杏吧论坛

Thursday, January 23, 2025, at 3:30 p.m. in MSPB 370

Chase Holcombe, 杏吧论坛

Title: TBC

Abstract: TBC                                                                                                                                                                                                                                                                                 

Thursday, November 7, 2024, at 3:30 p.m. in MSPB 370

Aparna Upadhyay, 杏吧论坛

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, 杏吧论坛

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 140

This 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, 杏吧论坛

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, 杏吧论坛

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, 杏吧论坛

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 脳 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鈥攖hese 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鈥檚 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, 鈥淵ou know what? You sound like just the type of person to help me with a schedule I鈥檓 trying to create!鈥�

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

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鈥檚 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

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

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

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)

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 鈥淥n Applications of Negative Dimensional Tensors鈥�. 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 鈥榲irtual鈥� 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.

Thursday, November 30, 2023 at 3:30 p.m. in MSPB 370

Title: Representations of groups and the importance of tensor products

Abstract: Representation theory is the study of non-linear algebraic objects, like groups or algebras, using methods from linear algebra (e.g. eigenvalues and Jordan decomposition). This provides an effective tool for understanding the structure of these algebraic objects, but the category of representations of a given group does not a priori determine that group up to isomorphism. In this talk, I will first explain some basic concepts of group representation theory, including the definition of tensor products of representations. Then I will discuss how any group can be recovered from the data of all of its representations and their tensor products (i.e. from the category of representations viewed as a tensor category).

Thursday, November 16, 2023 at 3:30 p.m. in MSPB 370

Title: Class numbers of imaginary quadratic fields

Abstract: Gauss was the first to count classes of binary quadratic forms with a fixed discriminant up to matrix equivalence, and information about the number of equivalence classes, called the class number, percolates into many branches of number theory, including the theory of L-functions via Dirichlet's class number formula, and elliptic curves in view of the work and conjecture of Birch and Swinnerton-Dyer. This talk will begin with an introduction to algebraic number theory and class numbers and a few important results in the history of their study. Then I will discuss a little of my work on the divisibility properties of class numbers.

Thursday, October 26, 2023 at 3:30 p.m. in MSPB 370

Title: Statistical Inference on Fractional Partial Differential Equations

Abstract: Researchers are often confronted with connecting theoretical models with real world data.  This is very common in the petroleum resources industry.  Specifically, they wish to understand the pressure of gas in a porous substance.  A wealth of mathematical models have been developed based on partial differential equations and have been studied extensively.  These modeling efforts have been extended to using Fractional Partial Differential Equations (FPDE) to better capture various attributes of the process.  The problem for researchers arises when one needs to estimate the fractional parameter.  As the fractional parameter is not a parameter in the traditional statistical sense, it is the fraction of the derivative used.  This work shows a Bayesian approach to solving this problem with associated simulation studies to illustrate how the method performs on a variety of cases.

Wednesday, October 25, 2023 at 6:00 p.m. in the Marx Library Auditorium (Room 110)

This talk is aimed at a general audience!

---8th Mishra Memorial Lecture---

Title: Chasing a pandemic鈥�

Abstract: The COVID-19 pandemic gave researchers in public health a huge opportunity to use well established models in a real world pandemic in real time.  While this opportunity allowed for researchers and policy makers the chance to make data driven decisions, early on it became very apparent that assumptions for these models needed to be adjusted and as the vaccine became available the models needed to be able to incorporate this.  This talk looks at how these models and assumptions changed during the pandemic and how various extensions of these models were created to help monitor the changes in the dynamics of the pandemic.  While mathematical models are discussed, mathematical details will be omitted.  For illustration the State of Qatar will be considered.

This event is sponsored by the Honor Society of Phi Kappa Phi.

Title: Special values of L-functions - a classical approach

Abstract: In 1734, Euler observed that the values of the Riemann zeta-function at even positive integers are rational multiples of powers of 蟺. However, the odd zeta-values remain a mystery to this day. In fact, it is widely believed that the odd zeta-values do not satisfy any polynomial relation over the rationals with 蟺. Almost three centuries after Euler, several different perspectives have emerged to study the general case of special values of L-functions. In this talk, we discuss a more analytic and classical approach and describe recent progress on related conjectures by Chowla, Erdos and Milnor.

Title: Playing With the Binary Icosahedral Group

Abstract:  For a little while, I have been trying to understand several things: (1) The 5-fold branched cover of the 3-sphere that is branched along the trefoil knot, (2) the five twist spin of the trefoil, (3) the group of (2-by-2) matrices that have entries in the integers mod 5, (4) the Poincare homology sphere, (5) the binary icosahedral group. The cognoscenti will observe that these are all related topics. In the talk, I will sketch why they are related and I'll preview the things that I have found out. 

My purpose is to educate myself and others because I feel that at the most basic level of instruction not enough is said on these examples that are key to the understanding of low dimensional topology. In particular, students are invited. I'll do my best to present things in an elementary fashion.

Title: Unpacking Gendered Differences in Approaches to Mathematical Problems: Implications and Potential Explanations

Abstract: Despite progress toward gender equity in past decades, gender differences in math problem-solving performance and career outcomes remain, with women pursuing careers in science, technology, engineering, and math less often than men. This talk will summarize extant theories about what contributes to gendered participation differences in mathematics, with a focus on the role socialization and environment may play. Within this, I introduce evidence that the strategies used on basic computation problems relate to problem solving performance on more complex mathematical tasks and introduce the construct of 鈥淏old Problem Solving鈥�, an approach to mathematics that involves risk-taking and inventiveness. A throughline of this work is that gender differences in mathematical performance are related to gendered differences in strategy use and bold problem solving, suggesting that girls and women learn how to do mathematics differently than their male peers. Implications for future work and teacher professional development are discussed. 

Title: Non-Hausdorff T鈧� Properties and Sociotechnical Infrastructure for Mathematics Research

Abstract: Several weakenings of the T鈧� property for topological spaces, including k-Hausdorff,  KC, weakly Hausdorff, semi-Hausdorff,  RC, and US, have been studied in the literature. Here we investigate how these properties do or do not relate to one another, and in doing so, illustrate and evaluate the role of technology in modern mathematics research.

Title: Let鈥檚 Color!

Abstract: Coloring graphs is a topic that was introduced in the 1850s.  Over the past 170 years, the study of graph coloring has led to the development and discovery of some rich and beautiful mathematics.  Also, the applications of graph coloring are vast, and applications can be found in scheduling, problems on social networks, design of seating plans, the assignment of frequencies to radio stations, the equitable distribution of resources, and much more!  Another intriguing property of graph coloring is that one does not need a lot of mathematical background to understand many graph coloring questions that remain a mystery to the human race today.  Since graph coloring is a relatively young research topic, it is also possible to make progress on some graph coloring questions through elementary ideas such as: extremal, inductive, and algorithmic techniques.  This makes graph coloring an ideal playground for both undergraduate and graduate students.

In this talk I will introduce graph coloring and discuss some of its historical development and famous questions.  Then, I will introduce a well-known generalization of classical graph coloring called list coloring which also has abundant applications.  The talk will end with some results that were recently obtained on list coloring as part of an undergraduate research project with students: Kennedy Cano, Emily Gutknecht, Gautham Kappaganthula, Ezekiel Thornburgh, and George Miller.

Hypothesis Testing on Wilcoxon-Mann-Whitney Effect for Clustered Data under Informative Cluster Size

Abstract: Under the general non-parametric settings where data are clustered and the size of each cluster is informative, we address the issue of comparing two groups of observations in terms of the Wilcoxon-Mann-Whitney effect (also known as the non-parametric relative effect). With the help of a technique called within-cluster resampling (WCR), several rank-based tests are proposed with the goal to assess whether the given two groups are tendentiously equivalent to each other. Computationally, results from simulation data have shown evidence that our tests maintain a reasonably high power and a correct size compared to existing methods in the literature. Furthermore, we provide interval estimates of the non-parametric relative effect by inverting our tests. These findings contribute to the development of more efficient and accurate methods for comparing the stochastic equality of two groups in the setting of clustered data under informative cluster size.

Nonparametric Rank Based Methodology in Cluster Data Analysis

Abstract: Statistical comparison of two independent groups is one of the most frequently occurring inference problems in scientific research. Most of the existing methods available in the literature are not applicable when measurements are taken with dependent replicates. In all these scenarios the replicates should neither be assumed to be independent nor come from different subjects. Furthermore, using a summary measure of the replicates would decrease the precision of the effect estimates. So, a solution is proposed for these two sample problems with correlated replicates. Furthermore, major attention will be given to the accuracy of the tests in terms of controlling the nominal Type-I error level as well as their powers when sample sizes are rather small.

Unavoidable Induced Subgraphs of 2-Connected Graphs

Abstract: Ramsey proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either a complete graph on r vertices or an independent set with r vertices. It is well known that every sufficiently large, connected graph contains an induced subgraph isomorphic to one of a large complete graph, a large star, and a long path. We prove an analogous result for 2-connected graphs. Similarly, for infinite graphs, every infinite connected graph contains an induced subgraph isomorphic to one of the following: an infinite complete graph, an infinite star, and a ray. We also prove an analogous result for infinite 2-connected graphs.

Two-Input and Two-Output Predictive Model for Multifunctional Materials with Hysteresis and Thermodynamic Compatibility

Abstract: Multifunctional materials have tremendous potential for engineering applications as they are able to convert mechanical to electromagnetic energy and vice-versa. One of the features of this class of materials is that they show significant hysteresis, which needs to be modeled correctly in order to maximize their application potential. A method of modeling multifunctional materials that exhibit the phenomenon of hysteresis and is compatible with the laws of thermodynamics was developed recently. The model is based on the Preisach hysteresis operator and its storage function and may be interpreted as a two-input, two-output neural net with elementary hysteresis operators as the neurons. The difficulty is that the parameters in the model appear in a non-linear fashion, and there are several constraints that must be satisfied by the parameters for thermodynamic compatibility. In this research, we present a novel methodology that uses the rate-independent memory evolution properties of the Preisach operator to split the parameter estimation problem into three numerically well-conditioned, linear least squares problems with constraints. The alternative direction method of multipliers(ADMM) algorithm and accelerated proximal gradient method are used to compute the Preisach weights. Numerical results are presented over data collected from experiments on a Galfenol and a Terfenol-D sample. We show that the model is able to fit not only experimental data for strain and magnetization over a wide range of magnetic fields and stress but also able to predict the response for stress and magnetic fields not used in the parameter estimation.

Thursday, March 30, 2023

On Embedding Problems for 3-Manifolds in 4-Space

Abstract: The problem of embedding one manifold into another has a long, rich history, and proved to be tremendously important for development of geometric topology since the 1950s. In this talk I will focus on the 3-manifold embedding problem(s) in 4-dimensional Euclidean space R⁴. Given a closed, orientable 3-manifold Y, it is of great interest but often a difficult problem to determine whether Y may be smoothly embedded in R⁴. This is the case even for integer homology spheres (where usual obstructions coming from homology disappear), and restricting to special classes such as Seifert manifolds, the problem is open in general, with positive answers for some such manifolds and negative answers in other cases. On the other hand, under additional geometric considerations coming from symplectic geometry (such as hypersurfaces of contact type in R⁴) and complex geometry (such as the boundaries of holomorphically or rationally or polynomially convex domains in complex Euclidean space C²), the problems become tractable and in certain cases a uniform answer is possible. For example, recent work shows for Brieskorn homology spheres: no such 3-manifold admits an embedding as a hypersurface of contact type in R⁴. This implies restrictions on the topology of rationally and polynomially convex domains in C². In this talk I will provide further context and motivations for these results, and give some details of the proof.

Online Monitoring of Image Sequences

Abstract: To monitor the Earth鈥檚 surface, the satellite of the NASA Landsat program provides us image sequences of any region on the Earth constantly over time. Gradual loss of water resource in the Salton Sea has got much attention from researchers for its damage to the local environment and ecosystems. To monitor the water resource of the lake, researchers usually obtain certain water resource indices manually from databases such as the satellite images of the region. We developed a new method to monitor the area of the Salton Sea automatically. By this method, the lake boundary is first segmented from each satellite image by an image segmentation procedure, and then its area is computed by a numerical algorithm. The sequence of lake areas computed from satellite images taken at different time points is then monitored by a control chart from the statistical process control literature. Because the lake area changes gradually over time, a new control chart designed for detecting process mean drifts is also proposed.

Automorphism Groups for Arithmetic Dynamical Systems

Abstract: Algebraic dynamics is the study of iteration of polynomial or rational functions. This talk focuses on dynamical systems with non-trivial automorphisms. Under the action of conjugation by fractional linear transformations, we can form a moduli space of dynamical systems of a certain degree. Certain elements in these moduli spaces have non-trivial automorphisms. This is analogous to the elliptic curves with complex multiplication in the moduli space of elliptic curves. These special maps have connections to many problems in arithmetic dynamics. I focus on two problems in this talk: identifying the locus of maps with non-trivial automorphisms and the realizability of subgroups of the projective linear group as automorphism groups. As time allows, I will use this automorphism locus to motivate a future project: the creation of a database of dynamical systems.

Dyadic Matroids with Spanning Cliques

Abstract: The Matroid Minors Project of Geelen, Gerards, and Whittle describes the structure of minor-closed classes of matroids representable by a matrix over a fixed finite field. To use these results to study specific classes, it is important to study the matroids in the class containing spanning cliques. A spanning clique of a matroid M is a complete-graphic restriction of M with the same rank as M. In this talk, we will describe the structure of dyadic matroids with spanning cliques. The dyadic matroids are those matroids that can be represented by a real matrix each of whose nonzero subdeterminants is a power of 2, up to a sign. A subclass of the dyadic matroids is the signed-graphic matroids. In the class of signed-graphic matroids, the entries of the matrix are determined by a signed graph. Our result is that dyadic matroids with spanning cliques are signed-graphic matroids and a few exceptional cases. The main results in this talk will come from joint work with Ben Clark, James Oxley, and Stefan van Zwam. This talk will include a brief introduction to matroids.

Modeling COVID-19 Pandemic using Graph Theory

Abstract: The coronavirus has affected many countries and taken the lives of several thousands of people since its outbreak in 2019. The spread (pandemic) pattern of this virus can be analyzed from graph theory perspective where the COVID-19 can be represented as a graph with each vertex representing an individual at any particular stage of infection (asymptomatic, presymptomatic, symptomatic, or vaccinated), and an edge indicating the transmission from person to person. This project considers the neighborhood prevalence of each individual, i.e., proportion of each individual's contacts who are either exposed or infected, and describe parameters for a threshold value R < 1 and R > 1 on the spread of the pandemic. This threshold value is similar to the reproduction number of the pandemic.

On Chromatic Polynomials, List Color Functions, and DP Color Functions

Abstract: Counting proper colorings of graphs is a fundamental topic in enumerative combinatorics that has been extensively studied since the early 20th century. Specifically, 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 widely studied generalization of classical coloring that was introduced in the 1970s, and DP-coloring is a generalization of list coloring that has received considerable attention since its introduction in 2015. In this talk we present an overview of both a list coloring analogue and DP-coloring analogue of the chromatic polynomial which are called the list color function and DP color function respectively. For a given graph G we will be primarily interested in comparing the list color function and DP color function of G to P(G,m). Specifically, we will present results and open questions related to the following questions. Does the list color function (resp. DP color function) of G eventually equal P(G,m) for sufficiently large m? When the answer to this question is yes, we will also ask: What is the smallest m at which the list color function (resp. DP color function) of G is nonzero, equals P(G,m), and stays equal to the chromatic polynomial of G for all integers greater than m? The results we present have proofs that utilize a diverse array of techniques from algebraic, extremal, and probabilistic combinatorics.

This talk will include joint works with Vui Bui, Samantha Dahlberg, Charlie Halberg, Hemanshu Kaul, Akash Kumar, Andrew Liu, Michael Maxfield, Patrick Rewers, Paul Shin, Seth Thomason, and Khue To.

Copula-Based Bivariate Zero-Inflated Poisson Time Series Models

Abstract: Count time series data are found in multiple applications such as environmental science, biostatistics, economics, public health, and finance. Such time series counts come with inflation and in a bivariate form that captures not only serial dependence within each time series but also interdependence between the two series. To accurately study such data, one needs to account for the two types of dependence that emerge from the observed data, and the inflation. A class of bivariate integer-valued time series models is constructed via copula theory. Each time series follows Markov chains with the serial dependence is captured using copula-based transition probabilities with the Poisson and the zero-inflated Poisson (ZIP) margins. The copula theory is also used again to capture the dependence between the two series using either the bivariate Gaussian or t-copula functions. Likelihood based inference is used to estimate the model parameters with the bivariate integrals of the Gaussian or t copula functions being evaluated using standard randomized Monte Carlo methods. To evaluate the proposed class of models, a comprehensive simulated study is conducted capturing both positive and negative cross correlations. Then, two real life datasets are analyzed assuming the Poisson and the ZIP marginals, respectively. The results show the promises of the proposed class of models. Extensions to other class of count time series will be presented.

This talk is aimed at a general audience!

Open Educational Resources in Mathematics Education

Abstract: PROSE (https://prose.runestone.academy) is the presenter's NSF-funded project scoping the development of an Open-Source Ecosystem surrounding software products that support the authoring and publishing of accessible Open Educational Resources in STEM. This presentation will overview the free technologies and educational resources involved in this ecosystem, and how they may be used to enhance mathematics instruction at the university and high school levels. This talk is aimed at current and future mathematics instructors, as well as undergraduate and graduate students who are considering industry careers related to software engineering.

This talk is directed especially toward undergraduate and graduate students in mathematics or related subjects.

An Introduction to Generalized Splines

Abstract: Splines are a fundamental tool in applied mathematics and analysis, classically described as piecewise polynomials on a combinatorial decomposition of a geometric object (a triangulation of a region in the plane, say) that agree up to a specified differentiability wherever two faces of the decomposition intersect. Generalized splines extend this idea algebraically and combinatorially: instead of certain classes of geometric objects, we start with an arbitrary combinatorial graph; instead of labeling faces with polynomials, we label vertices with elements of an arbitrary ring; and instead of applying degree and differentiability constraints, we require that the difference between ring elements associated to adjacent vertices is in a fixed ideal labeling the edge. Billera showed that generalized splines can be used to recover classical splines in most cases of real-world interest. Independently, a long and deep line of research into localization techniques in toric topology culminated in a result of Goresky, Kottwitz, and MacPherson showing that generalized splines can also be used to compute the torus-equivariant cohomology of suitable algebraic varieties.

In this talk, we describe generalized splines, how they extend classical splines (as well as ideas from other fields, from number theory to algebraic topology), and give some results on a longstanding open problem about the dimension of the space of splines on triangulations of the plane.

This talk is directed especially toward undergraduate and graduate students in mathematics or related subjects.

Scoring a Goal Optimally in a Soccer Game under Liouville-like Quantum Gravity Action

Abstract: In this paper, we present a new stochastic differential game-theoretic model of optimizing strategic behavior associated with a soccer player under the presence of stochastic goal dynamics by using a Feynman-type path integral approach, where the action of a player is on a 鈭�8/3-Liouville quantum gravity surface. Strategies to attack the oppositions have been used as control variables with extremes like excessive defensive and offensive strategies. As in a competitive tournament, all possible standard strategies to score goals are known to the opposition team, a player's action is stochastic, and they would have some comparative advantages to score goals. Furthermore, conditions like uncertainties due to rain, dribbling, and passing skills of a player, time of the game, home crowd advantages, and asymmetric information of action profiles are considered to determine the optimal strategy.

Tuesday, November 29, 2022

How Being a Math and Data Nerd Can Make You $uccessful!

Abstract: Brian will talk about his journey from graduating from USA with a degree in Mathematics to becoming a featured expert in the data and analytics world. He鈥檒l share how his education prepared him to get jobs and important skills that he picked up along the way to allow him to keep advancing during the data explosion. If time allows, Brian will share how companies are currently using data to understand their businesses and to make data driven decisions as well as what鈥檚 next.

Students in mathematics and related subjects are encouraged to attend.

Linking in 3.5 Dimensions

Abstract: Knots are circles embedded into Euclidean space and links are disjoint knots with multiple components. The classification of links is essential for understanding the fundamental objects in low-dimensional topology; every 3- and 4-manifold can be represented by a weighted link. When studying 3-manifolds, one considers isotopy as the relevant equivalence relation whereas when studying 4-manifolds, the relevant condition becomes knot and link concordance. In some sense, the nicest class of links are the ones called boundary links; like a knot, they bound disjointly embedded orientable surfaces in Euclidean space, called a multi-Seifert surface. The strategy to understand link concordance, starting with Levine in the 60's, was to first understand link concordance for boundary links and then to try to relate other links to boundary links. However, this point of view was foiled in the 90's when Tim Cochran and Kent Orr showed that there were links (with all known obstructions, i.e., Milnor's invariants, vanishing) that were not concordant to any boundary link. In this work, Chris Davis, Jung Hwan Park, and I consider weaker equivalence relations on links filtering the notion of concordance, called n-solvable equivalence. We will show that most links are 0- and 0.5-solvably equivalent but that for larger n, that there are links not n-solvably equivalent to any boundary link (thus cannot be concordant to a boundary link). I won't assume any knowledge of knot or link theory in this talk and there will be a lot of pictures! This is joint work with C. Davis and J.H. Park.

Students in mathematics or related subjects are encouraged to attend.

Seeking Better Perspectives of Symmetry

Abstract: To understand a problem in a given setting, it is often helpful to describe all symmetries. One technique to perceive symmetry is to imagine reversible actions according to a set of rules. For example, you can reflect a butterfly across a vertical axis and get the same picture. In this example, there is only one simple rule, but in general describing the (infinitely many) rules is a difficult problem that often leads to deep solutions and beautiful pictures. I will describe some symmetries important for physics and algebraic geometry, and indicate various perspectives developed over time to exploit extra structure. The talk will conclude with a computer calculation using some of the deepest known theory in algebraic geometry, combinatorics, and representation theory to show how certain smooth spaces need to collapse, while preserving symmetry, to describe singular spaces important for Lie theory.

Reference Intervals and Regions in Laboratory Medicine

Abstract: Reference intervals are data-based intervals that are meant to capture a pre-specified large proportion of the population values of a clinical marker or analyte in a healthy population. They can be one-sided or two-sided, and they are widely used in the interpretation of results of biochemical and physiological tests of patients. A population reference range is typically expected to include 95% of the population distribution, and reference limits are often taken to be the 2.5th and 97.5th percentiles of the distribution, which is especially meaningful if normality is appropriate. Usually the reference range is constructed based on a random sample and simply estimating the percentiles is clearly not satisfactory. This calls for the use of appropriate criteria for estimating the reference range from a random sample. When there are multiple biochemical analytes measured from each subject, a multivariate reference region is needed. Traditionally, under multivariate normality, reference regions have been constructed as ellipsoidal regions. This approach suffers from a major drawback: it cannot detect component-wise extreme observations. Thus rectangular reference regions need to be constructed based on appropriate criteria. The talk will review univariate reference intervals and multivariate reference regions, and the criteria that can be used in their construction. Both parametric and non-parametric scenarios will be addressed, and laboratory medicine examples will be used for illustration.

This talk is aimed at a general audience!

Seventh Satya Mishra Memorial Lecture

Cost-Effectiveness Analysis: A Statistical Overview

Abstract: Identifying treatments or interventions that are cost-effective (more effective at a reasonable cost) is clearly important in health policy decision making, especially in the allocation of health care resources. Various measures of cost-effectiveness that are informative, intuitive and simple to explain have been suggested in the literature. Popular and widely used measures include the incremental cost-effectiveness ratio (ICER), defined as the ratio between the difference of average costs and the difference of average effectiveness in two populations receiving two treatments. The ICER is interpreted as the additional cost per unit of effectiveness gained. Yet another measure proposed is the incremental net benefit (INB), which is the difference between the incremental cost and the incremental effectiveness after multiplying the latter with a "willingness-to-pay" amount. In the talk, I will provide a fairly non-technical review of the area of cost-effectiveness analysis, and its importance in health policy decision making. Some recently introduced cost-effectiveness measures will be discussed and examples will be given.

This talk is directed especially toward undergraduate and graduate students in mathematics or related subjects.

Gaussian Binomial Coefficients with Negative Arguments

Abstract: In the early 90's, Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We tell this remarkably little known story and show that all of it can be extended to the case of Gaussian binomial coefficients.

This talk is based on joint work with Sam Formichella, a former undergraduate student at South.