2020 Colloquia
Spring
Andrey Sarantsev – University of Nevada
Autoregression Modeling of the American Stock Market
High price-to-earnings ratios indicate overvaluation of the stock market due to irrational exuberance (a phrase by Alan Greenspan). We develop new measures of earnings and study whether the market is currently in a bubble. We use linear regression and time series modeling.
January 27, 2020 at 3:00 p.m. in Math 103
Nicolai A. B. Riis – Technical University of Denmark
Model Discrepancy Updates: Model calibration and uncertainty quantification applied to CT with uncertain view-angles
February 3, 2020 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Danny Crytser – St Lawrence University
Pure extension property for étale groupoid crossed products
A state on a \(C^*\)-algebra is a positive linear functional of norm \(1;\) it's pure if it can't be written as a convex combination of other states. The GNS construction relates states to the representation theory of \(C^*\)-algebras and is crucial in the study of operator algebras. If \(A \subset B\) is a unital inclusion of \(C^*\)-algebras, and every pure state on \(A\) has a unique extension to a state on \(B\), we say that \(A\) has the extension property, first identified by Kadison and Singer. In this talk I'll discuss the extension property for inclusions of the form \(A \subset A \rtimes \mathcal{G}\), where \(A \rtimes \mathcal{G}\) is an étale groupoid crossed product. This covers inclusions arising from étale groupoids as well as discrete group crossed products. This work builds off work of Zarikian and relates the extension property to a groupoid action on the spectrum. I'll also discuss the related question of the almost extension property, first defined by Nagy and Reznikoff.
February 13, 2020 3:00 p.m. in Math 103
Refreshments time 4:00 p.m. in Math Lounge 109
Juergen Kritschgau – Iowa State University
Few H copies of F saturated graphs
A graph is \(F\)-saturated if it is \(F\)-free but the addition of any edge creates a copy of \(F\). In this talk we will discuss the quantity \(\mathrm{sat}(n, H, F)\) which denotes the minimum number of copies of \(H\) that an \(F\)-saturated graph on \(n\) vertices may contain. This parameter is a natural saturation analogue of Alon and Shikhelman's generalized Turán problem, and letting \(H = K_2\) recovers the well-studied saturation function. We will focus on the cases where the host graph is either \(K_s\) or \(C_k\)-saturated.
February 24, 2020 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Algebra Seminar
Adam Chapman – Tel-Hai College in Israel
The u-Invariant and Friends
The maximal dimension of an anisotropic quadratic forms over a given field is an important arithmetic field invariant known as the u-invariant.
We will discuss the computation of the u-invariant in certain cases, and propose a finer invariant that tells apart fields the identical u-invariant based on the linkage properties of quaternion algebras over the fields.
The talk is partially based on joint work with Jean-Pierre Tignol.
February 25, 2020 at 4:00 p.m. in Math 211
Atish Mitra – Montana Tech
The Space of Persistence Diagrams on n points Coarsely Embeds in Hilbert Space
TDA (Topological Data Analysis) is a rapidly developing field that uses ideas from geometry and topology to get qualitative and quantitative information about the structure of data (finite sets of points in a metric space). One of the tools is the idea of Persistent Homology, which takes a one-parameter family of topological spaces and creates a signature called the persistence diagram that encodes useful information about the data set. For using existing kernel methods for analyzing such persistence diagrams, one needs to know how close the various metrics on the space on persistence diagrams can be to an inner product structure.
Using the methods of coarse geometry, we prove that the space of persistence diagrams on n points (with either the Bottleneck distance or a Wasserstein distance) coarsely embeds into Hilbert space. We also discuss various non-embeddability results when the number of points is not bounded.
This is joint work with Žiga Virk.
March 2, 2020 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Ellie Bayat-Mokhtari – University of Idaho
Information theoretic-based Biological Network Analysis
Biological networks are complex and often contain nonlinear interactions among a usually large number of species, genes, nutrients, metabolites, .... Correlation coefficients are widely used to analyze “omics” data as measures of linear interactions. However, how would we detect dependence when data is non-linear? In this talk, I will use mutual information based graph theory to analyze microbiome network and introduce a method to find a partition between contaminants and true bacteria that minimizes the loss of information. Among all the possible partitions of a network, this can be considered an optimal partition for characterizing the underlying structures of the network. Time permitting, I will briefly, discuss other projects that I have been working on.
March 9, 2020 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Canceled
Tracy Payne – Idaho State University
N-Graded Filiform Nilpotent Lie Algebras
We will start with a general discussion of Lie algebras, and nilpotent Lie algebras in particular, giving some examples and motivations for their study. Next we will focus on the subclass of N-graded filiform nilpotent Lie algebras, giving definitions, and examples. Finally we describe some recent results on the classification of N-graded nilpotent Lie algebras done jointly with Cameron Krome and John Edwards using iterated central extensions and a Python program.
March 23, 2020 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Canceled
Pedro Berrizbeitia – University of Colorado
March 30, 2020 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Joseph Hashisaki Memorial Scholarship
Kenton Ke
The Adams Scholarships
Junior: Burkleigh Yost
Senior: Kenton Ke
Anderson Mathematics Scholarship
Cory Emlen
Mac Johnson Family Scholarships
Esther Lyon Delsordo
Michael McKelligott
Jethro Thorne
Merle Manis Award
George Glidden-Handgis
Carolyn and Johnny Lott Elementary Education Scholarship
Haley VonGoedert
N. J. Lennes Competition
Alex Shepherd (1st)
Haley Wilson (2nd)
Undergraduate Research Scholars
Natalie Cole
Cory Emlen
George Glidden-Handgis
Kenton Ke
Martín Romero
Andi Wainwright
Burkleigh Yost
Undergraduate Tutorial Scholar
Martin Romero
William Myers Mathematics Scholarship
Anastasia Halfpap
Graduate Student Distinguished Teaching Awards
Daniel Gent
Caleb Huber
Graduate Student Summer Research Awards
Ian Derickson
Dakota Gray
Shurong Li
Van Magnan
Mohsen Tabibian
John A. Peterson Mathematics Education Awards
Kyra L. Mycroft
Mielle Posey Hubbard
Canceled
Jingjing Sun – University of Montana
Department of Teaching & Learning
Using Mixed Methods in Unlocking the Black Box of Classroom Learning
April 27, 2020 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Kevin Palencia Infante – University of Montana
PhD Candidate
Proper Mappings Between Balls
A continuous map f between topological spaces X and Y is proper if, for every compact subset K of Y, the preimage of K is compact in X. In this talk, we investigate the classification of proper holomorphic mappings between balls in complex Euclidean spaces. This investigation will lead to the study of two natural questions about homogeneous polynomials that are the subject of my dissertation.
April 27, 2020 at 3:00 p.m. online
Doctoral Dissertation Defense
Kevin Palencia Infante – University of Montana
A Natural Rank Problem for Homogeneous Polynomials and Connections with the Theory of Functions of Several Complex Variables
We study a natural extremal problem about the vector space consisting of all homogeneous polynomials of degree \(d\) in \(n+1\) variables with complex coefficients, together with the zero polynomial. We define the rank of a polynomial to be the number of distinct monomials appearing in the polynomial with non-zero coefficient. We are particularly interested in those homogeneous polynomials whose quotient with the homogeneous polynomial \(x_0+x_1+ \cdots +x_n\) is a polynomial of degree \(d-1\) with maximal rank. For each degree \(d\), we seek the minimum rank for an element of this subfamily and we seek to describe those polynomials with minimum rank. We call such polynomials sharp polynomials.
These problems have a simple solution for polynomials in one and two variables. The three-variable case is interesting and non-trivial, but well-understood. This research question has its roots in the study of proper polynomial mappings between balls in complex Euclidean spaces of different dimensions and the degree estimates problem. D'Angelo, Kos and Riehl and Lebl and Peters used a graph-theoretic approach to solve this problem in the case of proper monomial mappings. Lebl and Peters give a minimum rank estimate that answers our question in the three variable case. A family of sharp polynomials was described by D'Angelo and has been extensively studied. Brooks and Grundmeier provided a new proof of the minimum rank theorem in the three-variable case using a commutative algebra approach. They reformulate the problem as a question about homogeneous ideals and address it by studying the Hilbert function and the graded Betti number of certain ideals.
Using the same method as Brooks and Grundmeier, we give a sharp estimate for the minimum rank of homogeneous polynomials of our subfamily in four variables as well as a family of sharp polynomials. Moreover, we state a general result on the minimum rank for polynomials in \(n+1\) variables. Although this estimate is sharp in the three- and four- variable cases, the estimate is not sharp when the number of variables is greater than four.
May 6, 2020 at 10:00 a.m. in Zoom
Doctoral Dissertation Defense
Rick Brown – University of Montana
Semivariogram Methods for Modeling Whittle-Matérn Priors in Bayesian Inverse Problems
We present a new technique, based on semivariogram methodology, for obtaining point estimates for use in prior modeling for solving Bayesian inverse problems. This method requires a 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). We present a detailed mathematical description of this connection. We will show that there is an equivalence between these two Gaussian processes when the domain is infinite which breaks down when the domain is finite due to the effect of boundary conditions on Green's functions of PDEs. We show how this connection can be re-established using extended domains. We then introduce the semivariogram method for estimating the Matérn covariance parameters, which specify the Gaussian prior needed for stabilizing the inverse problem. Results are extended from the isotropic case to the anisotropic case where the correlation length in one direction is larger than another. The situation where the correlation length is spatially dependent rather than constant will also be considered. Finally, we compare and contrast the semivariogram method with a fully-Bayesian approach of finding estimates for and quantifying uncertainty in the hyperparameters. We implement each method in two-dimensional image inpainting test cases to show that it works on practical examples.
May 6, 2020 at 1:00 p.m. online
Fall
Ekaterina Smirnova – Virginia Commonwealth University
High dimensional regression with survey design: an application to the association between physical activity and mortality in NHANES
Understanding the mutual interactions between physical activity and health is crucial for developing clinical and public health intervention. Unfortunately, physical activity data used in public health research is often collected using self- or proxy-reports, which provide crude summaries of physical activity and are subject to substantial recollection, cognitive, and social desirability biases. To address this problem, a growing number of studies use accelerometers to objectively quantify physical activity in the free-living environment. Currently, the National Health Examination Study (NHANES) is the largest US-population study that contains publicly available physical activity data obtained from wearable accelerometers. Analyzing the association between physical activity in NHANES and mortality raises important and practical methodological challenges: (1) physical activity data in NHANES is high dimensional with one observation per minute for up to seven days for each study participant; (2) NHANES study participants are recruited form the US population according to a survey design procedure; and (3) the structure of the association between physical activity trajectories and outcomes is a-priori unknown. In this talk I will describe the problem of predicting health outcomes (five-year all-cause mortality) using high dimensional data (minute-level accelerometry summaries) while accounting for survey design and weights. I will also provide an easy to use tutorial for starting working with the NHANES physical activity data.
August 31, 2020 at 3:00 p.m. via Zoom
Cory Palmer – University of Montana
How do you grow an Erdős-Rado sunflower?
September 14, 2020 at 3:00 p.m. via Zoom
Cuneyt Gurcan Akcora – University of Manitoba
Topological Data Analysis on Networks – Applications and Scalability issues
Over the last couple of years, Topological Data Analysis (TDA) has seen a growing interest from Data Scientists of diverse backgrounds. TDA is an emerging field at the interface of algebraic topology, statistics, and computer science. The key rationale in TDA is that the observed data are sampled from some metric space and the underlying unknown geometric structure of this space is lost because of sampling. TDA recovers the lost underlying topology.
We aim at adapting TDA algorithms to work on networks and overcoming the scalability issues that arise while working on large networks. In this talk, I will outline our three alternative approaches in applying Persistent Homology and TDAMapper based Topological Data Analysis algorithms to Blockchain networks.
September 21, 2020 at 3:00 p.m. via Zoom
Belin Tsinnajinnie – Santa Fe Community College
Diversity, Inclusion, Equity, and Settler Colonialism in Mathematics
Diversity and inclusion projects to support aspiring and current mathematicians from marginalized communities are often framed through some need for diversity and inclusion to advance the fields of mathematics. However, these conversations often fail to center the needs and goals of marginalized and underrepresented communities. In this talk, I discuss diversity and inclusion initiatives through lens that is informed by frameworks that identify mathematics education as settler colonialism and my own experiences of inclusion/exclusion. I call for a shift in the ways we can frame conversations of justice, equity, diversity, and inclusion in mathematics by asking: How do diversity and inclusion efforts in mathematics and mathematics education directly empower marginalized communities?
September 28, 2020 at 3:00 p.m. via Zoom
David Freund – Harvard University
Intersections of Virtual Multistrings
Introduced by L. Kauffman (1999) and explicitly studied by V. Turaev (2004), a virtual multistring generalizes the notion of a collection of closed curves on a smooth surface. Considered up to virtual homotopy, virtual multistrings can be described both combinatorially and geometrically. Addressing a classical problem, we use the interplay between these perspectives to compute the minimal number of crossings for families of virtual multistrings. Along the way, we correct a previous misconception about "minimal" representatives of virtual multistrings. Throughout the talk, we will provide an overview of all requisite background.
October 12, 2020 at 3:00 p.m. via Zoom
Matt Roscoe – University of Montana
The City of Numbers: Unique Prime Factorization, Manipulative Development and the Big Apple.
The Fundamental Theorem of Arithmetic (FTA) states that every positive integer greater than 1 can be represented in exactly one way apart from rearrangement as a product of one or more primes (Hardy and Wright, 1979, pp. 2-3). Unique prime factorization is the basis for the multiplicative structure of the integers and, as such, is important for students of arithmetic. Nevertheless, research has shown that many students and their teachers fail to appreciate the uniqueness statement of the FTA. For more than five years I have pursued a quest to help young learners better understand the importance of prime numbers. This quest led me to develop a manipulative and an accompanying lesson which was studied in classrooms of 4th grade students. I will share the manipulative and what we learned when it was put to use, and, then, tell the story of how its use has led to new questions and even a trip to New York City.