Colloquia

Thursday, March 30, 2023 at 2:00 p.m. in MSPB 213

Note the different time and room!

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.

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.

Online Monitoring of Image Sequences

Abstract: To monitor the Earth’s 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 √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’ll 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’s 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.