Joey Lippert – University of Montana, PhD Candidate

Classifying Invariants of k-Graph C*-Algebras 

This talk will introduce higher-rank graphs and their C*-algebras. The goal will be to discuss the problem of classification and further discuss methods of computing classifying invariants. Highlights include tools from homological algebra called spectral sequences as well as combinatorial data such as "graph moves". These rather complicated objects take on some intuitive properties when viewed through the lens of C*-algebras which leads to surprising results for k-graphs. 

January 23, 2023 at 3:00 p.m. Math 103

Ian Derickson – University of Montana, PhD Candidate

Modeling Covid-19 in Montana

With the advent of Covid-19 upon the world and the pandemic that set in across the globe, a renewed interest in epidemiological modelling surfaced throughout the world as people sought answers. The attentive reader will observe the cessation of media interest in epidemiological modelling over the past couple of years. In this talk we will discuss the methodologies of epidemiological modelling with particular focus on the Susceptible-Infected-Recovered (SIR) model paradigm. Common issues in research today as well as the ways the SIR model can be modified to suit modelling needs will be discussed as well as how certain patterns can be seen in Covid data in the alpha wave for Montana.

January 30, 2023 at 3:00 p.m. Math 103

Anna Halfpap – University of Montana, PhD Candidate

Motivating Generalized Extremal Problems

In this talk, we discuss the history of extremal graph theory, in particular motivating several generalizations of classical extremal questions which have become popular in recent years. The foundational question of extremal graph theory is to determine the maximum number of edges in an n vertex graph which does not contain some "forbidden" graph F as a subgraph. After a century of work, this question has been well studied (though is still not well understood in some cases), revealing a rich underlying theory. But does the extremal question still make sense in a "colorful" graph setting? What about for hypergraphs? What if, instead of seeking to maximize the number of edges in an F-free graph, we wish to have an abundance of copies of some other subgraph H? We shall see that all of these directions naturally grow from the classical extremal question, and that we can often find "generalized" versions of "classical" theorems, e.g., supersaturation and stability results. However, we shall also highlight generalized phenomena which diverge from classical expectations. This talk assumes no background in extremal graph theory and is intended primarily to introduce history and motivation for generalized extremal questions. 

February 6, 2023 at 3:00 p.m. Math 103

Claire Seibold – FYR Diagnostics

An introduction to RNAseq 

High throughput sequencing data from Next Generation Sequencing (NGS) has enabled researchers to study the entire genome and the entire transcriptome with hypothesis-free experimental designs. In this talk, you will receive an overview of RNA sequencing, which leverages NGS to detect and quantify RNA in a biological sample. We will touch on lab preparation of samples (“libraries”), discover what happens inside a sequencer, and then learn about the different stages of analysis that occur afterward. Extra time will be spent on common practices to measure and reduce errors in the data, quantify alignment and mutations, and dive deeper into some more advanced analysis. There will be real-world examples of skills needed and challenges that face researchers working as bioinformaticians in the world of NGS, including anecdotes of my own experiences with a startup that recently purchased one of the newer sequencers on the market: the AVITI from Element Biosciences. 

March 13, 2023 at 3:00 p.m. Math 103

Blair Davey – Montana State University

A resolution to Landis' conjecture in the plane 

In the late 1960s, E.M. Landis made the following conjecture:  If u and V are bounded functions, and u is a solution to the Schrodinger equation \Delta u - V u = 0 in Euclidean space that decays faster than linear exponential, then u must be identically zero. In 1992, V. Z. Meshkov disproved this conjecture by constructing bounded, complex-valued functions u and V that solve the Schrodinger equation in the plane and decay at a much faster rate. The examples of Meshkov were accompanied by qualitative unique continuation estimates for solutions in any dimension. Meshkov's estimates were quantified in 2005 by J. Bourgain and C. Kenig. These results, and the generalizations that followed, have led to a fairly complete understanding of these unique continuation properties in the complex-valued setting. However, Landis' conjecture remains open in the real-valued setting. We will discuss a recent result of A. Logunov, E. Malinnikova, N. Nadirashvili, and F. Nazarov that resolves the real-valued version of Landis' conjecture in the plane. 

March 27, 2023 at 3:00 p.m. Math 103

Fred Peck – University of Montana

Bringing the outside in: Building with and strengthening community-based problem-solving practices in math classrooms 

Disrupting deficit ideologies around rural education, we suggest that local practices in rural communities are rich resources for learning mathematics. Through ethnographic fieldwork in rural communities in the U.S.A., we identified five community-based problem-solving practices: Resourcefulness, Care, Self-reliance, Use practical wisdom, and Try something and adjust. We describe these practices and discuss how they can be brought into mathematics classrooms, as assets that support mathematics learning. 

April 10, 2023 at 3:00 p.m. Math 103

Abdulla Al Mamun – Gonzaga University

Longitudinal Data Regression Analysis Using Semiparametric Modelling 

In the longitudinal data regression model, performing joint analysis of mean and covariance parameters simultaneously and accounting for the correlations improve statistical inference of the mean of interest. Zhang, Leng and Tang (2015) propose joint parametric modelling of the means, variances, and correlations by decomposing the correlation matrix via hyperspherical coordinates. In this presentation, I will talk about the methodology for joint estimation in semiparametric modelling of the means, the variances, and by decomposing the correlation matrix via hyperspherical coordinates. The method is illustrated to analyze one health data set. 

April 17, 2023 at 3:00 p.m. Math 103

Eric Hogle – Gonzaga University

Using Kronholm's Ideal to Compute Bredon Cohomology 

We are interested in computing the $RO(C_2)$-graded Bredon cohomology of equivariant spaces which can be constructed as $\text{Rep}(C_2)$-complexes. Although a theorem of Kronholm dictates that this cohomology must be free, the pages of the spectral sequences converging to these cohomologies are not free, nor practial to compute.

However certain of these spaces, such as the Grassmannian manifold of $k$-planes inside of a given $C_2$-representation, have multiple constructions and so multiple spectral sequences which must converge to the same place. Using these multiple angles of attack and a generalization of the Poincare polynomial, we present an algorithm to advance these calculations in the much friendlier setting of a polynomial ring. 

April 24, 2023 at 3:00 p.m. Math 103

Jennifer Brooks – Brigham Young University

Zeros of Complex-Valued Harmonic Functions 

BYU has a large number of students involved in mentored research. This past year, we had 362 math majors, and 119 of them were involved in research. Although it is easy to enumerate the benefits to students of doing such research, mentoring these students can be a challenge.  In particular, for many of us, our own research is not accessible to undergraduates.  In the past 4 years, I have added a new thread to my research program that has led to research projects for 15 undergraduates and 4 MS students. Students can begin with relatively little background, but there are still accessible open problems of interest to the larger community of complex analysts.

Specifically, we study the zeros of complex-valued harmonic polynomials. The Fundamental Theorem of Algebra states that for a polynomial $f$ in one complex variable of degree $n \geq 1$, the number of zeros (counting multiplicity) is exactly $n$. However, if $f=h+\overline{g}$ is the sum of an analytic polynomial and the conjugate of an analytic polynomial, the theorem no longer applies.  Strange things can happen; the number of zeros need not equal the degree and the number of zeros can vary with the coefficients.

In this talk, I give an overview of this research, highlighting the contributions of my students. I will also share ideas for making undergraduate research work.

May 1, 2023 at 3:00 p.m. Math 103

Jordan Broussard – Whitworth University

Template Arrays and Two-Dimensional Recurrence Relations 

In a two-dimensional recurrence relation, there is an underlying structure composed of the two-dimensional sequences (arrays) in which the set of indices is extended from an ordered pair where each entry comes from the set of natural numbers to an ordered pair where each entry comes from the set of integers. The recurrences we will look at have coefficients that come from a field. In this talk, we will look at a set of initial conditions sufficient to build a uniquely-determined array from a given recurrence and initial conditions, as well as look at how to construct a Schauder basis using elementary arrays for the set of arrays. 

May 8, 2023 at 3:00 p.m. Math 103

Leonid Kalachev – University of Montana

A simple modification to the classical SIR model to estimate the proportion of under-reported infections using case studies in flu and COVID-19 

Under-reporting and, thus, uncertainty around the true incidence of health events is common in all public health reporting systems. While the problem of under-reporting is acknowledged in epidemiology, the guidance and methods available for assessing and correcting the resulting bias are obscure. We present a simple method for the Susceptible – Infected – Removed (SIR) model for estimating the fraction or proportion of reported infection cases. The suggested modification involves rescaling of the classical SIR model producing its mathematically equivalent version with explicit dependence on the reporting parameter (true proportion of cases reported). We show how this rescaling parameter can be estimated from the data along with the other model parameters. The proposed method is then illustrated using simulated data with known disease cases and applied to two empirical reported data sets to estimate the fraction of reported cases in Missoula County, Montana, USA, using: (1) flu data for 2016 – 2017 and (2) COVID-19 data for fall of 2020. We demonstrate with the simulated and COVID-19 data that when most of the disease cases are presumed reported, the value of this additional parameter is close or equal to one, and the original SIR model is appropriate for the data analysis. Conversely, the flu example shows that the reporting parameter is close to zero, and the original SIR model is not accurately estimating the usual rate parameters. This research demonstrates the role of under-reporting of disease data and the importance of accounting for under-reporting when modeling simulated, endemic, and pandemic disease data. The role of correctly reporting the “true” number of disease cases will have downstream impacts on predictions of disease dynamics. A simple parameter adjustment to the SIR modeling framework can help alleviate bias and uncertainty around crucial epidemiological metrics (basic disease reproduction number) and public health decision making.

September 18, 2023 at 3:00 p.m. Math 103

Benjamin Moldstad – Montana State University

Circle Actions and Stratifications

An action of the circle group T on a presentable stable infinity category V was thought to be data of a differential map on V. In the category of chain complexes, this happens to the case. In this talk, I will talk about Circle Actions on a general V, and how we can use stratifications to solve this problem.

September 25, 2023 at 3:00 p.m. Math 103

Tyler Seacrest – Montana Western

Exotic number bases with application to combinatorics 

While it is well known that our traditional base 10 number system can be generalized to other bases such as binary or hexadecimal, such generalizations can be taken farther and be far more useful than many realize.  For example, they can give new insights into Fibonacci numbers, solve problems from combinatorics, solve and generalize the game of Nim, compute digits of pi, and even create fractals.

In this talk we'll give a smattering of results we've happened across over the last couple of years as we've pursued one question:  how far can you push the idea of a number base and still have the fundamental property where every number has exactly one representation?

October 9, 2023 at 3:00 p.m. Math 103

Tracy Payne – Idaho State University

Generalized Voronoi Diagrams and Lie Sphere Geometry 

The classical Voronoi diagram for a set S of points in the Euclidean plane is the subdivision of the plane into Voronoi cells, one for each point in the set.  The Voronoi cell for a point p is the set of points in the plane that have p as the closest point in S. This notion is so fundamental that it arises in a multitude of contexts, both in theoretical mathematics and in the real world. 

The notion of Voronoi diagram may be expanded by changing the underlying geometry, by allowing the sites to be sets rather than points, by weighting sites, by subdividing the domain based on farthest point rather than closest point, or by subdividing the domain based on which k sites are closest.

"Lie sphere geometry” can be used to describe many such "generalized Voronoi diagrams."

In this talk, we give overviews of generalized Voronoi diagrams and Lie sphere geometry, and we describe how they are related.

October 16, 2023 at 3:00 p.m. Math 103

Qian Mao – Whitworth University

Network Traffic Classification Using Deep Learning Neural Networks 

Network traffic can be classified into various types, i.e., web browsing, email, chat, streaming, file transfer, VoIP, TraP2P, etc. This technology has been extensively used in Quality-of-Service control, billing, malware detection, etc. The current network traffic classification methods basically have an accuracy of about 87%, and most of them rely on human knowledge to define specific patterns for classification. Our goal of this research includes 1) to increase the classification accuracy and 2) to increase the classification efficiency.

To classify network traffic accurately and automatedly, we are using neural networks, especially Deep Learning (DL) networks, in our research. Specifically, we have focused on three issues. First, neural networks require huge amount of training data. There are some public databases that offer network traffic data, but they are either not suitable for traffic classification or being too small. Therefore, building a comprehensive database for DL network training is our first focus. Another challenge is to determine what network traffic information should be used. The information carried by the raw network traffic data is extremely huge. We have designed various approaches to reduce the information amount while keep the classification accuracy relatively high. The third challenge is the neural networks architecture design and implementation, including DL architecture, the algorithms of each DL layer, the number of hidden layers, the number of neurons in each layer, etc. Through these approaches, we have achieved an accuracy of 92% for feature-based classification and 99% for raw-data-based classification. 

October 30, 2023 at 3:00 p.m. Math 103

Allechar Serrano Lopez – Montana State University

Counting number fields 

A guiding question in arithmetic statistics is: Given a degree $n$ and a Galois group $G$ in $S_n$, how does the count of number fields of degree $n$ whose normal closure has Galois group $G$ grow as their discriminants tend to infinity? In this talk, I will give an overview of the history and development of number field asymptotics and we discuss how we can obtain a count for dihedral quartic extensions over a fixed number field. 

November 6, 2023 at 3:00 p.m. Math 103

Breschine Cummins – Montana State University

Matching data from multiple experiments to a genetic network model

Modeling biological systems holds great promise for speeding up the rate of discovery in systems biology by predicting experimental outcomes and suggesting targeted interventions. However, this process is dogged by an identifiability issue, in which network models and their parameters are not sufficiently constrained by coarse and noisy data to ensure unique solutions. In this work, we evaluated the capability of a simplified yeast cell-cycle network model to reproduce multiple observed transcriptomic behaviors under genomic mutations. We matched time-series data from both cycling and checkpoint arrested cells to model predictions using an asynchronous multi-level Boolean approach. We showed that this single network model, despite its simplicity, is capable of exhibiting dynamical behavior similar to the datasets in most cases, and we demonstrated the drop in severity of the identifiability issue that results from matching multiple datasets. 

November 13, 2023 at 3:00 p.m. Math 103

Ryan Wood – University of Montana, PhD Candidate

A generalization of diversity for intersecting families

In this talk, we will introduce the flower base method toward solving a problem involving what we call “C-weighted diversity”. Beginning with a few fundamental problems of extremal set theory, most notably that of Erd\H{o}s-Ko-Rado, we hope to contextualize where this current problem fits in. Following this, we show the flower base in action. Though the talk will be technical at times, there will be fun pictures along the way.

November 27, 2023 at 3:00 p.m. Math 103

Zedong Peng – Computer Science

Examining Metamorphic Testing with Requirements Knowledge in Practical Settings 

Given a test input, not knowing the expected output of the software under test (SUT) is called the oracle problem. An emerging method of alleviating the oracle problem is metamorphic testing (MT). Rather than focusing on the correctness of output from a single execution of the SUT, MT exploits metamorphic relations (MRs) as derived oracles for checking the functional correctness of the code. Although researchers have argued that MT can be a simple and effective technique to help software developers, little is known about the actual cost of constructing MRs in real-world software and the relationship between MT and the already well-adopted method in software development. This talk will outline a range of methods for assessing the effectiveness of MT in the context of software development. 

December 4, 2023 at 3:00 p.m. Math 103