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.

Friday, January 11, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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).

Monday, February 4, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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.

Monday, February 11, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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).

Monday, February 25, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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.  

Monday, March 4, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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.

Monday, March 11, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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.

Monday, March 18, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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.

Thursday, March 21, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 10

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.

Monday, April 1, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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.

Monday, April 8, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

Esmaeil Parsa – University of Montana

“Distinguishing two notions of unique colorability for 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.

Wednesday, April 10, 2019 3:00 pm in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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.

Thursday, April 11, 2019 3:00 pm in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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. 

Monday, April 15, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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.

Monday, April 22, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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.

Wednesday, April 24, 2019 3:00 pm in Math 211
Refreshments at 4:00 p.m. in Math Lounge 109

Kimberly Ayers – Carroll College

A Skew Product Model for Hybrid Dynamical Systems

A hybrid system is a dynamical system that exhibits both continuous and discrete behavior.  Think of a bouncing ball: when the ball is in the air, its velocity and position are changing smoothly in time.  However, when the ball hits the ground, it instantaneous reverses directions, exhibiting a discrete change in its velocity.  In this talk, we’ll examine a particular type of hybrid system from the vantage point of skew products.  We will begin by isolating the discrete behavior and examining its dynamics, exploring ideas such as chain recurrence and chaos.  We’ll then examine the behavior of the product space, and explore different classical recurrence concepts within the context of the dynamics on the product space and projections.

Tuesday, September 17, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

Maria C. Quintana – University Carlos III of Madrid

On the Structure of Linearizations for Rational Approximations of Nonlinear Eigenvalue Problems

Given a nonlinear matrix-valued function \(F(\lambda):\mathbb{C} \longrightarrow \mathbb{C}^{m\times m},\) the Non-Linear Eigenvalue Problem (NLEP) consists in computing numbers \(\lambda \in \mathbb{C}\) (eigenvalues) and non-zero vectors \(v \in \mathbb{C}^{m}\) (eigenvectors) such that $$ F(\lambda) v = 0, $$ under the regularity assumption \(\det(F(z)) \not\equiv 0\). NLEPs arise in a variety of applications in Physics and Engineering. Nowadays, a useful approach to tackle them is based on Rational Approximation (RA), which leads to rational eigenvalue problems (REPs). Then, for solving REPs, linearizations of rational matrices are used, which is one of the most competitive methods for this task. In this talk we present the notion of local linearizations of rational matrices. A local linearization of a rational matrix \(R(\lambda)\) preserves the zeros and poles of \(R(\lambda)\) locally, that is, in subsets of \(\mathbb{C}\) and/or at infinity. By using this new notion of linearization, we study the structure of linearizations constructed in the literature for RAs of NLEPs on a target set. Moreover, we provide very simple criteria to determine when a linear polynomial matrix is one of these linearizations.

Monday, September 30, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

Andrew Gilbert – Pacific Northwest National Lab

Inverse methods for material quantification using neutron and X-ray radiography

The need for methods to complete non-destructive quantitative inspections remains an active one for a variety of applications, e.g., baggage and cargo scanning. Recently, there is a renewed interest in inspection of objects with a beam of neutrons. The way that neutrons travel through an object is unique to X-rays, potentially offering useful information beyond what would be available using a typical X-ray radiograph. We will discuss recent work at the Pacific Northwest National Lab on quantification of material composition of an object using radiography as well as the inverse algorithms that underpin the method. New work involving combining data from a complex neutron interrogation system, a so-called neutron associated particle imager, will also be discussed. This system presents interesting possibilities for developing new methods to combine multiple observables of multiple particles into a cohesive and meaningful output. 

Monday, October 7, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

Jake Downs – University of Montana

Inferring Holocene precipitation in west central Greenland using the Unscented Transform 

We investigate changing precipitation patterns in Greenland during a period of elevated  temperatures called the Holocene thermal maximum (~10,000 - 6,000 years ago), exploiting a new chronology of ice sheet extent through the Holocene and an inverse modeling approach based on the unscented transform (UT) . The UT is applied to estimate changes in annual precipitation in order to reduce the misfit between modeled and observed ice sheet margin positions. We discuss the basic theory of the UT and show how it can be applied to the problem of time dependent data assimilation. Our results indicate that Holocene warming coincided with elevated precipitation, without which modeled retreat in west Greenland is more rapid than suggested by observations. This result highlights the important role that changing precipitation patterns had in controlling ice sheet extent during the Holocene.

Monday, October 14, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

Neal Bushaw – Virginia Commonwealth University

Small Percolating Sets

Bootstrap percolation is a simple monotone cellular automaton which was originally introduced by Chalupa, Leath and Reich as a model of ferromagnetism in the late 1970s.   In this model, we think of some vertices of a graph as being initially *infected*.  Even worse, this infection can spread -- an *uninfected* vertex with many infected neighbors will itself become infected.  Does the infection spread to the entire graph?  Will it stop?  Can it be efficiently quarantined?

In this talk, we give an introduction to bootstrap percolation and its history, highlighting a few major breakthroughs, classic problems, and important variants.  Then, we'll proceed to a simple sounding extremal problem -- which graphs have a small set of vertices whose infection will eventually spread to the entire graph?  This question was the topic of this summer's Graph Brain Project; we will describe several results which came out of the summer's work, as well as that workshop's somewhat unusual characteristics. 

No background knowledge will be assumed -- the aim of this talk is to introduce you to the area and its problems, rather than to show complicated proofs.

Monday, October 21, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

Faculty Evaluations

Monday, October 28, 2019 at 3:00 - 5:00 p.m. in Math 103

Vilma Mesa – University of Michigan

Algebra Instruction at Community Colleges: Investigating the relationship with student outcomes

This project seeks to assess the connection between quality instruction in community college algebra courses and students' outcomes in these courses. It is a work in progress, so I will describe the context of the project, the research questions, and what we have found so far, and some connections to teaching undergraduate mathematics.

Tuesday, October 29, 2019 at 4:00 p.m. in Math 103
Refreshments at 3:30 p.m. in Math Lounge 109

Undergraduate Math Seminar
Chris Waters

A Survey of Cryptology: From World War II to the Present Day

The Enigma cipher machine produced in the early 1900s is one of the most famous cryptographic devices in history. Most Enigmas were destroyed during or after World War II. The surviving devices are worth hundreds of thousands of dollars. The Crypto Museum web site led a project to make the "electronic Enigma", or Enigma-E, kit available to collectors. We will demonstrate a working Enigma-E along with background about the real Enigma used in World War II. Attendees will be able encrypt and decrypt messages on the Enigma-E. Time permitting, we will fast forward to present-day cryptology and discuss the current state of affairs. A key fact about present-day cryptanalysis is the problems often occur in implementations or protocols rather than the cryptographic algorithms themselves.

Chris Waters is a Cyber Security Technical Lead and Cryptology Enthusiast.
He has a B. A. in Mathematics from the University of Utah.

Wednesday, October 30, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

Rick Brown – University of Montana
PhD Candidate

Semivariogram Methods for Modeling Whittle-Matérn Priors in Bayesian Inverse Problems

In this talk, I will briefly present a mathematical description of the connection between Gaussian processes with covariance operators defined by the Matérn covariance function and Gaussian processes with precision (inverse-covariance) operators defined by the Green's functions of a class of elliptic stochastic partial differential equations (SPDEs) in the isotropic case. I will show that this connection breaks down when the domain is finite due to the effect of boundary conditions and that it can be re-established using extended domains. I will then introduce the semivariogram method for obtaining point estimates of the Whittle-Matérn covariance parameters, which completely specifies the Gaussian prior needed for stabilizing the inverse problem. I will extend these results to the anisotropic case, where the correlation length in one direction is larger than another. Finally, I will consider the case where the the correction length is spatially dependent. Two-dimensional image examples will be presented throughout the talk.

Monday, November 4, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

David Pask – University of Wollongong

Undirected graphs and their C*-algebras

I will give a brief history of the topic and update on progress on K-theory computations. The talk will be non-technical, only basic mathematical knowledge will be assumed.

Wednesday, November 13, 2019 at 4:00 p.m. in Math 103
Refreshments at 3:30 p.m. in Math Lounge 109

Diego Martinez – University Carlos III of Madrid

Coarse Geometry and Inverse Semigroups

In this talk we will discuss, mainly, two seemingly disconnected notions in mathematics: coarse geometry and inverse semigroups.

Geometry often studies certain objects (such as sets or manifolds) equipped with a distance function. For instance, one classical problem would be to classify every compact manifold up to diffeomorphism. Coarse geometry shifts the point of view, and defines two sets to be coarse equivalent if they look the same from far away. In this way, for instance, a point and a sphere are indistinguishable from each other. Coarse geometry then studies properties that remain invariant under this weak equivalence relation, that is, properties of the space that only appear at infinity.

On the other hand, an inverse semigroup is a natural generalization of the notion of group, and is closely related to the idea of groupoid. Starting with one of these objects we will introduce how to construct a metric space, in the same fashion as the Cayley graph construction in the context of groups. We will then study its coarse structure, in particular its property A and its amenability. Time permitting, we will also relate these properties to analogue properties in some operator algebras.

Monday, November 25, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

Quy Cao – University of Montana
PhD Candidate

The Predictive Performance of Objective Measures of Physical Activity Derived From Accelerometry Data for 5-Year All-Cause Mortality in Older Adults: National Health and Nutritional Examination Survey 2003–2006.

Reliable measures of the frequency, duration and intensity of physical activity provided by wearable technology were used in the analysis of activity data. Accelerometry derived measures of physical activity were compared with established predictors of 5-year all-cause mortality in older adults, aged between 50 and 85 years from the 2003- 2006 National Health and Nutritional Examination Survey, in terms of individual, relative, and combined predictive performance. A total of 33 predictors of 5-year all-cause mortality, including 20 measures of objective physical activity, were compared using univariate and multivariate logistic regression. The results show that objective accelerometry-derived physical activity measures outperform traditional predictors of 5-year mortality, including age. This highlights the importance of wearable technology for providing reproducible, unbiased, and prognostic biomarkers of health.

Monday, December 2, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109