Colloquia

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Dave Futer – Temple University

Can you hear the shape of a 3-manifold?

In the 1960s, Marc Kac popularized the question, "Can you hear the shape of a drum?" In slightly more mathematical language, the question asks: "Can you determine the shape of a domain in the plane from the spectrum of frequencies at which it vibrates?" The study of this question has been extended to surfaces and manifolds of other dimensions.

We now know that the answer is usually "no." There exist planar domains (and surfaces, and 3-dimensional manifolds) that have the same spectrum but different shapes. However, essentially all known counterexamples are related by a rigid cut-and paste procedure called commensurability. I will explain how this works in the context of 3-dimensional manifolds, leading up to some recent joint work with Christian Millichap.

Ellie Bayat Mokhtari – University of Montana

Influenza-type illnesses and air pollutants of particulate matter < 2.5μm (PM2.5): an application of Archetypal Analysis to identify spatiotemporal structure

Particulate matter (PM2.5) readings are often included in air quality reports from environmental authorities as it can pose the most danger when it builds up in human respiratory system and increases the risk of respiratory infections and lung diseases. Understanding the spatio-temporal variability of  upper respiratory illness and its dependence upon air quality in Montana is an area of active research in the public health sphere.

Archetypal analysis (AA), Culter and Breiman 1994, is introduced as a method to decompose and characterize structures within spatio-temporal data. AA seeks to synthesize a set of multivariate observations through a few, not necessarily observed points (archetypes), which lie on the boundary of the data cloud. This method is new in climate science, although it has been around for more than two decades in pattern recognition.

The goal of this presentation is to examine the spatio-temporal variability of two sets of weekly influenza cases and PM2.5 across Montana between 2008-2018 through AA. Compared to other conventional methods, such as PCA, the results provide the direct link to the observations which facilitate the interpretation. The patterns exposed by AA in both cases are contrasted, as one data set is approximately spatially continuous (PM) and the other is not (Flu counts).

Derek Williams – Montana State University

Relationships Between Undergraduate Students' Engagement and Understanding

This presentation discusses results from a mixed methods study investigating student engagement, understanding of precalculus concepts, and associations between engagement, understanding, and instructional approaches as reported by community college precalculus students. Student- and classroom-level factors associated with precalculus students' engagement are identified, and task-based interviews reveal a relationship between affective and cognitive experiences. Implications for teaching, and the current/future directions of this research are shared.

Charlie Katerba – Montana State University

Searching for closed essential surfaces in knot complements

Culler-Shalen theory uses a 3-manifold’s (P)SL(2,C) character variety to construct essential surfaces in the manifold. It has been a fundamental tool over the last 35 years in low-dimensional topology. Much of its success is due to a solid understanding of the essential surfaces with boundary that can be constructed with the theory. It turns out, however, that not every surface with boundary is detected. One can also construct closed essential surfaces within this framework. In this talk, we will discuss a module-theoretic perspective on Culler-Shalen theory and apply this perspective to show that there are knot complements in S3 which contain closed essential surfaces, none of which are detected by Culler-Shalen theory. As a corollary, we will construct an infinite family of closed hyperbolic Haken 3-manifolds whose representations into PSL(2, C) have traces which are integral (over Z).

Ricela Feliciano-Semidei – University of Montana

The use of computer softwares and mathematics achievements of 8th grade students in Puerto Rico: using NAEP 2015 restricted data

Technology had dramatically changed people’s lives and education, including the teaching and learning of mathematics. This quantitative study explores the mathematics achievement patterns of 8th grade students in Puerto Rico and their relationship with the use of computer mathematical softwares (e.g. statistical, graphic, geometric, and spreadsheet programs). The emerged theoretical perspective is based on the Education Production Function and Critical Race Theory to acknowledge the unique Puerto Rican culture and to avoid comparisons with other group of students in the United States. The investigation analyzed 2015 National Assessment of Educational Progress (NAEP) data. This talk will present results from preliminary descriptive statistical analysis on non-restricted Math NAEP Data. The discussions of the results will help the mathematics education community to target improvements in the use of technology for the mathematics education of students in Puerto Rico.

Makini Beck – Rochester Institute of Technology

Mentoring STEM Women Faculty

Dr. Makini Beck is a Visiting Assistant Professor at the School of Individualized Study and the Department of Sociology at Rochester Institute of Technology. Her talk argues for improving mentoring practices for women faculty in higher education. Drawing on her research with STEM women faculty, a meta-synthesis of the literature and personal antidotes from within the academy, she calls for the necessity of mentoring approaches that are authentic humanistic, and critically disruptive of the institutional status quo.

Leonard Huang – University of Nevada, Reno

Generalizing a Real-Analysis Exam Problem: A Potpourri of Functional Analysis, Probability, and Topology

This talk is inspired by the following problem, which has tormented many a graduate student in real-analysis qualifying exams around the world:

Let $$(x_{n})_{n \in \mathbb{N}}$$ be a sequence in $$\mathbb{R}$$. If $$\displaystyle \lim_{n \to \infty} (2 x_{n + 1} - x_{n}) = x$$ for some $$x \in \mathbb{R}$$, then prove that $$\displaystyle \lim_{n \to \infty} x_{n} = x$$ also.

In the spirit of mathematical research, one may now ask: Is this result still true if we replace $$\mathbb{R}$$ by some other topological vector space? In this talk, we will show that the result is true for a wide class of topological vector spaces that includes all locally-convex ones, as well as some that are not locally convex, such as the $$L^{p}$$-spaces for $$p \in (0,1)$$. We will then construct, using basic probability theory, an example of a badly-behaved topological vector space for which the result is false.

Javier Perez-Alvaro – University of Montana

Nonlinear eigenvalue problems, a challenge for modern eigenvalue methods

Nonlinear eigenvalue problems arise in a variety of science and engineering applications and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This talk surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques.

Wend Werner – University of Muenster in Germany

The Mathematics of Shuffling Cards

Shuffling a deck of cards seems easy enough. However, analyzing this process from a more theoretical point of view we will discover that in a qualitative description of this process some more advanced mathematical techniques must be used and that not all questions about card shuffling could have been answered so far.

The mathematical topics that come up here stem from (a more abstract version of)  Fourier analysis, as applied to random walks on groups, variants of the (stochastic) central limit theorem, and, somewhat surprisingly, a problem from solid state physics is very similar in nature.

Tien Chih – Montana State University-Billings

Homotopy in the Category of Graphs

Homotopy Theory is the study of bending spaces into each other. While this is an essential part of the study of Topology, it does not translate immediately to discrete settings such as Graphs. However, we can use the language of Categories to abstract ideas from homotopy, and apply them to Graph Theory.

We begin by discussing the basic definitions of graph homotopies first established by Anton Dochtermann in 2008. We then discuss new results in homotopy of graphs, including a way to find a unique representative for each homotopy class of graphs. This work was done in collaboration with Dr. Laura Scull of Fort Lewis College.

Esmaeil Parsa – University of Montana

“Distinguishing two notions of unique colorabilityfor digraphs”

The study of homomorphisms is ubiquitous in mathematics. In graph theory, homomorphisms naturally generalize the notion of coloring. Many other important problems regarding the chromatic number, the clique number, the odd girth number, etc., can also be restated in terms of homomorphisms. In this talk we focus on a special directed graph homomorphism known as the “acyclic homomorphism” and study two ways of generalizing the notion of unique colorability using it. One natural way to do this is to deﬁne it in terms of partitions induced by acyclic homomorphisms, while a second way—mostly used in the literature— is done by automorphisms of digraphs. We show that these two approaches are not equivalent and study conditions under which they coincide. This mirrors analogous work by Bonato (2007) in the realm of (undirected) graphs.

Ted Owen – University of Montana

“An Introduction to the Local Pivotal Method and Variance Approximation Approaches”

Simple random sampling is a commonly used (and commonly taught) sampling method that is the extent of the knowledge of sampling theory for many people. Are there better ways of selecting a sample? If so, in what instances is one sampling design better than another? What does it even mean for one sampling design to be better than another? These questions will be explored through the introduction of some basic sampling designs, and through the definition of those designs more interesting questions about the way in which such complicated sampling designs can be carried out will be answered. Splitting methods will be introduced as a means of selecting unequal inclusion probability random samples. Two special splitting methods are the pivotal method and local pivotal method, the latter of which incorporates auxiliary information into the way that a sample is selected. One difficulty with these methods is that the variance of estimates can be a problem to calculate, so some of the current work being done on estimation of variance will be presented.

Marta Civil – University of Arizona

Reflections on mathematics education and equity considerations

In this talk I describe some of my research, which addresses equity in mathematics teaching and learning. Drawing on my work with parents, teachers, and students, I illustrate the importance of context, beliefs about mathematics, and language(s) in understanding and improving the mathematics education of all students. Implications for teacher education and undergraduate mathematics education will be presented.

Michelle Ghrist – Gonzaga University

Designer Multistep Methods

Multistep methods provide a computationally efficient way to approximate solutions to differential equations.  In general, there is a three-way tradeoff between the accuracy, stability, and computational cost of numerical methods.  The stability domain is a picture in the complex plane that shows the problems and stepsizes for which a given numerical method will give stable solutions (i.e., roundoff will not grow exponentially).

I will discuss the development and analysis of novel multistep methods created by introducing parameters that are allowed to vary.  Dahlquist's First Stability Barrier puts a cap on the maximum order of a stable method; we seek to maximize the order while maintaining stability. Applying Taylor series gives a linear system for the unknown coefficients of the multistep method.  Requiring stability gives bounds on the domains of the free parameters; varying the parameters within this domain results in changes in the size and shape of the stability domain, allowing us to produce methods that work better for a given differential equation, thus creating “designer” numerical methods.  I will also discuss staggered methods, some theoretical results, and some real-world applications of our methods.

Omid Khormali– University of Montana

“An Introduction to Extremal Problems for Forests in Graphs and Hypergraphs”

Extremal graph theory is the study of how the intrinsic structure of graphs ensures certain types of properties under appropriate conditions. One of the main problems in extremal graph theory is determining the Turán number for graphs. The Turán number, ex(n,H), of a graph H is deﬁned as the maximum number of edges in a graph on n vertices which does not contain H as a subgraph. A hypergraph is a generalization of a graph, except that instead of having edges that only made up of two vertices, their edges are sets of any number of vertices. Compared to what we know for graphs, there is much less known about hypergraph Turán problems. In this talk, we introduce the fundamentals of graph and hypergraph extremal theory. We present several classical results and conclude with the proof a new result on the extremal number for a particular hypergraph.