2024-25 Colloquia Talks
| Date | Speaker | Talk |
|---|---|---|
| Thursday, December 4, 2025, at 3:30 p.m. in MSPB 370 | Morgan Taylor, University of South Alabama | Title: Enumerative Chromatic-Choosability Abstract: Graph coloring was introduced in the 1850s with the famous Four Color Problem about coloring maps in such a way that any two regions sharing a border receive different colors. In this project, we study a variant of graph coloring called list coloring which was introduced in the 1970s. One notion in list coloring that has received significant attention is chromatic-choosability. A graph G is chromatic-choosable when its list chromatic number, χl(G), is equal to its chromatic number χ(G). In 1990, Kostochka and Sidorenko introduced the list color function of a graph G, denoted by Pl(G, m). The list color function of G is the list coloring analogue of P(G, m) where P(G, m) is the chromatic polynomial of G. A graph is said to be enumeratively chromatic-choosable if P(G, m) = Pl(G, m) whenever m ≥ χ(G). Additionally, we call a graph weakly enumeratively chromatic-choosable when Pl(G, χ(G)) = P(G, χ(G)). We prove that certain complete multipartite graphs are weakly enumeratively chromatic-choosable. This is joint work with Sarah Allred, Andy Fritz, Kaylee Hudleston, Aiden McCain, and Jeffrey A. Mudrock. This talk will be preceded by the Pi Mu Epsilon Honor Society Induction Ceremony. |
| Thursday, November 20, 2025, at 3:30 p.m. in MSPB 370 | Malavika Mukundan, Boston University | Title: Linking Topology, Geometry and Dynamics: The Story of Thurston Theory Abstract: A Thurston map is the topological analog of a post-singularly finite (PSF) meromorphic function of the Riemann sphere. Such maps were introduced by William Thurston as a tool for characterizing PSF functions, which are vital in the study of iteration of meromorphic functions. This theory provides a powerful tool linking topology, geometry and dynamics. However, several questions remain in Thurston's program, the most important among them a conjecture on when a Thurston map is 'equivalent' in a certain sense to a meromorphic one. In this talk we will give a brief introduction to this theory, and discuss emerging trends in the study of transcendental Thurston maps, including recent results in the field. This talk is part of a mini series featuring female early-career mathematicians supported by an MAA Tensor Women and Mathematics Grant awarded to Drs. Furno and E. Pavelescu. |
| Thursday, November 13, 2025, at 3:30 p.m. in MSPB 370 | Emily McMillon, Rice University | Title: Graph Homomorphisms from Trees Abstract: In this talk, we consider graph homomorphisms from trees into other graphs, with a particular emphasis on counting the number of homomorphisms from a tree into a given graph. We will visit a conjecture on the number of homomorphisms from a tree into a fixed graph - that the star graph is the unique maximizer and the path the unique minimizer among trees - and provide a specific counterexample with the property that for each sufficiently large n, there is a tree on n vertices that admits strictly fewer graph homomorphisms than the path on n vertices. This is based on joint work with David Galvin, JD Nir, and Amanda Redlich. This talk is part of a mini series featuring female early-career mathematicians supported by an MAA Tensor Women and Mathematics Grant awarded to Drs. Furno and E. Pavelescu. |
| Thursday, November 6, 2025, at 3:30 p.m. in MSPB 370 | Karlee Westrem, Appalachian State University | Title: New identities in the character table of symmetric groups Abstract: Tewodros Amdeberhan recently proposed certain equalities between sums in the character table of symmetric groups. These equalities are between signed column sums in the character table, summing over the rows labeled by partitions in Ev(lambda), a multiset containing 2^r partitions of size 2n. While we observe that these equalities are not true in general, we prove that they do hold in interesting special cases. In this talk, we will share these new equalities between sums of degrees of irreducible characters for the symmetric group and a new combinatorial interpretation for the Riordan numbers in terms of degrees of irreducible characters labeled by partitions with three parts of the same parity. This talk is from joint work with David Hemmer (Michigan Technological University) and Armin Straub (University of South Alabama). This talk is part of a mini series featuring female early-career mathematicians supported by an MAA Tensor Women and Mathematics Grant awarded to Drs. Furno and E. Pavelescu. |
| Thursday, October 23, 2025, at 3:30 p.m. in MSPB 370 | Kajal Lahiri, SUNY Albany | Title: ROC and PRC Approaches to Evaluate Rare Event Forecasts Abstract: We have studied the relationship between Receiver Operating Characteristics (ROC) and Precision-Recall Curve (PRC) both analytically and using a real-life empirical example of yield spread as a predictor of recession. We show that false alarm rate in ROC and inverted precision in PRC are analogous concepts, and their difference is determined by the interaction of sample imbalance and forecast bias. We found that in cases of severe class imbalance, the forecasts need to be adequately biased to mitigate the effect of sample imbalance. |
| Wednesday, October 22, 2025 at 6:00 p.m. in MSPB 140This talk is aimed at a general audience! | Kajal Lahiri, SUNY Albany | ---10th Mishra Memorial Lecture---Title: How good were the COVID-19 forecasters in America ?Abstract: Using data from the Centers for Disease Control and Prevention’s COVID-19 Forecast Hub spanning June 13, 2020, to May 2022, we examine the accuracy of weekly fatality forecasts across four consecutive horizons produced by nearly 50 leading research teams in the U.S. Our analysis evaluates how effectively these forecasts informed public health decision-making aimed at controlling and eliminating virus transmission. Although ensemble forecasts— which aggregate predictions from multiple teams—demonstrate the highest overall accuracy, most forecasts, including the ensemble, fail to meet key economic criteria of unbiasedness and efficiency in incorporating new information. We explore the underlying causes of these shortcomings and discuss potential pathways for improving infectious disease forecasting in the future. This event is sponsored by the Honor Society of Phi Kappa Phi. |
| Tuesday (!), September 16, 2025, at 3:30 p.m. in MSPB 360 (!) | Tien Chih, Oxford College of Emory University | Title: Team Based Inquiry Learning: an introduction Abstract: In this talk, we introduce `Team Based Inquiry Learning’, a pedagogical technique to introduce IBL into courses where traditional IBL can be challenging to implement: freshman and sophomore courses with mostly non-majors. We introduce the central tenants and philosophy of TBIL, how it runs in practice, and its impact on my teaching and classroom. |
| Thursday, September 11, 2025, at 3:30 p.m. in MSPB 370 | Neil Saunders, City University of London | Title: Maths, Memes & AI: An Exploration through Performance Abstract: Artificial Intelligence (AI) particularly generative AI, has exploded onto the scene reaching almost all parts of our lives - from our working lives, to our leisure time and everything in between. Many AI proponents say that it promises to revolutionise our education system and help us explore new frontiers of scientific research. Some others disagree, claiming that progress in AI is slowing, even suggesting that the AI market is in a bubble. With all of these competing views, we need to again ask the question: what does it really mean for an AI or computer to be genuinely intelligent? Does it even make sense to talk about computers in these terms? When we start to try and answer these questions, we are drawn to more fundamental questions around our own human capacities for intelligence and understanding. This talk will explore ideas of human and artificial intelligence from a range of viewpoints: philosophical, mathematical and artistic. Drawing on recent and historical work in the field, and with selected readings from David Harrower’s powerful play Knives in Hens, it will examine how language is central to intelligence and understanding, whether that be human or artificial. |
| Thursday, September 4, 2025, at 3:30 p.m. in MSPB 370 | Sarah Allred, University of South Alabama | Title: Enumerative Chromatic-Choosability Abstract: Counting proper (classical) colorings of graphs is a fundamental topic in enumerative combinatorics that has been extensively studied since the early 20th century. The chromatic polynomial of a graph G, denoted P(G, m), is equal to the number of proper m-colorings of G. List coloring is a well-studied generalization of classical coloring that was introduced in the 1970s. A graph G is chromatic-choosable when its list chromatic number χl(G) is equal to its chromatic number χ(G). Chromatic-choosability is a well-studied topic, and in fact, some of the most famous results and conjectures related to list coloring involve chromatic-choosability. In 1990, Kostochka and Sidorenko introduced the list color function of a graph G, denoted Pl(G, m). The list color function of G is the list analogue of P(G, m). A graph is said to be enumeratively chromatic-choosable if P(G, m) = Pl(G, m) whenever m ≥ χ(G). In this talk, I will present some results and open questions on enumerative chromatic-choosability. In particular, I give a characterization of enumeratively 2-chromatic-choosable graphs and explore the effect that joining a complete graph to an arbitrary graph has on enumerative chromatic-choosability. This is joint work with Jeff Mudrock, Morgan Taylor, Andy Fritz, and Aiden McCain. |
| Thursday, April 24, 2025, at 3:30 p.m. in MSPB 370 | Vasiliy Prokhorov, University of South Alabama | Title: On Rational Approximation of Analytic Functions Abstract: We discuss fundamental concepts in the rational approximation of analytic functions, including uniform approximation on compact subsets of the complex plane, best uniform rational approximation of order (n, m), the asymptotic behavior of the approximation error ρn,m, and its connection to the analytic continuation of the approximated function. Undergraduate and graduate students are encouraged to attend this talk. |
| Thursday, April 10, 2025, at 3:30 p.m. in MSPB 370 | Jörg Feldvoss, University of South Alabama | Title: Semi-Simple Leibniz Algebras Abstract: Leibniz algebras were introduced by Blo(c)h and Loday as non-anticommutative analogues of Lie algebras. Many results for Lie algebras have been proven to hold for Leibniz algebras, but there are also several results that are not true in this more general context. In my talk I will describe the structure of finite-dimensional (semi-)simple Leibniz algebras over a field of characteristic zero. They are hemi-semidirect products of a (semi-)simple Lie algebra and an irreducible (resp. completely reducible) module over this Lie algebra (without non-zero trivial submodules). The main result is an simplicity criterion for hemi-semidirect products that works for not necessarily finite-dimensional Lie algebras over arbitrary fields. If time permits, I will derive some applications, one of which is on the derivation algebras of finite-dimensional semi-simple Leibniz algebras over a field of characteristic zero that generalizes a classical result for Lie algebras obtained by Gerhard Hochschild in his Ph.D. thesis. I will explain everything from scratch so that some knowledge of linear algebra and basic abstract algebra should be sufficient to be able to follow my talk. |
| Thursday, March 27, 2025, at 3:30 p.m. in MSPB 370 | David Hemmer, Michigan Tech | Title: Some new identities in the character tables of symmetric groups Abstract: We will give an introduction to representation theory, in particular for finite symmetric groups. We also introduce the closely related area of symmetric polynomials. The irreducible characters of the symmetric group permuting n letters are labelled by partitions of n. For a partition of n, Tewadros recently defined an unusual set of partitions of 2n, and gave some conjectural identities in the character table of symmetric groups related to this set. We prove part of this conjecture. This leads to some fascinating combinatorics related to Motzkin paths, Riordan numbers and standard Young tableaux. This is joint work with Karlee Westrem. |
| Thursday, March 20, 2025, at 3:30 p.m. in MSPB 370 | Mathias Muia, University of South Alabama | Title: Kernel Smoothing for Bounded Copula Densities Abstract: Non-parametric estimation of copula density functions presents significant challenges. One issue is the unboundedness of certain copula density functions and their derivatives at the corners of the unit square. Another is the boundary bias inherent in kernel density estimation. This talk presents a kernel-based method for estimating bounded copula density functions, addressing boundary bias through the mirror reflection technique. Optimal smoothing parameters are derived via Asymptotic Mean Integrated Squared Error (AMISE) minimization and cross-validation, with theoretical guarantees of consistency and asymptotic normality. Two kernel smoothing strategies are proposed: the rule of-thumb approach and least squares cross-validation (LSCV). Simulation studies highlight the efficacy of the rule-of thumb method in bandwidth selection for copulas with unbounded marginal supports. The methodology is further validated through an application to the Wisconsin Breast Cancer Diagnostic Dataset (WBCDD), where LSCV is used for bandwidth selection. |
| Thursday, March 13, 2025, at 3:30 p.m. in MSPB 370 | Kevin Grace, University of South Alabama | Title: The Many Faces of Matroids Abstract: The concept of a matroid was independently introduced in the 1930s by Hassler Whitney and Takeo Nakasawa to unify common ideas of dependence in linear algebra and graph theory. However, matroid theory has connections to some concepts that have been studied since ancient times. One of these concepts is duality of polyhedra (including the Platonic solids). This is generalized by matroid duality, which also generalizes duality of planar graphs and orthogonality of vector spaces. This talk will be an introduction to some aspects of matroid theory, with an emphasis on matroid duality. |
| Thursday, February 27, 2025, at 3:30 p.m. in MSPB 370 | Andrei Pavelescu, University of South Alabama | Title: A survey of famous open questions in graph theory Abstract: According to Bollobas, Caitlin, and Erdös, the Hadwiger Conjecture is "one of the deepest unsolved problems in graph theory”. Made by Hugo Hadwiger in 1943, the conjecture states that the chromatic number of a loopless graph G which has no complete minor on t vertices satisfies the inequality χ(G) < t. In this talk we explore the connections to the Four Color Theorem, provide an overview on the progress towards obtaining a proof, and we discuss connected open problems/conjectures. This talk should be accessible to students and panicky non-tenured faculty. |
| Thursday, February 6, 2025, at 3:30 p.m. in MSPB 370 | Chase Holcombe, University of South Alabama | Title: The Building Blocks of Classical Nonparametric Two-Sample Testing Procedures:
Statistically Equivalent Blocks Abstract: Statistically equivalent blocks are not frequently considered in the context of nonparametric two-sample hypothesis testing. Despite the limited exposure, I hope to show that a number of classical nonparametric hypothesis tests can be derived on the basis of statistically equivalent blocks and their frequencies. Far from a moot historical point, this allows for a more unified approach in considering the many two-sample nonparametric tests based on ranks, signs, placements, order statistics, and runs. Perhaps more importantly, this approach also allows for the easy extension of many univariate nonparametric tests into arbitrarily high dimensions that retain all null properties regardless of dimensionality and are invariant to a wide number of transformations. These generalizations do not require depth functions or the explicit use of spatial signs or ranks and may be of use in various areas such as life-testing and quality control. Here, an overview of statistically equivalent blocks and tests based on these blocks are provided. This is followed by reformulations of some popular univariate tests and generalizations to higher dimensions. Comments comparing proposed methods to those based on spatial signs and ranks are offered along with some conclusions. |
| Thursday, January 30, 2025, at 3:30 p.m. in MSPB 370 | Jeff Mudrock, University of South Alabama | Title: Introducing… The Coloring Research Group of South Alabama Abstract: In January 2024, Haley Broadus and Gabriel Sharbel, both South Alabama undergraduate students, established the Coloring Research Group of South Alabama (CRGSA). The group aims to cultivate a diverse and vibrant research community among South Alabama students, dedicated to producing original contributions in the field of graph coloring and broadening participation in math research. Graph coloring was introduced in the 1850s with the famous Four Color Problem about coloring the regions of a map in such a way that any two regions sharing a border receive different colors. Over the past 170 years, the study of graph coloring has led to the development and discovery of beautiful and useful mathematics. Applications of graph coloring can be found in computer science, scheduling, problems on social networks, and the equitable distribution of resources. In this talk, marking the group’s one-year anniversary, I will share a brief history of the CRGSA and highlight its accomplishments. Specifically, I will discuss two recently completed research projects focused on generalizations of graph coloring: DP coloring and flexible list coloring. This is joint work with Timothy Bennett, Michael Bowdoin, Haley Broadus, Daniel Hodgins, Adam Nusair, Gabriel Sharbel, and Josh Silverman. |
| Thursday, November 7, 2024, at 3:30 p.m. in MSPB 370 | Aparna Upadhyay, University of South Alabama | Title: Combinatorial representation theory Abstract: Do you know that matrices can really be revolutionary in simplifying abstract mathematical concepts? Representation theory is an area of mathematics that helps us translate problems from the world of abstract algebra to the world of matrices. In combinatorial representation theory, combinatorial objects are used to model these representations. In particular, certain combinatorial objects provide a very elegant description of representations of symmetric groups. In this case, the interplay between the algebra and the combinatorics is refined enough to provide combinatorial answers to fundamental questions in representation theory of symmetric groups. We will describe these combinatorial objects and see how they can be used to derive important information about representations of symmetric groups. |
| Thursday, October 24, 2024, at 3:30 p.m. in MSPB 370 | Dan Silver, University of South Alabama | Title: The Four Color Theorem and the Penrose Polynomial Abstract: The Four Color Theorem states that no more than four colors are required to color the regions of any map so that no two contiguous regions receive the same color. The theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken with the aid of a computer. A non-electronic proof remains elusive. In this joint work with Susan Williams, we give a new description of a polynomial implicit in a 1969 paper by Roger Penrose. A complete understanding of the polynomial would result in a proof of the Four Color Theorem. |
| Thursday, October 17, 2024, at 3:30 p.m. in HUMB 170 | William H. Woodall, Virginia Tech | Title: Monitoring and Improving Surgical Quality Abstract: Some statistical issues related to the monitoring of surgical outcome quality will be reviewed in this presentation. The important role of risk-adjustment in healthcare, used to account for variations in the condition of patients, will be described. Some of the methods for monitoring quality over time will be outlined and illustrated with examples. The National Surgical Quality Improvement Program (NSQIP) will be described, along with a case study demonstrating significant improvements in surgical infection rates and mortality. |
| Wednesday, October 16, 2024 at 6:00 p.m. in MSPB 140This talk is aimed at a general audience! | William H. Woodall, Virginia Tech | ---9th Mishra Memorial Lecture---Title: The Systematic Manipulation of the Scientific Publication Process Abstract: I became aware of the extent of the attacks on the scientific publication process though Taylor & Francis, the publisher of Quality Engineering for which I am currently editor-elect. All areas of science are being affected, from mathematics to medicine. The following are some of the primary concerns: authorships being sold on fake papers produced by paper mills, sham reviews, unethical behavior of guest editors for special issues, bribes being offered to editors, the rise of predatory journals and conferences, citation cartels, plagiarism, and the fabrication of data and images. These and other issues, illustrated with numerous examples, will be discussed in this presentation. A related issue is the proliferation of junk science. Efforts to protect the scientific literature will be discussed, such as the contributions by individual sleuths and the use of the STM Integrity Hub that was established by the major academic publishers. There will also be some discussion of the article retraction process and the effect of AI on publishing. Over ten thousand scientific papers were retracted in 2023, a record number. This event is sponsored by the Honor Societies of Sigma Xi and Phi Kappa Phi. |
| Thursday, October 3, 2024, at 3:30 p.m. in MSPB 360 | Steven Clontz, University of South Alabama | Title: Modeling and Verifying Mathematics by Computer Abstract: While for many years the role of computers in mathematics has been limited to brute-force computation, the age of computer-verified mathematical arguments is fast approaching, with mathematicians such as Kevin Buzzard ("What is the point of computers? A question for pure mathematicians") and Terence Tao ("Machine Assisted Proof") prognosticating how the act of mathematics will change in the near, not distant, future. But when exactly will the rank and file theoretical mathematician need more than their chalkboard and an outdated installation of LaTeX to share their research with the world? What are these new software tools exactly? And how might we start using them today?This talk is appropriate for undergraduate and graduate students. |
| Thursday, September 26, 2024, at 3:30 p.m. in MSPB 370 | Arik Wilbert, University of South Alabama | Title: Categorification without Categories - On the book that I will probably never
finish... Abstract: In this talk, I will tell you about a book I am trying to write. The book is supposed to contain fancy pictures of mathematical knots. To create a knot, take a piece of string or rope, tie a knot in it, and then glue the ends together. So far, I have had a hard time deciding what knots I would like to feature in my book. I will introduce you to the Jones polynomial and explain how it has been a helpful tool in making decisions. However, at this point, I still do not know how to finish the book. In graduate school, I learned about a mathematical field called categorification. Categorification is an active area of research. Maybe categorification, or one of you guys, can help me finish my book?This talk is geared towards undergraduate and graduate students. |
| Thursday, September 5, 2024, at 3:30 p.m. in MSPB 370 | Bach Nguyen, Xavier University of Louisiana | Title: On the upper nilpotent completion problem for matrices Abstract: The matrix completion problem is a well known problem with exciting applications in various areas such as matrix theory, signal processing, probability theory, and even representation theory. The matrix completion problem for upper nilpotent matrix was stated by Rodman and Shalom and later solved by Krupnik and Leibman. However, the work of Krupnik and Leibman didn't provide, in practice, an effective or numerically stable way to construct the desired upper nilpotent matrix. In this talk, we present a numerically stable and highly efficient algorithm which constructs explicit (binary) solutions to the upper nilpotent completion problem. Moreover, our algorithm also provides an explicit solution for the Deligne-Simpson problem for Coxeter connections on the Riemann sphere. This is a joint work with Neal Livesay and Dan Sage. |
| Thursday, April 11 2024, at 3:30 p.m. in MSPB 370 | Vasiliy Prokhorov, University of South Alabama | Title: Generating Series of Lattice Paths, Resolvents of Difference Operators, Random
Polynomials, and Matrix Continued Fractions Abstract: In this presentation, we investigate the generating series of weighted lattice paths, with a particular focus on the sets denoted by D[n,i,j], where i and j are nonnegative integers. Each path within D[n,i,j] is composed of n steps, allowing for movement upwards (up to q units), rightwards (1 unit), or downwards (up to p units), starting at (0, i) and ending at (n, j). Paths in D[n,i,j] never go below the x-axis. Our main result establishes a matrix continued fraction representation for a matrix constructed from generating series associated with the sets of lattice paths D[n,i,j]. This result extends the notable contributions previously made by P. Flajolet and G. Viennot in the scalar case p = q = 1. The generating series can also be interpreted as resolvents of difference operators of finite order. Additionally, we analyze a class of random banded matrices H, which have p + q + 1 diagonals with entries that are independent and bounded random variables. These random variables have identical distributions along diagonals. We investigate the asymptotic behavior of the expected values of eigenvalue moments for the principal n × n truncation of H as n tends to infinity. |
| Thursday, April 4 2024, at 3:30 p.m. in MSPB 370 | Scott Baldridge, Louisiana State University | Title: Using quantum states to understand the four-color theorem Abstract: The four-color theorem states that a bridgeless plane graph is four-colorable, that is, its faces can be colored with four colors so that no two adjacent faces share the same color. This was a notoriously difficult theorem that took over a century to prove. In this talk, we generate vector spaces from certain diagrams of a graph with a map between them and show that the dimension of the kernel of this map is equal to the number of ways to four-color the graph. We then generalize this calculation to a homology theory and in doing so make an interesting discovery: the four-color theorem is really about all of the smooth closed surfaces a graph embeds into and the relationships between those surfaces. The homology theory is based upon a topological quantum field theory. The diagrams generated from the graph represent the possible quantum states of the graph and the homology is, in some sense, the vacuum expectation value of this system. It gets wonderfully complicated from this point on, but we will suppress this aspect from the talk and instead show a fun application of how to link the Euler characteristic of the homology to the famous Penrose polynomial of a plane graph. This talk will be hands-on and the ideas will be explained through the calculation of easy examples! My goal is to attract students and mathematicians to this area by making the ideas as intuitive as possible. Topologists and representation theorists are encouraged to attend also—these homologies sit at the intersection of topology, representation theory, and graph theory. This is joint work with Ben McCarty. |
| Tuesday (!), March 26, 2024 at 3:30 p.m. in MSPB 370 | Matt Noble, Middle Georgia State University | Title: On Graphs with Rainbow 1-Factorizations Abstract: For a finite, simple graph G, a proper edge coloring of G is an assignment of colors to the graph’s edges so that no two edges sharing a vertex receive the same color. A 1-factor of G is a collection of edges, no two of them sharing a vertex, which together span G. Any subgraph H of G is said to be rainbow if all edges of H are colored different colors. In this talk, I will elaborate on what can go wrong and what can go right in attempting to construct small order graphs G which are of order 2n, are regular of degree n, and can be properly edge colored with n colors and then decomposed into rainbow 1-factors. This subject matter is accessible to just about anyone, regardless of what prior graph theoretic knowledge they possess. Lots of pictures will drawn, and lots of questions will be posed. Amazingly, the speaker was introduced to this line of inquiry when he and his wife were in a bar, and upon hearing that we were mathematicians, a friend of a friend said, “You know what? You sound like just the type of person to help me with a schedule I’m trying to create!” |
| Thursday, March 14, 2024 at 3:30 p.m. in MSPB 370 | Scott Brown, United States Geological Survey | Title: An Introduction to Gaussian Process Regression Abstract: We will introduce the basic foundations for Gaussian Process Regression, discuss recent advances in approximating methods and applications in Ecology. We will span topics from Statistics and Probability, Linear Algebra, Computer Science and Biology. The talk will be broad rather than deep and accessible to an undergraduate audience with a mathematics background - There will be lots of pictures, and some derivations and hand-waving, but any proofs will be left as an exercise. |
| Tuesday, February 6, 2024 at 3:30 p.m. in MSPB 370 | Huiling Liao, University of Minnesota | Title: Inferring Causal Direction Using Coefficient of Determination R2 Abstract: In the framework of Mendelian randomization, two single SNP-trait Pearson's correlation-based methods have been developed to infer the causal direction between an exposure (e.g. a gene) and an outcome (e.g. a trait), including the widely used Steiger’s method and its recent extension called Causal Direction-Ratio (CD-Ratio). Steiger's method uses a single SNP as a single instrumental variable (IV) for inference, while CD-Ratio combines the results from each of multiple SNPs. In this study, we first propose an approach based on R2, the coefficient of determination, to simultaneously combine information from multiple (possibly correlated) SNPs to infer a causal direction between an exposure and an outcome. The proposed method can be regarded as a generalization of Steiger's method from using a single SNP to multiple SNPs as IVs. It is especially useful in transcriptome-wide association studies (TWAS) with typically small sample sizes for gene expression data, providing a more flexible and powerful approach to inferring causal directions. It can be easily extended to use GWAS summary data with a reference panel. We also discuss its robustness to invalid IVs. We compared the performance of Steiger's method, CD-Ratio and the new R2-based method in simulations to demonstrate the advantage of the proposed method. Finally we applied the methods to identify causal genes for high/low-density lipoprotein cholesterol (HDL/LDL) using the GTEx (V8) gene expression data and UK Biobank GWAS data. The proposed method was able to confirm some well-known causal genes, such as LPL, LIPC and TTC39B for HDL, and identified some novel gene-trait relationships, suggesting the power gains of our proposed method through its use of multiple correlated SNPs as IVs. |
| Thursday, February 1, 2024 at 3:30 p.m. in MSPB 370 | Chase Holcombe, University of Alabama | Title: A Novel Distribution-Free Multivariate Control Chart with Simple Post-Signal
Diagnostics Abstract: Multivariate statistical process control (MSPC) charts are particularly useful when the need arises to simultaneously monitor several quality characteristics of a process. Most control charts in MSPC assume that the quality characteristics follow some parametric multivariate distribution, such as the normal. This assumption is almost impossible to justify in practice. Distribution-free MSPC charts are attractive, as they can overcome this hurdle by guaranteeing stable in-control (IC) or null performance of the control chart without the assumption of a parametric multivariate process distribution. Utilizing an existing distribution-free multivariate tolerance interval, we construct and propose a simple Phase II Shewhart-type distribution-free MSPC chart for individual observations, with control limits based on some Phase I sample order statistics. In addition to being easy to interpret, the proposed charting methodology preserves the original scale of measurements and can easily identify out-of-control variables after a signal, which are both important practical advantages, particularly in the multivariate setting. The exact in-control performance based on the conditional and unconditional perspectives is presented and examined. Determination of the control limits is discussed. The out-of-control (OOC) performance of the chart is studied by simulation for data from several multivariate distributions. Illustrative examples are provided for chart implementation, using both real and simulated data, along with a summary and conclusions. The proposed control chart is attractive as it fills a gap in the literature for multivariate control charts for individual data. It is easy to construct, visualize, and interpret, is exactly distribution-free, requires no complex parameter estimation calculations for implementation, comes with a natural and simple post signal diagnostic mechanism, and only requires a modestly large reference sample size for small to moderate dimensions. Continuing and future areas of work related to autocorrelation, Phase I contamination, and high dimensionality are also briefly discussed. |
| Tuesday, January 30, 2024 at 3:30 p.m. in MSPB 370 | Mathias N. Muia, University of Mississippi | Title: Dependence and Mixing for Perturbations of Copula-Based Markov Chains Abstract: This presentation explores the impact of perturbations of copulas on dependence and mixing properties for stationary ergodic Markov chains. We focus on two types of copula perturbations and establish a connection between these perturbations and the resulting mixing properties of the Markov chains they generate. Examples are provided to illustrate the different perturbations explored. Furthermore, we delve into the analysis of a discrete Markov chain that is based on the Frechet family of copulas. The objective behind this study being to explore the impact of discrete marginals on copula-based Markov chains. We analyse the mixing properties of such models to emphasize the difference between continuous and discrete state-space Markov chains. The Maximum likelihood approach is applied to derive estimators for model parameters in the case of a discrete-state space Markov chain with Bernoulli marginal distribution. A stationary case and a non-stationary case are considered. The asymptotic distributions of parameter estimators are provided. A simulation study showcases the performance of different estimators for the Bernoulli parameter of the marginal distribution. Some statistical tests are provided for model parameters. |
| Thursday, January 25, 2024 at 3:30 p.m. in MSPB 370 | Yongli Sang, University of Louisiana at Lafayette | Title: Asymptotic Normality of Gini Correlation in High Dimension with Applications
to the K-sample Problem Abstract: The categorical Gini correlation proposed by Dang et al. is a dependence measure between a categorical and a numerical variable, which can characterize independence of the two variables. The asymptotic distributions of the sample correlation under the dependence and independence have been established when the dimension of the numerical variable is fixed. However, its asymptotic distribution for high dimensional data has not been explored. In this talk, we show the central limit theorem for the Gini correlation for the more realistic setting where the dimensionality of the numerical variable is diverging. We then construct a powerful and consistent test for the K-sample problem based on the asymptotic normality. The proposed test not only avoids computation burden but also gains power over the permutation procedure. Simulation studies and real data illustrations show that the proposed test is more competitive to existing methods across a broad range of realistic situations, especially in unbalanced cases. |
| Thursday, January 18, 2024 at 3:30 p.m. in MSPB 213 (or on Zoom) | Louis H. Kauffman, University of Illinois at Chicago | Title: From Map Coloring to Multiple Virtual Knot Theory Abstract: Around 1880 Peter Guthrie Tait showed that the Four Color Theorem is equivalent to the statement that a bridgeless, trivalent (cubic) plane graph can be edge-colored with three colors so that three distinct colors appear at every node of the graph. We begin by discussing the beautiful method of counting the number of three colorings of trivalent plane graphs due to Roger Penrose in his paper “On Applications of Negative Dimensional Tensors”. The Penrose method has a graphical skein relation and an interpretation in terms of tensor networks. It has the drawback that it does not count the colorings of non-planar graphs. We explain our fix for this problem that uses a new tensor and two types of ‘virtual” crossings in the diagrams. One type of virtual crossing arises from immersing a non-planar map in the plane. The other comes from the skein expansion. Next, we generalize the evaluations to (perfect matching) polynomials that are assigned to graphs with associated perfect matchings and we show how in certain cases the coloring count extends to n colors where n is any natural number. This part of the talk is joint work with Scott Baldridge and Ben McCarty. Then we use these combinatorial ideas to discuss an extension of virtual knot and link theory that uses an arbitrary cardinality of distinct virtual crossings. The transition occurs by re-interpreting the perfect matching polynomials as generalized Kauffman bracket polynomials that are invariants of virtual knots and links with two types of virtual crossings. It is natural to go from two types of virtual crossings to arbitrarily many of them. In this generalization, all virtuals detour with respect to one another and with respect to classical crossings and graphical crossings. There are corresponding generalizations of cobordisms of virtual knots and links, welded knot theory, flat virtual theory and free knot and link theories. We discuss basic invariants and their generalizations such as the Jones polynomial, Kauffman bracket, Arrow polynomial, fundamental group, augmented fundamental groups, quandles and biquandles, Sawollek polynomials and other invariants. This work is motivated by specific invariants for two virtual crossings that generalize the bracket polynomial and give knot theoretic versions of generalized Penrose evaluations for trivalent graphs. There are many new questions about this generalization of virtual knot theory both knot theoretic and graph theoretic. |