Colloquia

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Spring 2024

Ryan James, PhD – Reddit

Multivariate Information Theory: Difficulties and Recent Progress

Information theory has been widely applied within the sciences. Much of the appeal comes from it providing a substraight-agnostic method of quantifying the interactions within a system. In the bivariate case, many measures have reasonably robust interpretations stemming from their status as solutions to operational problems. In the multivariate case, however, most measures are extensions of their bivariate cousin, but lack the operational interpretation. In this talk, we review many multivariate information measures that have been employed in data analysis, their interpretational difficulties, and some recent progress in producing interpretable, robust multivariate measures.

February 12, 2024 at 3:00 p.m. Math 103

Anna Halfpap – Iowa State University

Proper Rainbow Saturation Numbers 

A graph G is F-saturated if G does not contain F as a subgraph, and is edge-maximal with regards to this property. That is, for any edge e that G is "missing", the graph G + e obtained by adding e to G contains one or more subgraphs isomorphic to F. The study of F-saturated graphs informs the core questions in extremal graph theory. The extremal number ex(n, F) is the maximum number of edges possible in an n-vertex graph which does not contain F -- that is, ex(n, F) is the maximum number of edges in an n-vertex, F-saturated graph. On the other hand, the saturation number sat(n, F) is the minimum number of edges in an n-vertex, F-saturated graph. Both ex(n, F) and sat(n, F) are extensively studied, and naturally generalize to a variety of settings.

In this talk, we will discuss a variation on saturation numbers which arises in an edge-colored setting. An edge coloring of a graph is an assignment of colors (typically, some subset of the positive integers) to the graph's edges. We say that an edge coloring is proper if any two edges which share an endpoint receive distinct colors, and is rainbow if any two edges receive distinct colors. In 2007, Keevash, Mubayi, Sudakov, and Verstraete introduced the rainbow extremal number, which combines extremal graph theory questions with edge coloring. The rainbow extremal number of F is the maximum number of edges in a graph G such that, under some proper edge-coloring, G does not contain a rainbow copy of F. Rainbow extremal numbers have received substantial attention over the last fifteen years, but the corresponding rainbow saturation question was only posed very recently. In this talk, we will introduce and motivate rainbow F-saturated graphs and share some new results on rainbow saturation for cycles. 

March 11, 2024 at 3:00 p.m. Math 103

Kelvin Rivera-Lopez – Gonzaga

An algebraic approach to scaling limits of up-down chains

An up-down chain is a Markov chain in which each transition can be decomposed into a growth step followed by a reduction step. In general, these two steps can be unrelated, but if they satisfy a natural commutation relation, the up-down chain turns out to be particularly amenable to analysis.

In this talk, we will present a general framework for analyzing these special up-down chains. This approach will mainly be algebraic but will lead to convergence results. If time permits, we will discuss an example in the context of permutations and permutons.

Based on joint work with Valentin Féray.

March 25, 2024 at 3:00 p.m. Math 103

Van Magnan – University of Montana, PhD Candidate

 

 

 

April 1, 2024 at 3:00 p.m. Math 103

Regina Souza – University of Montana

Active Learning in College Algebra, College Trig and Precalc: A Demonstration 

About one and a half years ago, with the support of Rick Darnell, Fred Peck, and Josh Herring, I started exploring the 'Building Thinking Classrooms' practices (by Peter Liljedahl) in my own classroom. Since last semester, all sections of College Algebra, Trig, and Precalc have been using this approach. The learning assistants program has been vital to this implementation: the program trains and supports these wonderful undergraduate students, who have been very important in creating activities, fostering community, and energizing both students and instructors.

This will be an 'untalk', designed for you to experience a few of the practices, hear from the instructors and learning assistants who have been implementing them, and form your own opinion about the pros and cons.

April 8, 2024 at 3:00 p.m. Math 103

Lucia Williams – Computer Science

 

 

April 15, 2024 at 3:00 p.m. Math 103

Ellen Weld – Sam Houston State University

A Gentle and Brief Introduction to \(L^p\)-Operator Algebras 

Originally defined by Herz in the 1970s, \(L^p\)-operator algebras are Banach algebras which can be isometrically represented on an \(L^p\)-space for \(p\in [1,\infty)\) and in many ways generalize the notion of operator algebras. However, \(L^p\)-operator algebras did not receive wider interest until Phillips' 2013 paper computing the K-theory of analogs of Cuntz algebras after which a number of authors have explored what well known operator algebra properties do and do not extend to \(L^p\)-operator algebras.

In this talk, we will gently introduce \(L^p\)-operator algebras and provide motivating examples suitable for non-experts as well as discuss exciting results and trends in this area of research. Knowledge of operator algebras is not required. 

April 22, 2024 at 3:00 p.m. Math 103

José Martinez – University of Montana, PhD Candidate

 

 

 

April 29, 2024 at 3:00 p.m. Math 103

Available Dates

February 26
March 4

Fall 2023

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

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