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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 or Calvin Berry with questions or suggestions about the colloquium series.

Our colloquia are normally held on Thursday at 3:30 p.m. in room 211 of Maxim Doucet Hall. Some colloquia may be presented via zoom instead. Refreshments are served at 3:15 in room 211. 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 2024

  • 10 October 2024 (ON ZOOM)
    (If you are not on our mailing list, please contact Calvin Berry to request the zoom link)
    Balázs Boros
    Andronov-Hopf bifurcations in mass-action systems
    Balázs Boros
    Department of Mathematics
    University of Wisconsin-Madison
    Abstract: Mass-action dynamical systems are probably the most common mathematical models in biochemistry, cell biology, and population dynamics. Further, they represent a large class of polynomial dynamical systems that are very important both theoretically and from the point of view of applications.
    Since oscillatory phenomena are ubiquitous in nature, developing a theory that ensures periodic solutions in mass-action differential equations is desirable. Probably the most powerful tool for showing the existence of an oscillation in a parameterized family of differential equations is the concept of bifurcations of equilibria, and in particular, Andronov-Hopf bifurcation.
    This talk presents the list of the smallest bimolecular mass-action systems that admit an Andronov-Hopf bifurcation. The analysis of these networks allows us to fully classify which networks admit a supercritical, a subcritical, or a degenerate Andronov-Hopf bifurcation, and thereby we get information about the stability of the emerging limit cycles. A deeper study of the degenerate cases led us to a classification of the smallest bimolecular mass-action systems that admit a Bautin bifurcation, a codimension-two bifurcation which produces two small limit cycles. In case the outer limit cycle is asymptotically stable, we obtain a bistable system, a property that is important in applications. We also comment on the computational aspects.
    Finally, we discuss how the investigation of small, simplified (and therefore unrealistic) models of reaction networks can contribute to the understanding of the behavior of larger, more realistic models. Namely, we give a brief introduction to the theory of inheritance of equilibria, periodic orbits, and bifurcations in mass-action systems.
  • 17 October 2024
    Deep Learning: A Mathematical Perspective with Applications
    Krishna Rauniyar
    Krishna Rauniyar
    Center for Advanced Computer Studies
    University of Louisiana at Lafayette
    Abstract:Deep learning has emerged as a very powerful subset of machine learning, revolutionizing image recognition, natural language processing, scientific discovery, and independent systems. Deep learning rests on mathematical principles enabling the automatic extraction of complex patterns in high-dimensional data. We will go through some basic mathematical ideas related to neural networks, from the simple to more advanced ideas around linear algebra, optimization, and function approximation.
    We will introduce neural networks as universal function approximators and discuss how optimization techniques, like gradient descent, enable the model to "learn" from data. Not going too deep into technical details, we are going to provide an intuitive understanding of how these algorithms work and simply refer to the Universal Approximation Theorem and the mathematical problems emerging during the training of deep learning models.
    These will also involve practical applications, such as image classification and natural language processing, demonstrating the use of these mathematical ideas through real-life success stories. We will then be able to indicate some open mathematical problems in deep learning and, through them, build a case for an interdisciplinary collaboration between mathematics and machine learning.
  • 31 October 2024
    Jin (Veronica) Liu
    New Perspectives on Modeling Nonlinear Growth Trajectories and Time-Varying Covariate Effects in Longitudinal Data
    Jin (Veronica) Liu
    Takeda Pharmaceuticals
    Boston, Massachusetts
    Abstract:Longitudinal data analysis is pivotal for understanding developmental processes and individual differences over time. Traditional models often fall short when dealing with nonlinear trajectories, unstructured measurement occasions, and the complex effects of time-varying covariates (TVCs). This presentation introduces innovative methodologies to address these challenges, drawing from three interconnected studies.
    First, we propose a novel specification for modeling nonlinear growth curves by conceptualizing change over time as the area under the curve (AUC) of the rate of change versus time graph. This approach accommodates unequally spaced and individual-specific measurement occasions. It allows for the simultaneous estimation of meaningful change-related parameters and other substantive parameters.
    Second, we tackle the limitations of traditional longitudinal analysis with a TVC by decomposing the impact of the TVC into baseline effects and temporal effects. By viewing the baseline value of the TVC as an initial trait influencing random growth coefficients, and characterizing temporal states through interval-specific slopes or changes, we provide a more nuanced understanding of how a TVC affects longitudinal outcomes. This decomposition enables the estimation of unbiased and precise effects, facilitating deeper insights into the joint development of related processes.
    We then extend these concepts to the finite mixture modeling framework to explore the heterogeneity of the TVC effects on the longitudinal outcome.
    Simulation studies and real-world data analyses demonstrate that these new methodologies yield unbiased and accurate estimates with target coverage probabilities, even in complex scenarios involving unstructured measurement occasions and nonlinear growth patterns. To facilitate the application of these models, code for the proposed methods and relevant extensions is available in the R package nlpsem.
  • 7 November 2024
    Yehenew Kifle
    Comparison of Local Powers of Exact Tests for a Common Normal Mean in Statistical Meta-Analysis
    Yehenew Kifle
    Department of Mathematics and Statistics
    University of Maryland Baltimore County
    Baltimore, Maryland
    Abstract: The inferential problem of making inferences about a common mean μ of several independent normal populations with unequal variances has drawn universal attention, and there are many exact and asymptotic tests for testing a null hypothesis H0 : μ = μ0 against two-sided alternatives. In this talk, I will provide a review of some of these exact and asymptotic tests and present theoretical expressions of local powers of the exact tests and a comparison. It turns out that in the case of equal sample sizes, a uniform comparison and ordering of the exact tests based on their local power can be carried out even when the variances are unknown.
  • 14 November 2024
    Jun Liu
    Parallel-in-time iterative methods for pricing American options
    Jun Liu
    Department of Mathematics and Statistics
    Southern Illinois University Edwardsville
    Edwardsville, Illinois
    Abstract: In finance, American options allow holders to exercise the option rights at any time before and including the day of expiration. For pricing such American options by PDE models, a sequence of linear complementarity problems (LCPs) is discretized in space and time. The resulting LCPs at each time step are solved through the policy iteration sequentially. In this talk, we propose to solve all the LCPs simultaneously by using the policy iteration for an “all-at-once” form of LCP. The designed parallel-in-time (PinT) implementation of policy iteration will be described in detail.
    Numerical examples confirm the effectiveness of our proposed methods.
  • 21 November 2024
    Greg Friedman
    Unitary equivalence of normal matrices over topological spaces
    Greg Friedman
    Department of Mathematics
    Texas Christian University
    Fort Worth, Texas
    Abstract: By the classical spectral theorem, normal complex matrices are unitarily equivalent to diagonal matrices, and so any two with the same collection of eigenvalues are unitarily equivalent to each other. However, this is not true in general for normal matrices with coefficients in other rings, in particular the ring of continuous functions C(X) over a topological space X (which we think of as bundles of complex matrices that vary continuously over the points of X). Such matrices can have quite interesting behavior, including a monodromy action that permutes eigenvalues and eigenspaces when traveling around a loop of a non-simply connected space. We show that, nonetheless, there are simple cohomological obstructions that determine unitary equivalence, at least for matrices with distinct eigenvalues at each point of a CW complex X, and these can determine the number of unitary equivalence classes for a given collection of eigenvalues in terms of the topology of X.
    This is joint work with Efton Park.

Spring 2024

  • 8 February 2024

    A quick introduction to Goedel's Theorem and its proof
    Arturo Magidin
    Department of Mathematics
    UL Lafayette
    Abstract: Goedel's Theorem is a famous result in formal logic from the first half of the 20th Century, and it led to Godel being dubbed "the most significant logician since Aristotle." It arose in the context of the Hilbert Programme and an ongoing philosophical argument among mathematicians about the foundations of mathematics.
    In this talk, we will talk about the context in which the theorem came about. We will describe the basic idea of the proof and exactly what it does (and what it does not) prove.
    (This is an expository talk; it will expand on a talk on this subject I gave at the UL Lafayette Undergraduate Seminar eight years ago. No formal background in logic will be required, and it should be accessible to all.)
  • 29 February 2024
    Nir Gadish
    Letter-braiding invariants of words in groups
    Nir Gadish
    Department of Mathematics
    University of Michigan
    Abstract: How can we tell if a group element is a k-fold nested commutator? I suggest seeking computable invariants of words in groups that detect k-fold commutators. We introduce the novel theory of letter-braiding invariants: these are elementarily defined functions on words, inspired by the homotopy theory of loop-spaces and carrying deep geometric content. They give a universal finite-type invariant for arbitrary groups, extending the influential Magnus expansion of free groups that already had countless applications in low dimensional topology. As a consequence we get new combinatorial formulas for braid and link invariants, and a way to linearize automorphisms of general groups that specializes to the Johnson homomorphism of mapping class groups.
  • 7 March 2024
    Thoa Thieu
    Stochastic modeling with applications in complex biological systems
    Thoa Thieu
    Department of Mathematics
    Indiana University
    Abstract: Stochastic models have been extensively used for the description of problems arising from biological systems and life sciences. In this talk, we discuss the applications of stochastic modeling across the domains of neuroscience and bio-molecular dynamics. In the first part, we study the coupled effects of channels and synaptic dynamics under the stochastic influence of healthy brain cells with applications to Parkinson's disease (PD). We also discuss the applications of a Wilson-Cowan-type system, modeling the dynamics of two interacting populations of excitatory and inhibitory neurons. In this work, we rigorously establish the well-posedness of the system modeling excitatory and inhibitory populations. As main working techniques, we use compactness methods and Skorokhod's representation of solutions to SDEs posed in bounded domains to prove the well-posedness of the system. In the second part, we introduce a stochastic model describing glucose-stimulated insulin secretion (GSIS) dynamics in pancreatic $\beta$ cells. Recognizing the critical regulatory role of the cell's microtubule (MT) cytoskeleton in insulin secretion, we construct a 3D computational model to explore how transport along the MT cytoskeleton influences insulin availability near the plasma membrane. Our study would potentially contribute to the development of an alternative therapeutic strategy for diabetes by targeting specific MT regulators.
  • 13 March 2024 (WEDNESDAY at 4:00)
    Thoa Thieu
    Nonlinear Asymptotic Stability of 3D Relativistic Vlasov-Poisson systems
    Jiaxin Jin
    Department of Mathematics
    The Ohio State University
    Abstract: Motivated by solar wind models in the low altitude, we explore a boundary problem of the nonlinear relativistic Vlasov-Poisson systems in the 3D half space in the presence of a constant vertical magnetic field and strong background gravity with the inflow boundary condition. As the main result, we construct stationary solutions and establish their nonlinear dynamical asymptotic stability.
  • 11 April 2024

    Multi-Scale Models of Infectious Disease Dynamics and Validating with Data
    Hayriye Gulbudak
    Department of Mathematics
    UL Lafayette
    Abstract:The bidirectional feedback induced through population and individual-level infectious disease and host immune dynamics requires development of innovative multi-scale models. In this talk, I will introduce structured nonlinear partial differential equation models linking immunology and epidemiology, along with novel stability analysis and computational tools for simulating ODE-PDE hybrid systems to understand the nonlinear dynamics and apply them to biological data. Applying the modeling framework to dengue virus, we first demonstrate how intermediate levels of antibodies enhance infection severity within a host, and scale up to population wide antibody level distributions evolving through multiple infections by distinct strains and waning immunity. Then, to test the theoretical results, we fit primary and secondary dengue infection data to provide evidence of antibody dependent enhancement. These results have critical implications for optimal vaccination policy, and the modeling framework is currently being applied to examine the emergence of COVID-19 variants.
  • 18 April 2024

    Transformer-Based Models for Time Series Forecasting
    Thu Nguyen
    Department of Mathematics and Statistics
    University of Maryland Baltimore County
    Abstract: Time series forecasting is essential for various real-world applications, typically relying on domain expertise and engineered features, which can be time-consuming and require extensive prior knowledge. Deep learning has emerged as a promising alternative, enabling data-driven approaches to capture temporal dynamics efficiently. This talk introduces a new class of Transformer-based models for time series forecasting. These models leverage attention mechanisms and incorporate ideas from classical time series methods to enhance their ability to learn complex patterns and dynamics. They are versatile and can handle both univariate and multivariate time series data effectively. Empirical evaluations demonstrate notable improvements over conventional benchmarks, showcasing the practical utility of the proposed models.
  • 25 April 2024

    Some multivariate equivalence problems
    Thomas Mathew
    Department of Mathematics and Statistics
    University of Maryland Baltimore County
    Abstract: In many applications involving hypothesis testing, the null hypothesis is often a statement of the equality of two parameters; the two sample t-test is perhaps the most widely known example. In some applications, it is more appropriate to test equivalence rather than equality; i.e., to test if the parameters are close according to a specified threshold. An important application that calls for equivalence testing is that of assessing the bioequivalence of a generic drug with a brand name drug, where the statement of equivalence is the alternative hypothesis, so that equivalence is concluded by rejecting the null hypothesis. In the talk I will discuss some multivariate equivalence testing problems. These will include testing the equivalence of mean vectors and that of covariance matrices. All of the problems will be motivated and illustrated using practical applications.

Fall 2023

  • 7 September 2023
    Local control of rationality properties for finite group representations
    Michael Geline
    Department of Mathematical Sciences
    Northern Illinois University
    Abstract: The ultimate goal of the modular representation theory of finite groups is to show that various properties of a group’s representations are “locally controlled.” Regarding basic things like the number of irreducible representations (in characteristic p), this remains a long way off. Rationality properties, by contrast, can be shown to “be local” with an application of the Green correspondence. This is what we shall discuss.
  • Postponed

    Deep Learning based Approaches for Time Series Forecasting
    Thu Nguyen
    Department of Mathematics and Statistics
    University of Maryland Baltimore County
  • 2 November 2023

    A Journey into Global Stability: From Monotone to Mixed Monotone and from autonomous to nonautonomous 
    Saber Elaydi
    Trinity University
    Abstract:The study of global stability of fixed and periodic points of monotone maps and triangular maps in one or higher-dimensional spaces has been successful. For general maps, the use of Liapunov functions has had limited success. Recently, Liapunov functions have been successful in obtaining global stability of the disease-free equilibrium of epidemic models. In this talk, we extend some of these results to mixed monotone maps. A special property of these maps is that they can be embedded in symmetric monotone maps in higher-dimension spaces. The aim here is to investigate the global stability of the interior fixed points of mixed monotone autonomous systems. The study is then extended to non-autonomous systems that are asymptotically autonomous, and to periodic difference equations. For the periodic systems, we show that a periodic cycle is globally asymptotically stable. These results are then applied to single and multi-species evolutionary competition models such as the Ricker model and the Leslie-Gower model with one trait or multi-traits

Spring 2023 Schedule

  • 10 January 2023 (Tuesday)
    Graph Neural Networks for Disease Classification and Feature Selection
    Tiantian Yang
    Department of Biostatistics
    Boston University
    Abstract: Omics data, such as genomics, transcriptomics, proteomics, and metabolomics, have crucial roles in discovering disease pathways and predicting clinical outcomes. However, the large p and small n issue is a great challenge for building predictive models for omics data analysis. In addition, the sparsity of the biological network and the unknown correlation structures between features also bring more challenges. With the recent emergence of Graph Neural Network (GNN), incorporating known functional relationships over a graph could potentially solve these issues. However, those GNN methods only utilize graphs in either an external feature graph or a generated feature graph alone, which will limit the classification performance. We proposed an external and generated Graph Neural Network (engGNN) model to aggregate the information of feature data in both external and generated feature graphs. The engGNN model utilized not only external knowledge to construct a feature graph but also eXtreme Gradient Boosting (XGBoost) to build a directed feature graph. We conducted extensive simulations and real data applications to validate the performance of the engGNN model. Compared with other baseline deep learning and machine learning models, engGNN has demonstrated a good classification performance. Moreover, by extending the connection weights method to rank the importance of features, engGNN can select important features with a corresponding meaningful interpretation for real biomedical data.
  • 17 January 2023 (Tuesday)
    Statistical Methodologies and Applications for Discrete Data in Climate and Microbiome Studies
    Mo Li
    Johns Hopkins University
    Baltimore, Maryland
    Abstract: There has been a remarkable resurgence in the need for novel statistical methodologies for analyzing discrete data, such as categorical divisions of the rainfall amount, and count in high-throughput sequencing studies. This talk covers two projects within this general framework. In the first project, we introduce a changepoint detection method for serially correlated categorical time series in climate studies. Climate time series usually contain sudden changes in their marginal distribution or correlation structure. A cumulative sum (CUSUM)-type test is devised to test for a single changepoint in a correlated categorical data sequence constructed from a latent Gaussian process through clipping techniques. A sequential parameter estimation method is proposed to estimate the parameters in this model. The method is applied to a real categorized rainfall time series from Albuquerque, New Mexico. In the second project, we focus on the high-dimensional microbiome count data. Bias is ubiquitous in microbiome sequencing studies and leads to invalid statistical conclusions even if the studies are well-designed. We will discuss the bias generation process through the experimental and analysis pipelines. We utilize and modify a log-linear compositional model to understand the bias generation procedure in microbiome sequencing through the analysis of data from a mock community (i.e., a lab-generated bacterial mixture in which the true relative abundances are all known) and real microbiome data from tobacco samples. Finally, a novel semi-parametric rank-based regression approach will be briefly discussed for the bias-resistant modeling task.
  • Monday 20 March 2023 (Monday 3:45 coffee, 4:00 talk)
    Yangwen Zhang
    An improved incremental SVD and its applications
    Yangwen Zhang
    Department of Mathematical Sciences
    Carnegie Mellon University
    Abstract: In 2002 an incremental singular value decomposition (SVD) was proposed by Brand to efficiently compute the SVD of a matrix. The algorithm needs to evaluate thousands or millions of orthogonal matrices and to multiply them together. Rounding error may destroy the orthogonality. Hence many reorthogonalization steps are needed in practice. In [Linear Algebra and its Applications 415 (2006) 20–30], Brand said: ``It is an open question how often this is necessary to guarantee a certain overall level of numerical precision.'' In this talk, we answer this question: by modifying the algorithm we can avoid computing the most of those orthogonal matrices and hence the reorthogonalizations are not necessary. We prove that the modification does not change the outcome of Brand's algorithm. We have successfully applied our improved scheme to snapshot-based POD model order reduction, time fractional PDEs and integro-differential equations and time dependent optimal control problems. Numerical analysis and experiments are presented to illustrate the impact of our modified incremental SVD on these important problems.
  • Wednesday 22 March 2023 (Wednesday 3:45 coffee, 4:00 talk)
    Zhaopeng Hao
    Efficient Numerical Methods for the Nonlocal Laplacian
    Zhaopeng Hao
    Department of Mathematics
    Purdue University
    Abstract: Due to its extraordinary modeling capabilities, the nonlocal fractional-order Laplacian has recently attracted increasing scientific and engineering attention. However, the nonlocal operators bring up new challenges in discretization and computation. To cope with difficulties, we discuss two efficient numerical methods, the spectral and currently popular deep neural-network methods. We present optimal error estimates for the spectral methods based on the sharp prior regularity estimates. For the neural network methods, we provide convergence analysis. Finally, we demonstrate numerical examples to show their efficiency and accuracy.
  • 23 March 2023

    The ADI, the LOD and the Operator Splitting - A New Western Story about Splitting Methods
    Qin (Tim) Sheng
    Department of Mathematics and
    Center for Astrophysics, Space Physics and Engineering Research
    Baylor University
    Waco, TX
    Abstract: Splitting methods have been used for solving a broad spectrum of mathematical problems. They are for the numerical solution to not only differential equations, but also statistical and optimization procedures. A splitting method separates the original mathematical model into several subproblems, separately computes the solution to each of them, and then combines all sub-solutions to form an approximation of the solution to the original problem. A canonical example is splitting of diffusion and convection terms in a convection-diffusion partial differential equation. The splitting idea generalizes in a natural way to problems with multiple operators too. In all cases, the computational advantage is that it is faster to compute the solution of the split components separately, than to compute the solution directly when they are treated together. However, this comes at the cost of an error introduced by the splitting, so strategies must be devised for controlling the error. This presentation introduces splitting methods via symbolic operations. It also surveys recent developments in the area. An interesting investigation is given in global error analysis of popular exponential splitting formulations. One recent development, adaptive splitting, deserves and receives special mention in this talk: it is a new splitting approach to collaborate with an otherwise parallel computational strategy.
    This talk is suitable for all graduate students. It may be accessible to undergraduate students who have studied differential and integral calculus.
    About the Speaker: Dr. Sheng received his PhD from the University of Cambridge under the supervision of Professor Arieh Iserles. After his postdoctoral research with Professor Frank T. Smith, FRS, in University College London, he joined National University of Singapore in 1990. Since then, Dr. Sheng was on the faculty of several major universities.  He joined Baylor University in 2005. He is associated with the Mathematics Department and the Center for Astrophysics, Space Physics and Engineering Research (CASPER).
    Dr. Sheng has been interested in splitting and adaptive numerical methods for solving linear and nonlinear partial differential equations. He is also known for the Sheng-Suzuki theorem in numerical analysis. He has published over 110 refereed journal articles as well several research monographs. His projects have been supported by the National Science Foundation, the Simons Foundation and the Department of Defense. He has been Editor-in-Chief of the SCI journal, International Journal of Computer Mathematics, published by Taylor and Francis, London, since 2010. Dr. Sheng currently advises two doctoral students.
  • 30 March 2023

    Discovering higher categories
    Martina Rovelli
    University Massachusetts Amherst
    Amherst, Massachusetts
    Abstract: Several algebraic structures of interest that we learn and study in math, such as groups or vector spaces, consist of endowing a set with one or more operations subject to axioms. There are many interesting phenomena in mathematical physics, algebraic geometry, and type theory that seem formalizable by one of such algebraic structures at first glance, although with a closer look the axioms are only met up to a "higher isomorphism". In order to accommodate those situations, one needs to acknowledge the presence of higher structures, and replace the role played by the usual equality relation with the relation induced by higher isomorphisms. In this talk we will describe a couple of different situations across math where the presence of higher structures is observable, discuss the complications that arise, and give an idea of the different approaches that can be taken to work with higher categories rather than ordinary categories.

Fall 2022 Schedule

  • 25 August 2022
    Bootstrap approximations to check binary regression models
    Gerhard Dikta
    FH Aachen, Campus Jülich
    Dept. Medical Engineering and Technomathematics
    Abstract: Suppose we observe a series of binary data along with explanatory variables and we suspect that these observations belong to a binary regression model.
    To test this model assumption, we use Kolmogorov-Smirnov and Cramér von Mises like test procedures based on a cumulative residual process introduced by Stute. The associated p-values are determined via a model-based bootstrap procedure. The theoretical results are based on a joint work with van Heel and Braekers, and are also presented more generally in a textbook together with Scheer. Bootstrap Methods With Applications in R by Gerhard Dikta and Marsel Scheer.
    In a final simulation study, we compare this approach with the Hosmer-Lemeshow test in a logistic setup. The implementation in R is done with the R package bootGof developed for the book in a joint work with Scheer.
  • 1 September 2022
    Contrasting Methods of Including Uncertainty: Philosophy and Practicalities
    R. Baker Kearfott
    University of Louisiana at Lafayette
    Abstract: We present a broad elementary overview of uncertainty, from the point of view of models and applications. In particular, we point out where and how uncertainty occurs and why its study is important. We then present the underlying principles behind three different paradigms for handling uncertainty. We give hypothetical examples of use of each method, then contrast their strengths and weaknesses. Technical details are covered in courses, both undergraduate and graduate. If there is sufficient interest, technical details may be discussed in future colloquia or seminars.
  • 8 September 2022
    Eco-evolutionary dynamics in prey-predator networks applied to HIV immune escape
    Cameron Browne
    University of Louisiana at Lafayette
    Abstract: Population dynamics and evolutionary genetics underly the structure of ecosystems. For example, during HIV infection, the virus escapes several immune response populations via resistance mutations at distinct epitopes (proteins coded in viral genome), precipitating a dynamic network of interacting virus and immune variants. I will talk about recent work to link virus population genetics with dynamics theoretically and through data to characterize their evolution. We analyze a resource-prey-predator differential equation network model to characterize the emergence of distinct stable equilibria and persistence of different diverse collections of virus and immune populations. Using binary sequences to code viral strain resistance to immune responses, we prove that bifurcations are determined and simplified by a certain evolutionary genetics measure of nonlinearity in the map from viral sequences to fitness (reproductive rate) landscape. The results generalize to decipher stability, structure and invasion of ecological networks based on the linear algebra of prey binary sequences encoding predation and fitness trade-offs. Finally, numerical simulations and an interdisciplinary application to virus-immune data illustrate our eco-evolutionary modeling framework.
  • 15 September 2022
    Regression Model under Skew-Normal Error with Applications in Predicting Groundwater Arsenic Level in the Mekong Delta Region
    Nabendu Pal
    University of Louisiana at Lafayette
    Abstract: Recently there has been some renewed interest in the skew-normal distribution (SND) because it provides a nice and natural generalization (in terms of accommodating skewed data) over the usual normal distribution. In this study we have used the SND error in a regression set-up, discussed a step by step approach on how to estimate all the model parameters, and show how naturally the resultant SND-based regression model can lead to a superior fitting to a given dataset. This generalization enhances the precision in predicting the future value of the response variable when the values of the independent (or input) variables are available. We validate the applicability of our proposed SND-based regression model by using a recently acquired dataset from the Mekong Delta Region (MDR) of Vietnam which had necessitated this study from a public health perspective. Using the existing survey data our proposed model allows all the stakeholders to better predict the groundwater arsenic level at a site easily, based on its geographic characteristics, in lieu of costly chemical analyses, which can be very beneficial to developing countries due to their resource constraints.
  • 22 September 2022
    The classification program for simple, nuclear, separable C*-algebras
    Leonel Robert
    University of Louisiana at Lafayette
    Abstract: I will describe the goals of the classification program described in the title, its history, and recent results arguably amounting to its complete resolution.
  • 29 September 2022
    Career Panel Discussion
    UL Lafayette AMS Graduate Student Chapter
    University of Louisiana at Lafayette
    Abstract: Panelists with varying levels of professional experience in industry will join us to share their career experiences, give advice on making and achieving career goals, and answer any questions you may have. The event is geared towards both undergraduate and graduate students.
    Please join us for this casual event - light refreshments will be provided, and the event will be followed by a social hour at Olde Tyme Grocery.
  • 6 October 2022
    No colloquium
    UL Lafayette Fall Break
  • 13 October 2022
    Singularity Formation in a Mean Field Neural Network
    Phillip Whitman
    University of Texas at Austin
    Abstract: Idealized networks of integrate-and-fire neurons with impulse-like interactions obey McKean-Vlasov diffusion equations in the mean-field limit. These equations are prone to singularities: for a strong enough interaction coupling, the mean-field rate of interaction diverges in finite time with a finite fraction of neurons spiking simultaneously, thereby marking a macroscopic synchronous event. Characterizing these blowup singularities analytically is the key to understanding the emergence and persistence of spiking synchrony in mean-field neural models. Here, we introduce a delayed Poissonian variation of the classical integrate-and-fire dynamics for which blowups are analytically well defined in the mean-field limit. Albeit fundamentally nonlinear, we show that this delayed Poissonian dynamics can be transformed into a noninteracting linear dynamics via a deterministic time change. This formulation also reveals that the fraction of simultaneously spiking neurons can be determined via a self-consistent, probability-conservation principle about the time-changed linear dynamics.
  • 20 October 2022
    Multi-Scale Structured Models of Infectious Disease Dynamics
    Hayriye Gulbudak
    University of Louisiana at Lafayette
    Abstract: A current challenge for disease modeling and public health is to understand pathogen dynamics across infection scales from within-host to between-host. Viral and immune response kinetics upon infection impact transmission to other hosts and feedback into population-wide immunity, all of which influence the severity, trajectory, and evolution of a spreading pathogen. In this talk, I will introduce structured partial differential equation models linking immunology and epidemiology in order to investigate coevolution of virus and host, multi-scale data fitting, and impacts of dynamic host immunity from an individual to the whole population. We apply the models to vector-borne diseases, such as Rift Valley fever (RVF) and dengue (DENV), with immunological and epidemiological data. Using invasion dynamics analysis and multi-scale numerical methods, we characterize different scenarios of virus-host evolution and coexistence of viral strains under waning host cross-immunity. In the case of DENV, we recapitulate how intermediate levels of pre-existent antibodies enhance infection within a host, and how to scale up to distributions of antibody levels among epidemiological classes in the host population to determine risk of severe DENV prevalence. These results have implications for optimal vaccination policy, and the modeling framework developed here is currently being applied to examine the emergence of COVID-19 variants partially resistant to antibodies induced by host infection or vaccination.
  • 27 October 2022
    Mathematical modeling of actin and myosin dynamics in the cell cortex during confined bleb-based migration
    Emmanuel Asante-Asamani
    Clarkson University
    Potsdam, New York
    Abstract: The actin cortex is very dynamic during cell migration, permitting the rapid formation of leading-edge protrusions that facilitate the forward motion of cells. When cells move in confined environments, they can use excess intracellular pressure to detach and expand small regions of their membrane, completely disassemble the cortex beneath the protruded membrane (actin scar) and form a new cortex at the expanded membrane. This mode of migration, known as blebbing, has been observed in metastatic cancer cells moving in highly confined tumor environments, and may be used to evade cancer drugs that target non-blebbing modes of motility. The mechanism by which cells completely disassemble their cortex during blebbing is not fully understood and could shed light on the dynamic regulation of the actin cortex during cell migration. Recent experimental data in Dictyostelium discoideum cells reveals a local accumulation of myosin II, an essential motor protein capable on inducing contractile stress, in the actin scar prior to any significant disassembly of the cortex. Thus, our data suggests a role for the motor protein in cortex disassembly. In this talk, I will discuss our efforts to use mathematical models to explain the accumulation of myosin in the cortex during blebbing. In particular, I will present a minimal data-driven linear dynamical system (ODEs) of actin-myosin interaction whose parameters show that the accumulation is a result of an increase in the relative binding rate of myosin to the cortex. Stability analysis of the model and exploration of its parameter space reveals that the accumulation is robust to perturbations in the model parameters. I will also introduce a partial differential equation coupled with a system of ODEs that model the intracellular signaling pathway regulating the binding of myosin to the cortex. Numerical simulations of the model suggests that the increase in myosin’s binding rate is induced by the separation of the cell membrane from the cortex which occurs at the onset of blebbing. Together, our experiments and theory elucidate the regulation of cortex disassembly and more broadly shed light on how cells can use mechanical events, such as membrane separation, to regulate internal cell process during confined migration.
  • 3 November 2022 (Special Time 3:45-4:30)
    Global dynamics of a cholera model with two nonlocal and delayed transmission mechanisms
    Xiang-Sheng Wang
    Abstract: A nonlocal and delayed cholera model with two transmission mechanisms in a spatially heterogeneous environment is derived. We introduce two basic reproduction numbers; one is for the bacterium in the environment and the other is for the cholera disease in the host population. If the basic reproduction number for the cholera bacterium in the environment is strictly less than one and the basic reproduction number of infection is no more than one, we prove globally asymptotically stability of the infection-free steady state. Otherwise, the infection will persist and there exists at least one endemic steady state. For the special homogeneous case, the endemic steady state is unique and globally asymptotically stable. Under some conditions, the basic reproduction number of infection is strictly decreasing with respect to the diffusion coefficients of cholera bacteria and infectious hosts. When these conditions are violated, numerical simulation suggests that spatial diffusion may not only spread the infection from the high-risk region to the low-risk region but also increase the infection level in the high-risk region.
  • 10 November 2022
    Algebraically, what can be said about the functions on a nonempty set S into itself; or do we need a left and a right distributive law?
    Gary Birkenmeirer
    Abstract: We discuss the evolution of algebraic systems motivated by the questions in the title of the talk.
  • 17 November 2022
    On Zoom. contact Calvin Berry to request the link.
    Direct Parallel in Time Solvers for Two Inverse PDE Problems
    Jun Liu
    Southern Illinois University Edwardsville
    Abstract: In this talk, I will briefly introduce the diagonalization-based parallel in time (PinT) algorithms, which show promising parallel efficiency for solving time-dependent differential equations (ODEs and PDEs). However, such PinT algorithms were not applied to inverse PDE problems in literature. Within the framework of quasi-boundary value regularization methods, we will present direct PinT solvers for solving two classical inverse PDE problems: Backward Heat Conduction Problem and Inverse Source Problem. The novel idea is to maneuver the flexibility of regularization methods for better structured linear systems that enable direct PinT solvers. The high efficiency of the proposed algorithms is illustrated by 1D and 2D numerical examples.
    More technical details can be found in the paper: Fast parallel-in-time quasi-boundary value methods for backward heat conduction problems and the arxiv preprint. For more PinT algorithms, I recommend the Parallel-in-Time website and also a report on related (iterative) ParaDIAG algorithms at arxiv.
  • 1 December 2022
    This is a continuation of the colloquium of 11 November 2022
    Algebraically, what can be said about the functions on a nonempty set S into itself; or do we need a left and a right distributive law?
    Gary Birkenmeirer
    Abstract: We discuss the evolution of algebraic systems motivated by the questions in the title of the talk.

Spring 2022 Schedule

  • 20 January 2022
    The Benson-Solomon fusion systems
    Justin Lynd
    UL Lafayette
    Abstract: A fusion system can be thought of as a "finite group at a prime p". After some historical remarks around the Classification of the Finite Simple Groups, I will explain what a p-fusion system is. Most of the examples of fusion systems we know of come from finite groups, but there do exist many exotic examples, chiefly at odd primes. In fact, we know of just one infinite family of simple exotic fusion systems at the prime 2, the Benson-Solomon fusion systems. In the second part of the talk, I will explain some coincidences that allow the Benson-Solomon systems to exist and, time permitting, discuss some of what is known about them.
  • 10 March 2022
    On Zoom. contact Leonel Robert to request the link
    Coarse geometry and rigidity
    Ilijas Farah
    York University
    Abstract: Coarse geometry is the study of metric spaces when one forgets about the small scale structure and focuses only on the large scale. For example, this philosophy underlies much of geometric group theory. To a coarse space one associates an algebra of operators on a Hilbert space, called the uniform Roe algebra. No familiarity with coarse geometry, operator algebras, or logic is required. After introducing the basics of coarse spaces and uniform Roe algebras, we will consider the following rigidity questions:
    (1) If the uniform Roe algebras associated to coarse spaces X and Y are isomorphic, when can we conclude that X and Y are coarsely equivalent?
    (2) The uniform Roe corona is obtained by modding out the compact operators. If the uniform Roe coronas of X and Y are isomorphic, what can we conclude about the relation between the underlying uniform Roe algebras (or about the relation between X and Y)?
    The answers to these questions are fairly surprising. This talk is based on a joint work with F. Baudier, B.M. Braga, A. Khukhro, A. Vignati, and R. Willett.
  • 17 March 2022
    On Zoom. contact Leonel Robert to request the link
    Acoustic and quantum scattering on the line
    Peter Gibson
    York University
    Abstract: The one-dimensional Schrödinger equation, with its connection to quantum scattering experiments, is among the most intensively studied models in quantum mechanics. This talk presents a new approach to scattering theory motivated by applications in acoustic imaging. The central idea is a natural approximation scheme that leads to new theoretical and computational methods that allow one to analyze forward and inverse scattering problems beyond the scope of previous theory. In addition, the talk will describe some newly discovered mathematical connections between scattering theory, orthogonal polynomials and a remarkable Riemannian manifold. The talk is intended for a general audience; no background in mathematical physics is required.
  • 28 April 2022
    On Zoom. contact Leonel Robert to request the link
    Infinitely many sign-changing solutions to a conformally invariant integral equation
    Mathew Gluck
    Towson University

Fall 2021 Schedule

We have a few speakers lined up already but still have room for more so let us know (contact Leonel Robert) if you want to speak or have a suggestion for a speaker. Check back for updates.

  • 14 October 2021
    On Zoom. contact Leonel Robert to request the link
    Geometries of topological groups
    Christian Rosendal
    University of Maryland
    Abstract: We will discuss how topological groups (of which Banach spaces are a particular example) come equipped with inherent geometries at both the large and small scale. In the context of Banach spaces, the ensuing study is part of geometric nonlinear analysis and we shall present various results and fundamental concepts dealing both with Banach spaces and more general topological groups appearing in analysis, dynamics and topology.
  • 15 October 2021 (1:00 pm Friday)
    On Zoom. contact Leonel Robert to request the link
    Bayesian jackknife empirical likelihood
    Yichuan Zhao
    Georgia State University
    Abstract: Empirical likelihood is a very powerful nonparametric tool that does not require any distributional assumptions. Lazar (2003) showed that in Bayesian inference, if one replaces the usual likelihood with the empirical likelihood, then posterior inference is still valid when the functional of interest is a smooth function of the posterior mean. However, it is not clear whether similar conclusions can be obtained for parameters defined in terms of U-statistics. We propose the so-called Bayesian jackknife empirical likelihood, which replaces the likelihood component with the jackknife empirical likelihood. We show, both theoretically and empirically, the validity of the proposed method as a general tool for Bayesian inference. Empirical analysis shows that the small-sample performance of the proposed method is better than its frequentist counterpart. Analysis of a case-control study for pancreatic cancer is used to illustrate the new approach.
  • 21 October 2021 (In person!)
    Almost finite almost groups
    Andrew Chermak
    Kansas State University
  • 4 November 2021 (In Person!)
    Accounting for Lack of Trust in Optional RRT Models
    Sat Gupta
    Professor and Head
    Department of Mathematics and Statistics
    University of North Carolina at Greensboro
    Abstract: When conducting face-to-face surveys containing sensitive questions, Social Desirability Bias (SDB) often leads to low response rate or worse, untruthful responding. Randomized Response Techniques (RRT) circumvent SDB by allowing respondents to provide scrambled responses. However, if respondents do not trust the RRT model, data accuracy will be compromised as shown and addressed in Young et al. (2019) and Lovig et al. (2021) for binary RRT models. Yet, no quantitative RRT model currently accounts for respondent lack of trust. We review the Young et al. (2019) and the Lovig et al. (2021) models, and propose an Optional Enhanced Trust (OET) Quantitative RRT model to mitigate respondent lack of trust by allowing additional noise to respondents who do not trust the basic Warner Additive Model. Using a combined measure of respondent privacy and model efficiency, we demonstrate both theoretically and empirically that the proposed OET model is superior to existing models.
    This talk is based on joint work with Maxwell Lovig (University of Louisiana at Lafayette), Joia Zhang (University of Washington), and Sadia Khalil (Lahore College for Women University)

Spring 2021 Schedule

  • 22 April 2021
    on Zoom at 3:30 pm -- social time at 3:00 pm contact Philip Hackney to request the link

    Informative censoring
    Gerhard Dikta
    FH Aachen University of Applied Sciences
     
    Based on an identifying Volterra-type integral equation for a lifetime distribution F and randomly right censored observations, we solve the corresponding estimating equation by an explicit and implicit Euler scheme. Depending on the assumptions we make about the conditional expectation of the censoring indicator given the observation time, we derive the well know Kaplan-Meier and other established estimators of F under the explicit Euler scheme. Moreover, under the implicit Euler scheme, we obtain new pre-smoothed and semi-parametric estimators of F. Some properties of the new semi-parametric estimator are discussed and a real data application finalizes the presentation.
  • 15 April 2021
    on Zoom at 3:30 pm -- social time at 3:00 pm contact Philip Hackney to request the link

    Showcase on Undergraduate Research

    This week we have four presentations by some of our undergraduate mathematics majors
     
    A Study of Natural Predator's Effectiveness at Controlling a Tick Population
    Maxwell Lovig
    (Faculty Advisors: Amy Veprauskas and Ross Chiquet)
     
    Harvesting Strategies to Control Invasive Species
    Madeleine Angerdina and Ian Bonin
    (Faculty Advisors: Amy Veprauskas and Ross Chiquet)
     
    Party Investment and Citizens' Willingness to Vote
    Maxwell Reigner Kane
    (Faculty Advisor: Ross Chiquet)
     
    Realizing Finite Groups as Internal Automorphism Groups
    Andrew Bayard
    (Faculty Advisor: Justin Lynd)
  • 8 April 2021
    on Zoom at 3:30 pm -- social time at 3:00 pm contact Philip Hackney to request the link

    Topology and Azumaya Algebras
    Ben Williams
    University of British Columbia
     
    The algebra Mat_d(C) of d x d matrices over the complex numbers is a familiar one. A topological Azumaya algebra of degree d is a bundle of algebras, each isomorphic to the matrix algebra, i.e., it is a twisted family of matrix algebras. I will use the algebra structure of Mat_d(C) to produce a universal example of a topological Azumaya algebra: informally, an example that is as twisted as possible.
    Classical Azumaya algebras are also twisted forms of matrix algebras, but here the twisting is purely algebraic rather than topological. They generalize central simple algebras over fields: a venerable area of study.
    Topological Azumaya algebras are intimately related to “classical" Azumaya algebras. I will attempt to explain this, and show how you can use homotopy theory to produce examples of Azumaya algebras that behave very differently from central simple algebras. This talk is intended to be accessible to a general mathematical audience.
  • 25 March 2021
    on Zoom at 3:30 pm -- social time at 3:00 pm contact Philip Hackney to request the link

    The hot spots conjecture can be false: Some numerical examples using boundary integral equations
    Andreas_Kleefeld
    Jülich Supercomputing Centre
    Institute for Advanced Simulation
     
    The hot spots conjecture is only known to be true for special geometries. It can be shown numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.
  • 25 February 2021
    on Zoom at 3:30 pm -- social time at 3:00 pm contact Philip Hackney to request the link

    Algebraic K-theory for 2-categories
    Niles Johnson
    Ohio State University Newark
     
    Quillen recognized the higher algebraic K-groups of a commutative ring R as homotopy groups of a certain topological space, BGL(R)^+. We review some of the basic definitions and computations via categorical algebra. We then describe how a 2-categorical extension of this theory leads to a new model for K_3(R), together with more general applications. We will give a mild sampling of key technical details and close with some of the problems we're currently working on. The work we present is joint with Gurski-Osorno, Fontes, and Fontes-Gurski.

Fall 2020 Schedule

  • 15 October 2020
    Classification of purely infinite C*-algebras
    James Gabe
    University of Southern Denmark
     
    I will talk about the basics of the Elliott programme for classifying amenable (aka nuclear) C*-algebras. I will focus on the case where my C*-algebras are purely infinite.
  • 12 November 2020
    on Zoom at 3:30 pm -- social time at 3:00 pm contact Philip Hackney to request the link

    Transfer systems and weak factorization systems
    Angélica Osorno
    Reed College
     
    N-infinity operads over a group G encode homotopy commutative operations together with a class of equivariant transfer (or norm) maps. Their homotopy theory is given by transfer systems, which are certain discrete objects that have a rich combinatorial structure defined in terms of the subgroup lattice of G. In this talk, we will show that when G is finite Abelian, transfer systems are in bijection with weak factorization systems on the poset category of subgroups of G. This leads to an involution on the lattice of transfer systems, generalizing the work of Balchin-Bearup-Pech-Roitzheim for cyclic groups of squarefree order. This is joint work with Evan Franchere, Usman Hafeez, Peter Marcus, Kyle Ormsby, Weihang Qin, and Riley Waugh.

Spring 2020 Schedule

  • 13 February 2020
    Some applications of topology to the analysis of data
    Jose Perea
    Michigan State
     
    Many problems in modern data science can be phrased as topological questions: e.g., clustering is akin to finding the connected components of a space, and tasks such as regression, classification and dimensionality reduction can be thought of as learning maps between structured spaces. I will describe in this talk how tools from classical algebraic topology can be leveraged for the analysis of complex data sets. Several illustrative examples will be provided, including applications to computer vision, machine learning and computational biology.
  • 20 February 2020
    Equivalence relations and algebras of generalized matrices
    Hung-Chang Liao
    University of Ottawa
     
    To an equivalence relation one can naturally associate a collection of  "generalized matrices". These matrices typically have infinitely many rows and columns, but otherwise form an algebra just like ordinary matrices. Since the fundamental work of F. Murray and J. von Neumann, the study of these algebras has evolved into a huge mathematical entity with applications to many other fields. A natural and important question is how these algebras interact with the original equivalence relations. We will give an introduction to some of the most important work regarding this question, with a focus on equivalence relations arising from group actions. The talk does not assume any background beyond (the ordinary) matrix algebras, a (tiny) bit of topology, and a (tiny) bit of measure theory.
  • 12 March 2020
    The Wald Method versus The Score Method
    Jie Peng
    St Ambrose University
     
    The Wald method and the score method are well-known classical large sample methods of obtaining inference on parametric distributions. They are commonly used in hypothesis testing and interval estimation. In this talk, we describe these methods for finding confidence intervals and prediction intervals for the binomial, hypergeometric and Poisson distributions. For each distribution, we show via numerical comparison that the score method is better than the Wald method. The score method also provides satisfactory results even for somewhat small samples. Furthermore, We illustrate the construction of confidence intervals and prediction intervals using some practical examples.

Colloquia Archive