in topology and algebra much like the Taylor series approximates an ordinary analytic function. The goal of this talk is to give an introduction to calculus of functors and indicate how one constructs the Taylor tower that mimics the Taylor series. The talk will start with a brief introduction to categories and functors, and, after an overview of the general setup for functor calculus, special attention will be devoted to one of its brands, namely manifold calculus. In particular, we will explain how this theory has been applied with great success to embedding spaces, and specifically to spaces of knots and links.

  • 28 February 2019
    Dynamical Systems on Networks and their Applications: Perspectives from Population Dynamics
    Zhisheng Shuai
    University of Central Florida

    Many large-scale dynamical systems arising from different fields of science and engineering can be regarded as coupled systems on networks. Examples include biological and artificial neural networks, nonlinear oscillators on lattices, complex ecosystems and the transmission models of infectious diseases in heterogeneous populations. Of particular interest is to investigate in what degree and fashion the dynamical behaviors are determined by the architecture of the network encoded in the directed graph. We will address this from population dynamics perspectives.
    Specifically, many recent outbreaks and spatial spread of infectious diseases have been influenced by human movement over air, sea and land transport networks, and/or anthropogenic-induced pathogen/vector movement. These spatial movements in heterogeneous environments and networks are often asymmetric (biased). The effects of asymmetric movement versus symmetric movement will be investigated using several epidemiological models from the literature, and the analytical tools employed are from differential equations, dynamical systems to matrix theory and graph theory. These investigations provide new biological insights on disease transmission and control, and also highlight the need of a better understanding of dynamical systems on networks.

  • 14 March 2019
    Some recent developments on linear processes and linear random fields
    Hailin Sang
    University of Mississippi

    The linear processes and linear random fields are tools for studying stationary time series and stationary random fields. One can have a better understanding of many important time series and random fields by studying the corresponding linear processes and linear random fields. In this talk we survey some recent developments on linear processes and linear random fields. One part is the moderate and large deviations under different conditions. This part research plays an important role in many applied fields, for instance, the premium calculation problem, risk management in insurance, nonparametric estimation and network information theory. We also study the memory properties of transformations of linear processes which have application in econometrics and financial data analysis when the time series observations have non-Gaussian heavy tails. Entropy is widely applied in the fields of information theory, statistical classification, pattern recognition and so on since it is a measure of uncertainty in a probability distribution. At the end, we focus on the estimation of the quadratic entropy for linear processes. With a Fourier transform on the kernel function and the projection method, it is shown that, the kernel estimator has similar asymptotical properties as the i.i.d. case if the linear process has the defined short range dependence. Part of the results are confirmed by simulation studies. We also obtain very promising results in some real data analysis.

  • 21 March 2019 (Oliver Hall auditorium (room 112))
    Gini Distance Correlation and Feature Selection
    Xin Dang
    University of Mississippi

    Big data is becoming ubiquitous in the biological, engineering, geological and social sciences, as well as in government and public policy. Building an interpretable model is an effective way to extract information and to do prediction. However, this task becomes particularly challenging for the scenario of big data, which are large scale and ultra-high dimensional with mixed-type features being both structured and unstructured. A common practice in tackling this challenge is to reduce the number of features under consideration via feature selection by choosing a subset of features that are "relevant" and useful. The work in this talk aims at proposing a new dependence measure in feature selection. The features having strong dependence with the response variable are selected as candidate features. We proposes a new Gini correlation to measure dependence between categorical response and numerical feature variables. Compared with the existing dependence measures, the proposed one has both computational and statistical efficiency advantages that improve the feature selection procedure and therefore the resulting prediction model.

  • 28 March 2019
    Linda Allen
    Texas Tech University

  • 4 April 2019
    Ergodicity and loss of capacity for a family of concave random maps
    Peter Hinow
    University of Wisconsin-Milwaukee

    Random fluctuations of an environment are common in ecological and economical settings. We consider a family of concave maps on the unit interval, f_\lambda(x)=x(1+\lambda-x), that model a self-limiting growth behavior. The maps are parametrized by an independent, identically distributed random variable \lambda with values in the unit interval. We show the existence of a unique invariant ergodic measure of the resulting random dynamical system for arbitrary parameter distributions supported on certain subintervals of [0,1]. Moreover, there is an attenuation of the mean of the state variable compared to the constant environment with the averaged parameter. We also provide an example of a family of just two maps such that the invariant probability measure is supported on a Cantor set.

  • 11 April 2019
    Numerical methods for anomalously diffusive hyperbolic models: efficiency analysis and pattern formation
    J. E. Macías-Díaz
    Universidad Autónoma de Aguascalientes, Mexico

    In this talk, we will depart from a generalized two-dimensional hyperbolic system that appears in epidemic models. The model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. The presence of two-dimensional diffusive terms consisting of fractional operators of the Riesz type is considered here, using spatial differentiation orders in the set (0,1)U(1,2]. We impose initial conditions on a closed and bounded rectangle, and a finite-difference methodology based on the use of fractional centered differences is proposed. Among the most important results of this work, we prove the existence and the uniqueness of the solutions of the numerical method, and establish analytically the second-order consistency of our scheme. Moreover, the discrete energy method is employed to prove the stability and the quadratic convergence of the technique. Some numerical simulations obtained through our method show the appearance of Turing patterns and wave instabilities, in agreement with some previous reports by the speaker on superdiffusive hyperbolic systems. As a new application, we show that Turing patterns are also present in subdiffusive scenarios.

  • 25 April 2019
    Rank-based estimating equation with non-ignorable missing responses
    Yichuan Zhao
    Georgia State

    In this talk, a general regression model with responses missing not at random is considered. From a rank-based estimating equation, a rank-based estimator of the regression parameter is derived. Based on this estimator's asymptotic normality property, a consistent sandwich estimator of its corresponding asymptotic covariance matrix is obtained. In order to overcome the over-coverage issue of the normal approximation procedure, the empirical likelihood based on the rank-based gradient function is defined, and its asymptotic distribution is established. Extensive simulation experiments under different settings of error distributions with different response probabilities are considered, and the simulation results show that the proposed empirical likelihood approach has better performance in terms of coverage probability and average length of confidence intervals for the regression parameters compared with the normal approximation approach and its least-squares counterpart. A data example is provided to illustrate the proposed methods.

  • 30 April 2019 (Oliver Hall auditorium (room 112))
    Contact Invariants and Reeb Dynamics
    Jo Nelson
    Rice University

    Contact topology is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability. The associated one form is called a contact form and uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will give some background on this subject, including motivation from classical mechanics. I will also explain how to make use of J-holomorphic curves to obtain a Floer theoretic contact invariant, contact homology, whose chain complex is generated by closed Reeb orbits. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.

Fall 2018 Schedule

  • 13 September 2018
    A Case Study of Green Tree Frog Population Size Estimation by Repeated Capture-Mark-Recapture Method with Individual Tagging: A Parametric Bootstrap Method vs. Jolly-Seber Method
    Nabendu Pal
    UL Lafayette

    This talk deals with estimation of a green tree frog population in an urban setting using repeated capture-mark-recapture (CMR) method over several weeks with an individual tagging system (ITS) which gives rise to a complicated generalization of the hypergeometric distribution. Based on the maximum likelihood estimation, a parametric bootstrap approach is adopted to obtain interval estimates of the weekly population size which is the main objective of our work. The method is computation based; and programming intensive to implement the algorithm for resampling. This method can be applied to estimate the population size of any species based on repeated CMR method at multiple time points. Further, it has been pointed out that the well known Jolly-Seber method, which is based on some strong assumptions, produces either unrealistic estimates, or may have situations where its assumptions are not valid for our observed data set.

  • 20 September 2018
    The Maximum Dimension of a Lie Nilpotent Matrix Algebra
    Leon Van Wyk
    Stellenbosch University

    In 1905 Schur obtained the maximum dimension of a commutative subalgebra of the n x n matrix algebra over the complex numbers. We discuss the maximum dimension of a Lie nilpotent (index m) subalgebra of the n x n matrix algebra over any field.

  • 11 October 2018
    Reflection Positivity: Representation Theory meets Quantum Field Theory
    Gestur Olafsson
    Louisiana State University

    Reflection positivity is one of the axioms of constructive quantum field theory as they were formulated by Osterwalder and Schrader 1973/1975. The goal is to build a bridge from a euclidean quantum field to a relativistic quantum field by analytic continuation to imaginary time. In terms of representation theory this can be formulated as transferring representations of the euclidean motion group to a unitary representation of the Poincare group via c-duality of symmetric pairs.
    We start by recalling the idea of reflection positivity and how it relates to representation theory. We discuss the case of one-parameter subgroups and then give several examples from representation theory. In particular we discuss the duality between the sphere and hyperbolic space.

  • 25 October 2018
    Reducing Mathematical Models for Wolbachia Transmission in Mosquitoes to Control Mosquito-borne Diseases
    Zhuolin Qu
    Tulane University

    We develop and analyze a reduced model for the spread of Wolbachia bacteria infection in wild mosquitoes. Wolbachia is a natural parasitic microbe that can reduce the ability of mosquitoes to spread mosquito-borne viral diseases such as dengue fever, chikungunya, and Zika. It is difficult to sustain an infection of the maternally transmitted Wolbachia in a wild mosquito population because of the reduced fitness of the infected mosquitoes and cytoplasmic incompatibility limiting maternal transmission. The infection will only persist if the fraction of the infected mosquitoes exceeds a minimum threshold. This threshold can be characterized as a backward bifurcation for a system of nine ordinary differential equations modeling the complex maternal transmission of the bacteria infection in a heterosexual mosquito population. Although the large system of differential equations capture the detailed transmission dynamics, they are difficult to extend to account for the spatial heterogeneity of Wolbachia infection when releasing the infected mosquitoes into the wild. We derive a seven-equation, a four-equation and a two-equation system of differential equations that are formulated in terms of the more accurate nine-equation model and capture the important properties of the original system. The reduced models preserve the key dimensionless numbers, such as the basic reproductive number, and accurately capture the backward bifurcation threshold.

  • 30 October 2018 (TUESDAY)
    Cohomology of Cantor minimal systems and a model for Z^2-actions
    Thierry Giordano
    University of Ottawa

    In 1992, Herman, Putnam and Skau used ideas from operator algebras to present a complete model for minimal actions of the group Z on the Cantor set, i.e. a compact, totally disconnected, metrizable space with no isolated points. The data (a Bratteli diagram, with some extra structure) is basically combinatorial and the two great features of the model are that it contains, in a reasonably accessible form, the orbit structure of the resulting dynamical system and also cohomological data provided either from the K-theory of the associated C*-algebra or more directly from the dynamics via group cohomology. This led to a complete classification of such systems up to orbit equivalence. This classification was extended to include minimal actions of Z^2 and then to minimal actions of finitely generated abelian groups. However, what was not extended was the original model and this has made difficult the general understanding of these actions.
    In this talk I will indicate how we can associate to any dense subgroup H of R^2 containing Z^2 a minimal action of Z^2 on the Cantor set, such that its first cohomology group is isomorphic to H. Joint work with Ian F. Putnam and Christian F. Skau.

  • 1 November 2018
    Projected Tests for High-Dimensional Covariance Matrices
    Tung-Lung Wu
    Mississippi State University

    The classic likelihood ratio test for testing the equality of two covariance matrices breakdowns due to the singularity of the sample covariance matrices when the data dimension is larger than the sample size. In this talk, we present a conceptually simple method using random matrices to project the data onto a one-dimensional random subspace so that conventional methods can be applied. Both one-sample and two-sample tests for high-dimensional covariance matrices are considered. A transformation using the precision matrix is used to help maintain the information on the off-diagonal elements of the covariance matrices. Multiple projections are used to improve the performance of the proposed tests. An extremal type theorem is established and used to estimate the significance level. Simulations and an application to the Acute Lymphoblastic Leukemia (ALL) data are given to illustrate our method.

  • 8 November 2018
    Model Selection without penalty using Generalized Fiducial Inference
    Jan Hannig
    University of North Carolina, Chapel Hill

    R. A. Fisher, the father of modern statistics, developed the idea of fiducial inference during the first half of the 20th century. While his proposal led to interesting methods for quantifying uncertainty, other prominent statisticians of the time did not accept Fisher's approach as it became apparent that some of Fisher's bold claims about the properties of fiducial distribution did not hold up for multi-parameter problems. Beginning around the year 2000, the authors and collaborators started to re-investigate the idea of fiducial inference and discovered that Fisher's approach, when properly generalized, would open doors to solve many important and difficult inference problems. They termed their generalization of Fisher's idea as generalized fiducial inference (GFI). The main idea of GFI is to carefully transfer randomness from the data to the parameter space using an inverse of a data generating equation without the use of Bayes theorem. The resulting generalized fiducial distribution (GFD) can then be used for inference. After more than a decade of investigations, the authors and collaborators have developed a unifying theory for GFI, and provided GFI solutions to many challenging practical problems in different fields of science and industry. Overall, they have demonstrated that GFI is a valid, useful, and promising approach for conducting statistical inference.
    Standard penalized methods of variable selection and parameter estimation rely on the magnitude of coefficient estimates to decide which variables to include in the final model. However, coefficient estimates are unreliable when the design matrix is collinear. To overcome this challenge an entirely new perspective on variable selection is presented within a generalized fiducial inference framework. This new procedure is able to effectively account for linear dependencies among subsets of covariates in a high-dimensional setting where $p$ can grow almost exponentially in $n$, as well as in the classical setting where $p \le n$. It is shown that the procedure very naturally assigns small probabilities to subsets of covariates which include redundancies by way of explicit $L_{0}$ minimization. Furthermore, with a typical sparsity assumption, it is shown that the proposed method is consistent in the sense that the probability of the true sparse subset of covariates converges in probability to 1 as $n \to \infty$, or as $n \to \infty$ and $p \to \infty$. Very reasonable conditions are needed, and little restriction is placed on the class of possible subsets of covariates to achieve this consistency result.
    (Joint work with Jonathan Williams)

  • 15 November 2018
    Constructing homotopies of least complexity
    Gregory R. Chambers
    Rice University

    Can we replace a homotopy of curves on a surface with an isotopy of curves without increasing lengths? Can we replace an isotopy of curves with one composed of pairwise disjoint curves also without increasing lengths? If we have a map between finite simplicial complexes of Lipschitz constant L, and if this map is null-homotopic, then what is the minimal Lipschitz constant of a null-homotopy? All of these questions seek to understand the minimal geometric or topological complexity of a homotopy under constraints. In addition to discussing the solutions to these problems, I will describe some applications to bounding the complexity of null-cobordisms, and to proving the existence of minimal surfaces in noncompact manifolds.

Spring 2018 Schedule

  • 25 January 2018
    Quantitative uniqueness of partial differential equations
    Jiuyi Zhu
    Louisiana State University

    Motivated by the study of eigenfunctions, we consider the quantitative uniqueness of partial differential equations. The quantitative uniqueness is characterized by the order of vanishing of solutions, which describes quantitative behavior of strong unique continuation property. Strong unique continuation property states that a solution vanishes identically if the solution vanishes of infinite order at a point . It is interesting to know how the norms of the coefficient functions control the order of vanishing. We will report some recent progresses about quantitative uniqueness in different spaces for elliptic equations and parabolic equations. Part of work is joint with Blair Davey.

  • 8 February 2018
    Examining the effect of environmental disturbances on population and evolutionary dynamics
    Amy Veprauskas
    University of Louisiana at Lafayette

    Environmental disturbances, such as oils spills and other toxicants, may impact populations through reductions in vital rates, resulting in population declines or even extinction. In this talk, we examine how a disturbance may affect both population and evolutionary dynamics. We first develop an evolutionary game theoretic version of a general nonlinear matrix model that includes the dynamics of a vector of mean phenotypic traits subject to natural selection. Using bifurcation analysis, we establish the existence and stability of a branch of positive equilibria that bifurcates from the extinction equilibrium when the inherent growth rate passes through one. We apply these results to demonstrate how the evolution of resistance to a toxicant may change persistence scenarios for species of daphniids. We show that, in certain situations, it is possible for a daphniid population to evolve toxicant resistance whereby it is able to persist at higher levels of the toxicant than it would otherwise. These results highlight the complexities involved in using surrogate species to examine toxicity. We then consider a nonautonomous matrix model to examine the possible long-term effects of environmental disturbances on population recovery. We focus on the recovery time following a single disturbance, which is defined to be the time it takes for the population to return to its pre-disturbance population size. Using methods from matrix calculus, we derive explicit formulas for the sensitivity of the recovery time with respect to properties of the population or the disturbance. We apply the results of this model to examine the possible response of a sperm whale population to an environmental disturbance.

  • 22 February 2018
    Bruce Wade
    University of Wisconsin - Milwaukee

  • 23 February 2018 (FRIDAY 12:00 ROOM 201)
    q-Series: a Bridge between Analysis and Discrete Mathematics
    Mourad E. H. Ismail
    University of Central Florida

    We discuss the connection between partitions and allied areas of combinatorics and the q-series identities. We shall illustrate this interaction by several examples.

  • 6 March 2018 (TUESDAY)
    A Brief Summary of Research Interests and a Vision for the Future of the Department of Mathematics
    Kamel Rekab
    University of Missouri - Kansas City

    In this talk, I provide a summary of my research interests. I present in very broad terms the main ideas on work I have done in areas such as design for manufacturing, software testing, cybersecurity, breast cancer, aspiration pneumonia prevention for stroke patients, microRNA classification and sleep apnea. I will also present descriptions of my contributions to mathematical statistics and the design of experiments. In the second part of the talk, I give my vision and how I view my responsibilities as head of the department.

  • 8 March 2018
    Hyper-Lie algebras
    Benjamin Ward
    Stockholm University

    I will introduce a higher dimensional analog of the notion of a Lie algebra. Roughly speaking, a hyper-Lie algebra is a vector space along with a multi-linear operation for each homology class in the moduli space of punctured Riemann spheres. I will explain where such algebras arise, what they measure, and how I became interested in them.

  • 12 March 2018 (MONDAY)
    The structure of cohomology operations
    Martin Frankland
    Institut für Mathematik
    Universität Osnabrück

    Algebraic topology studies topological spaces using algebraic invariants, such as cohomology groups. One obtains a richer structure by taking into account the operations acting on cohomology. This structure is encoded by the Steenrod algebra and the Adams spectral sequence, powerful computational tools in homotopy theory. The generalized and motivic Adams spectral sequences have also proved fruitful.
    In this talk, I will survey some classical results, recent developments, and open problems related to cohomology operations. The focus will be on structural features rather than computations.

  • 20 March 2018 (TUESDAY)
    Operads and their ilk, up to coherent homotopy
    Philip Hackney
    Macquarie University

    Operads are algebraic gadgets which control various types of algebras. For example, there is an operad L so that the set of actions of L on a vector space V is in bijection with the set of Lie algebra structures on V (similarly, there are operads controlling associative and commutative algebra structures). Variations on the concept of operad (such as colored operad, prop(erad), cyclic operad, etc) allow one to model other types of structures, such as Hopf algebras and maps of associative algebras.
    In the past decade, homotopy-coherent versions of (generalized) operads have become increasingly important for applications. In this talk, we will discuss some frameworks that have emerged (including in my work with Bergner, Robertson, Yau) to make this notion precise.

  • 22 March 2018
    Persistence measures for populations in river environments
    Yu Jin
    University of Nebraska-Lincoln

    Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem impacts. We develop process-oriented reaction-diffusion-advection equations that couple hydraulic flow to population growth and dispersal in the flow, and we analyze them to assess the effect of water flow on population persistence. Then we consider the situation where a population grows on the benthos, drift in water, and transfer between the water column and the benthos, and we use reaction-diffusion-advection equations coupled with ordinary differential equations to describe the dynamics of a single species and of two competitive species. We present a mathematical framework of persistence measures based on the net reproductive rate and related measures as well as eigenvalues of corresponding eigenvalue problems. We apply all the persistence measures under various flow regimes to investigate the influences of various factors on population persistence in rivers. The theory developed here provides the basis for effective decision-making tools for water managers.

  • 29 March 2018
    Variational method and periodic solutions of N-body problem
    Zhifu Xie
    Department of Mathematics
    University of Southern Mississippi

    N-body problem concerns the motion of celestial bodies under universal gravitational attraction. Although it has been a long history to apply variational method to N-body problem, it is relatively new to make some important progress in the study of periodic solutions. We develop the variational method with Structural Prescribed Boundary Conditions (SPBC) and we apply it to study some well-known periodic solutions in the 3-body problem with equal masses such as Schubart orbit (1956) and Broucke-Henon orbit (1975). Simulations for some new orbits discovered by this method will also be presented. The presentation will be accessible to undergraduate and graduate students.

  • 19 April 2018
    Higher dimensional knots
    Victor Turchin
    Kansas State University

    A higher dimensional knot is an embedding of a sphere S^m in a Euclidean space R^n, m 2 such knots are much easier to classify. I will explain some results of Haefliger and will also briefly describe more recent results about the spaces of such knots. With the new approach one can not only compute the isotopy classes of knots, but also the homotopy groups of such knot spaces.

  • 20 April 2018 (FRIDAY)
    Kibble-Slepian formulas for univariate and multivariate Hermite polynomials
    Plamen Simeonov
    University of Houston-Downtown
    We will review several Kibble-Slepian formulas for univariate and multivariate Hermite polynomials. These formulas include the Kibble-Slepian formula for the classical Hermite polynomials, and some recently derived formulas for the univariate Ito polynomials, the multivariate real Hermite polynomials, and the multivariate Ito polynomials, which are special cases of the multivariate complex Hermite polynomials. Some special cases such as Mehler type formulas and generating functions will be discussed. We will also briefly discuss several derivation techniques and proofs for such formulas.

  • 26 April 2018
    Global dynamics of discrete dynamical systems/difference equations: Application to population dynamics and economics
    Saber Elaydi
    Trinity University
    San Antonio, Texas
    Global dynamics of difference equations/discrete dynamical systems are the most challenging problems in these disciplines. In this talk, we will explore some of the recent breakthroughs and advances in this area. The global dynamics of two types of discrete dynamical systems (maps) have been successfully established. These are triangular difference equations (maps) and monotone discrete dynamical systems (maps). We establish a general theory of triangular maps with minimal conditions. Smith’s theory of planar monotone discrete dynamical systems is extended via a new geometric theory to any finite dimension. Then we show how to establish global stability for maps that are neither monotone nor triangular via singularity theory and the notion of critical curves.
    Applications to models in biology and economics will be discussed.

Fall 2017 Schedule

  • 24 August 2017
    Mathematical modeling of interactive wild and paratransgenic mosquitoes
    Jia Li
    Department of Mathematical Sciences
    University of Alabama in Huntsville

    We formulate homogeneous and stage-structured models for the interactive wild mosquitoes and paratransgenic mosquitoes carrying genetically-modified bacteria which are resistant to malaria transmissions. We establish conditions for the existence and stability of equilibria for the model systems and provide numerical examples to demonstrate our findings. We then investigate how the bacteria uptake rate of wild mosquitoes and the bacteria loads of paratransgenic mosquitoes carrying affect their interactive dynamics.

  • 21 September 2017
    Computed Flow and Fluorescence Over the Ocular Surface
    Longfei Li
    Mathematics Department
    UL Lafayette

    Fluorescein is perhaps the most commonly used substance to visualize tear film thickness and dynamics; better understanding of this process aids understanding of dry eye syndrome which afflicts millions of people. We study a mathematical model for tear film flow, evaporation, solutal transport and fluorescence over the exposed ocular surface during the interblink. Transport of the fluorescein ion by fluid flow in the tear film affects the intensity of fluorescence via changes in concentration and tear film thickness. Evaporation causes increased osmolarity and potential irritation over the ocular surface; it also alters fluorescein concentration and thus fluorescence. Using thinning rates from in vivo measurements together with thin film equations for flow and transport of multiple solutes, we compute dynamic results for tear film quantities of interest. We compare our computed fluorescent intensity distributions with in vivo observations. A number of experimental features are recovered by the model.

  • 28 September 2017
    Discrete-time Structured Model for Malaria Transmission with constant releasing sterile mosquitoes
    Yang Li
    Mathematics Department
    UL Lafayette

    To incorporate the interactive mosquitoes into malaria transmissions, we formulate susceptible-exposed-infective-recovered (SEIR) compartmental discrete-time models, which are of high dimensions, and then include the interactive mosquito models into these disease models. We derive formulas for the reproductive number R_0 of infection for the malaria models with or without sterile mosquitoes and explore the existence of endemic Fixed points as well. We then study the impact of sterile mosquitoes releases on the disease transmissions by investigating the effects of varying the releases of sterile mosquitoes. We use numerical simulations to verify our results for all cases and finally give brief discussions of our findings.

  • 3 October 2017 (TUESDAY)
    Z^d-Odometers : a very interesting class of free minimal actions on the Cantor set
    Thierry Giordano
    University of Ottawa

    Z-odometers form a very rigid class of minimal Cantor systems: any two orbit equivalent Z-odometers are conjugate. The situation is totally different in higher dimensions. In this talk, I will review the construction of Z^d-odometers and show that their first group of cohomology is a complete algebraic invariant of conjugation. Examples of orbit equivalent Z^2-odometers which are not conjugate, but orbit equivalent will be presented! (This work is joint with I. Putnam and C. Skau.)

  • 12 October 2017
    Diagonalizability and the Pythagorean Theorem
    Jireh Loreaux
    Southern Illinois University, Edwardsville

    There is a natural interpretation of the Pythagorean Theorem in terms of elementary linear algebra. While this interpretation is straightforward, it leads to a much more general statement in operator theory with surprising consequences. In this talk, we explore the relationship between these more general statements and the diagonalizability of matrices by means of a unitary which is a small (Hilbert-Schmidt) perturbation of the identity.

  • 2 November 2017
    Hyperrectangular Tolerance and Prediction Regions for Setting Multivariate Reference Regions in Laboratory Medicine
    Derek S. Young
    University of Kentucky

    Reference regions are widely used in clinical chemistry and laboratory medicine to interpret the results of biochemical or physiological tests of patients. There are well-established methods in the literature for reference limits for univariate measurements, however, only limited methods are available for the construction of multivariate reference regions. This is because traditional multivariate statistical regions (e.g., confidence, prediction, and tolerance regions) are not constructed based on a hyperrectangular geometry. We address this problem by developing multivariate hyperrectangular nonparametric tolerance regions for setting the reference regions. Our approach utilizes statistical data depth to determine which points to trim and then the extremes of the trimmed dataset are used as the faces of the hyperrectangular region. We also specify the number of points to trim based on previously-established asymptotic results. Extensive coverage results show the favorable performance of our algorithm provided a minimum sample size criterion is met. Our procedure is used to obtain reference regions for addressing two important clinical problems: (1) characterizing insulin-like growth factor concentrations in the serum of adults and (2) assessing kidney function in adolescents. This is a joint work with Thomas Mathew (UMBC).

  • 9 November 2017
    Algebraic K-theory and polynomial maps
    Saul Glasman
    University of Minnesota

    Assuming no prior background, I'll give a brief introduction to algebraic K-theory, an elusive invariant of rings with profound connections to algebraic topology and number theory. Many properties of K-theory are best understood via /categorified algebra,/ where algebraic constructions are performed at the level of categories rather than individual objects. Armed with this doctrine, and a very concrete version of the theory of polynomial functors, we'll see that such functors give rise to polynomial maps on zeroth K-groups and thence uncover some of the arcane algebraic structure present on these groups. Insofar as time permits, I'll formally discuss lambda-rings and the theory of spectral lambda-rings developed in joint work in progress with C. Barwick, A. Mathew and T. Nikolaus.

  • 16 November 2017
    How to Write a Research Paper for Publication
    T. Wu
    Southern Illinois University Edwardsville
    Edwardsville, Illinois

    To write a research paper is not an easy task. It is even more difficulty to write a research paper for publication. For a beginning or junior faculty she or he will face some promotion pressure, job security, family pressure, and social problem. In a short time period she or he must at least publish one paper in a prestige journal. To do this she or he must have good English background in reading and writing and thoroughly understanding the subject matter and wide range of related disciplines. This talk is based upon the university requirements, department environment, personal interest, and individual ability. The speaker will split entire teaching career into three stages beginning stage, mature period, and enjoy time. The speaker will provide one example for each time period respectively. (1) The speaker provides a paper that is published in the Journal of Number Theory and that is used of five pages to prove of two theorems while the original proof used thirteen pages. (2) The speaker published a paper in the ACM Transactions on Mathematical Software, the reviewer comments on the paper “this is a very good paper that I ever read in many years.” (3) The third example is a paper that is developed to present at an IEEE annual conference that is completed in one week. Hope all audiences will enjoy this talk.

  • 28 November 2017 (TUESDAY)
    Identifiability Issues of an Immuno-Epidemiological Model: The case of Rift Valley Fever Virus
    Maia Martcheva
    University of Florida

    We discuss the structural and practical identifiability of a nested immuno-epidemiological model of arbovirus diseases, where host–vector transmission rate, host recovery, and disease-induced death rates are governed by the within-host immune system. We incorporate the newest ideas and the most up-todate features of numerical methods to fit multi-scale models to multi-scale data. For an immunological model, we use Rift Valley Fever Virus (RVFV) time-series data obtained from livestock under laboratory experiments, and for an epidemiological model we incorporate a human compartment to the nested model and use the number of human RVFV cases reported by the CDC during the 2006–2007 Kenya outbreak. We show that the immunological model is not structurally identifiable for the measurements of time-series viremia concentrations in the host. Thus, we study the scaled version of the immunological model and prove that it is structurally globally identifiable. After fixing estimated parameter values for the immunological model derived from the scaled model, we develop a numerical method to fit observable RVFV epidemiological data to the nested model for the remaining parameter values of the multi-scale system. For the given (CDC) data set, Monte Carlo simulations indicate that only three parameters of the epidemiological model are practically identifiable when the immune model parameters are fixed. Alternatively, we fit the multi-scale data to the multi-scale model simultaneously. Monte Carlo simulations for the simultaneous fitting suggest that the parameters of the immunological model and the parameters of the immuno-epidemiological model are practically identifiable. (Research performed in collaboration with Necibe Tuncer, Hayriye Gulbudak, and Vincent Cannataro.)

Spring 2017 Schedule

  • 26 January 2017
    Nonstandard Finite Difference Methods for Dynamical Systems in Biology
    Hristo Kojouharov
    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
    Abdhi Sarkar
    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
    Abhishek Kaul
    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
    Yongli Sang
    Department of Mathematics
    The University of Mississippi
    University, 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
    Shibin Dai
    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
    Seth Oppenheimer
    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
    Moon-Jin Kang
    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
    Miles Wheeler
    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
    Jia Zhao
    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
    Wei Guo
    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
    David Goluskin
    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
    Hayriye Gulbudak
    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
    Michael Gurski
    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
    Mentor Stafa
    Indiana-Purdue University

    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
    Robin Koytcheff
    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
    Longfei Li
    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
    Qingwen Hu
    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.

Colloquia Archive