Jarek Kwapisz – Montana State University

Markov Partitions: from Decimal Expansions to Nilmanifolds

A deterministic dynamical system (like the weather) can be chaotic: its long term behavior is so unpredictable (under tiniest of the initial data uncertainty) that it is best understood as a stochastic process. The proverbial rolling of a dice (repeatedly) is one such process; and replacing the dice with a finite state automaton yields ubiquitous Markov chains (or sofic shifts). Finding the right automaton for a given dynamical system can be tricky and involves partitioning the dynamical space into carefully designed (fractal) subsets called Markov boxes. For the flagship class of (uniformly hyperbolic) chaotic systems called Anosov maps, existence of such partitions has been known for over 40 years but their design methods lagged and only touched the simplest subclass, the maps of tori. We develop a construction applicable to all known Anosov maps (up to a covering). Its validation is the first ever Markov partition for Smale's famous 1967 example on a six dimensional nilmanifold, the simplest non-toral example. I will explain the key ideas and how the whole story is a far reaching extension of the concept of the ordinary decimal expansion.

February 1, 2021 at 4:10 p.m. via Zoom

Patricia Cahn – Smith College

Knot Theory and the Fourth Dimension

A mathematical knot is a positioning of a circle in space--imagine taking a piece of string, tangling it up somehow, and then glueing the ends together.  We'll learn how knots can be used to represent 3-dimensional manifolds, using a combinatorial invariant called a Fox coloring, and then discuss analogous constructions in dimension 4 using knotted surfaces.

February 8, 2021 at 3:00 p.m. via Zoom

Alexander Turbiner – ICN-UNAM, Mexico and Stony Brook University

Choreography in Nature
(towards theory of dancing curves, superintegrability) 

 By definition the choreography (dancing curve) is a closed trajectory on which \(n\) classical bodies move chasing each other without collisions. The first choreography (the so-called Remarkable Figure Eight) at zero angular momentum was discovered in physics  unexpectedly by C Moore (Santa Fe Institute) in 1993 for 3 equal masses in \(R^3\) Newtonian gravity numerically and independently in mathematics by Chenciner(Paris)-Montgomery(Santa Cruz) in 2000. At the moment about 6,000 choreographies in \(R^3\) Newtonian gravity are found, all numerically, for different \(n > 2\). All of them are represented by transcendental curves. It manifests the major discovery in celestial mechanics, next after H Poincare chaotic nature of \(n\) body problem.

Does exist (non)-Newtonian gravity for which dancing curve is known analytically? - Yes, a single example is known - it is the algebraic lemniscate by Jacob Bernoulli (1694) - and it will be the subject of the talk. Astonishingly, the Figure Eight trajectory in \(R^3\) Newtonian gravity coincides with algebraic lemniscate with \(\chi^2\) deviation \(\sim 10^{-7}\). Both choreographies admit any odd numbers of bodies on them. Both 3-body choreographies define maximally superintegrable trajectory with 7 constants of motion.

Talk will be accompanied by numerous animations.

February 22, 2021 at 3:00 p.m. via Zoom

Puck Rombach – University of Vermont

Expressing graphs as symmetric differences of cliques of the complete graph 

Any finite simple graph \(G = (V,E)\) can be represented by a collection \(\mathcal{C}\) of subsets of \(V\) such that \(uv\in E\) if and only if \(u\) and \(v\) appear together in an odd number of sets in \(\mathcal{C}\). We are interested in the minimum cardinality of such a collection. In this talk, we will discuss properties of this invariant and its close connection to the minimum rank problem. This talk will be accessible to students. Joint work with Calum Buchanan and Christopher Purcell.

March 1, 2021 at 3:00 p.m. via Zoom

Ling Xiao – University of Connecticut

Translating solitons in Euclidean space 

In this talk, I will present the following results. First, we prove any complete immersed two-sided mean convex translating soliton in \(R^3\) for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in \(R^3\) is the axisymmetric “bowl soliton”. Moreover, if the mean curvature of the translating soliton tends to zero at infinity, then this translating soliton can be represented as an entire graph and so it is the “bowl soliton”.  Finally, we classify all locally strictly convex graphical translating solitons defined over strip regions. This is a joint work with Joel Spruck.

March 8, 2021 at 3:00 p.m. via Zoom

Katherine Moore – Wake Forest University

Communities in Data 

Although clustering is a crucial component of human experience, there are relatively few methods which harness the richness of a social perspective. Here, we introduce a probabilistically-interpretable measure of local depth from which the cohesion between points can be obtained, via partitioning. The PaLD approach allows one to obtain graph-type community structure (with resulting clusters) in a holistic manner which accounts for varying density and is entirely free of extraneous inputs (e.g., number of communities, neighborhood size, optimization criteria, etc.). Some theoretical properties of cohesion are included. Joint work with Kenneth Berenhaut.

March 15, 2021 at 3:00 p.m. via Zoom

Therese-Marie Landry – UC Riverside

Metric Convergence of Spectral Triples on the Sierpinski Gasket and Other Fractal Curves 

Many important physical processes can be described by differential equations. The solutions of such equations are often formulated in terms of operators on smooth manifolds. A natural question is to determine whether differential structures defined on fractals can be realized as a metric limit of differential structures on their approximating finite graphs. One of the fundamental tools of noncommutative geometry is Alain Connes' spectral triple. Because spectral triples generalize differential structure, they open up promising avenues for extending analytic methods from mathematical physics to fractal spaces. The Gromov-Hausdorff distance is an important tool of Riemannian geometry, and building on the earlier work of Marc Rieffel, Frederic Latremoliere introduced a generalization of the Gromov-Hausdorff distance that was recently extended to spectral triples. The Sierpinski gasket can be viewed as a piecewise \(C^1\)-fractal curve, which is a class of fractals first formulated by Michel Lapidus and Jonathan Sarhad for their work on spectral triples that recover the geodesic distance on these spaces. In this talk, we will motivate and describe how their framework was adapted to our setting to yield approximation sequences suitable for metric approximation of spectral triples on piecewise \(C^1\)-fractal curves.

March 22, 2021 at 3:00 p.m. via Zoom

Rachael M. Norton – Fitchburg State University

Cartan subalgebras of non-principal twisted groupoid \(C^*\)-algebras 

An algebraic structure called a \(C^*\)-algebra can be built from a group or groupoid (a generalization of a group). In this talk we focus on a special subalgebra, called a Cartan subalgebra, of a particular type of groupoid \(C^*\)-algebra whose multiplication is twisted by a circle-valued \(2\)-cocycle. We identify sufficient conditions on a subgroupoid \(S \subset G\) so that the twisted \(C^*\)-algebra generated by \(S\) is a Cartan subalgebra of the twisted \(C^*\)-algebra generated by \(G\). We then describe (in terms of \(G\) and \(S\)) the so-called Weyl groupoid and twist that J. Renault defined in 2008, which give us a different groupoid model for our Cartan pair. Time permitting, we discuss ongoing efforts to apply these results to \(C^*\)-algebras of higher rank graphs. This is joint work with A. Duwenig, E. Gillaspy, S. Reznikoff, and S. Wright.

March 29, 2021 at 3:00 p.m. via Zoom

Katharine Shultis – Gonzaga University

Reducibility of parameter ideals in low powers of the maximal ideal 

It is well-known that a commutative, local, noetherian ring \(R\) is Gorenstein if and only if every parameter ideal of the ring is irreducible. A less well-known result due to Marley, Rogers, and Sakurai gives that there is an integer \(\ell\) such that \(R\) is Gorenstein if and only if there exists an irreducible parameter ideal in the \(\ell\)-th power of the maximal ideal. The proof of this result gives that \(\ell\) is the smallest integer such that a certain map of Ext modules is surjective after taking socles. Our work investigates upper bounds on this integer \(\ell\). In this talk, we'll focus on historical context and examples where the ring \(R\) is a quotient of a power series ring.

April 5, 2021 at 3:00 p.m. via Zoom

Ted Owen – University of Montana PhD Candidate
Doctoral Defense

Variance Approximation Approaches For The Local Pivotal Method

The problem of estimating the variance of the Horvitz--Thompson estimator of the population total when selecting a sample with unequal inclusion probabilities using the local pivotal method is discussed and explored. Samples are selected using unequal inclusion probabilities so that the estimates using the Horvitz--Thompson estimator will have smaller variance than for simple random samples. The local pivotal method is one sampling method which can select samples with unequal inclusion probability without replacement. The local pivotal method also balances on other available auxiliary information so that the variability in estimates can be reduced further.

A promising variance estimator, bootstrap subsampling, which combines bootstrapping with rescaling to produce estimates of the variance is described and developed. This new variance estimator is compared to other estimators such as naive bootstrapping, the jackknife, the local neighborhood variance estimator of Stevens and Olsen, and the nearest neighbor estimator proposed by Grafstrom.

For five example populations, we compare the performance of the variance estimators. The local neighborhood variance estimator performs best where it is appropriate. The nearest neighbor estimator performs second best and is more widely applicable. The bootstrap subsample variance estimator tends to underestimate the variance.

April 9, 2021 at 2:00 p.m. via Zoom

William Duncan – Montana State University

Equilibria in Networks with Steep Sigmoidal Nonlinearities

In differential equation models of gene regulatory networks, interactions between genes are often modeled by nonlinear sigmoidal functions. If these sigmoidal functions are replaced by piecewise constant or switching functions, the dynamics of the resulting system are completely determined by a finite number of inequalities between parameters and can be computed efficiently. The expectation is that the equilibria of the switching system correspond to equilibria of steep sigmoidal systems. However, the sigmoidal system will have additional equilibria not present in the switching system. In this talk, I will discuss results which show that all equilibria of steep sigmoidal systems can be determined from the switching system inequalities. In the case of ramp systems, a subclass of sigmoid systems, I discuss bifurcations of these equilibria as the steepness of the functions decrease and give explicit bounds on their slopes that guarantee the equilibria maintain their stability and numbers that are predicted by the switching system.  

April 12, 2021 at 3:00 p.m. via Zoom

Mohsen Tabibian – University of Montana PhD Candidate

Weighted Neural Networks for Predicting Daily Covid-19 Death Counts 

Covid-19 is a highly contagious virus that has almost frozen the world. This virus is more likely to be moved from one county to adjacent counties. Accurate predictions of disease trajectory in the near term are critical. Thus, spatial contagion is an important aspect of the Covid-19 spread and the death counts attribute to Covid-19 in the adjacent counties are spatially correlated. The task poses the challenge that the dataset is spatially and temporally correlated. Artificial neural networks (ANNs) are presently the single best class of predictive functions but cannot handle this kind of dataset. To overcome this and attempt to exploit information induced by spatial and temporal dependencies, we modified ANNs by adding observation weights to the conventional neural networks referred to as a weighted neural network. The performance of the model is quantified by the mean absolute error.

April 13, 2021 at 3:00 p.m. via Zoom

Jen Berg – Bucknell University

The geometric nature of Diophantine equations 

Does there exist a box such that the distance between any two of its corners is a rational number? Which integers can be expressed as the sum of three cubes? These questions and many others can be reframed as Diophantine problems, that is, questions of existence of rational or integer solutions to polynomial equations. Each such Diophantine problem has a geometric manifestation called an algebraic variety whose properties often shed light on why these questions don't have elementary answers. In this talk I'll give an introduction to the guiding principle that geometry influences arithmetic, and describe work on the existence of (and obstructions to) rational solutions to equations that define algebraic surfaces. 

April 19, 2021 at 3:00 p.m. via Zoom

Mohsen Tabibian – University of Montana PhD Candidate
Doctoral Defense

Extending Bootstrap Aggregation of Neural Networks for Prediction with an Application to COVID-19 Forecasting 

The aim of the research discussed herein to improve the forecasting accuracy of artificial neural networks. The focus on forecasting for epidemiological purposes, and in particular, the problem of predicting case and death counts from seven to n days in the future for a spatially contiguous region such as a county. The task poses several challenges: the data are both spatially and temporally correlated, and the data sets are quite small for the intended purpose. To overcome these challenges, the methods attempt to exploit information induced by spatial and temporal dependencies. More importantly, we have developed a fusion of artificial neural networks and bootstrap methods.

Bootstrap aggregation (bagging) is an ensemble technique used for (1) reduction prediction function variance and a concurrent improvement in the predictive accuracy (2) construction of prediction intervals. Note that random forests extend bagging by sampling predictor variables in addition to sample observations with the result of often dramatic improvement in accuracy compared to the base prediction function (binary recursive trees). The method developed herein resembles random forests though there are important differences. To improve predictive accuracy and to construct prediction intervals, we apply the bagging mechanism to create a collection of fitted neural networks from a single data set. A forecast is the mean of the forecasts computed from each prediction function in the collection. We refer to this new approach as extended bagging.

Covid-19 is a highly contagious virus that has almost frozen the world and its economy. Accurate predictions of disease trajectory in the near term are critical for the efficient allocation of resources for combating the disease. Artificial neural networks are presently the single best class of predictive functions. Recurrent neural networks (RNNs) are a subclass that exploits temporal data structures; however, they are problematic in use and remain poorly understood by both researchers and practitioners. Hence, we propose a simple alternative referred to as Weighted Neural Network (WNN) and use this new neural network with extended bagging. To investigate and compare these innovations with standard neural network approaches, we apply the methods to Covid-19 datasets using counties as the spatial units.

The predictive functions forecast the number of deaths for two weeks in the future using four of the most populous counties in the United States: Los Angeles County in California, Cook County in Illinois, Harris County in Texas, and New York County in New York State. The performance of neural network-based models is quantified by the mean absolute error (MAE) between predicted and observed numbers of deaths. In the majority of cases, the extended bagging of GRU and WNN models yield highly informative predictions and outperformed the other prediction models. Our proposed technique, extended bagging improved the results of both GRU and WNN models. The assessment of constructed prediction intervals is measured by coverage probability (CP) which is the percentage of target values covered by the constructed prediction intervals. The extended bagging GRU models performed best for building prediction intervals with a CP of 84.2%. Our results show that extended bagging enhanced prediction accuracy, extended bagging of GRU can be exploited for pandemic prediction for better planning and management. These methods can be applied to a wide variety of other situations from Ebola outbreak mitigation to intraand inter-day stock price forecasting.

April 23, 2021 at 3:00 p.m. via Zoom

Jeff Boersema – Seattle University

K-Theory: Algebraic Topology and Non-commutative Algebraic Topology 

This will be a gentle introduction to K-theory, first in the context of topological spaces and then in the context of operator algebras. We will discuss both real K-theory and complex K-theory, and the interplay between them. At the end, I will present two important classification theorems for real C*-algebras using K-theory. I will assume no prior knowledge of K-theory.

September 13, 2021 at 3:00 p.m. in Math 103

Leonid Hanin – Idaho State University 

Mathematical Discovery of Natural Laws in Biomedical Sciences with Application to Metastasis

Mathematical modeling of systemic biomedical processes faces two principal challenges: (1) enormous complexity of these processes and (2) variability and heterogeneity of individual characteristics of biological systems and organisms. As a result, in the grand scheme of things, mathematical models have played so far an auxiliary role in biomedical sciences. I propose a new methodology of mathematical modeling that would allow mathematics to give, in certain cases, definitive answers to important biomedical questions that elude empirical resolution. The new methodology is based on two ideas: (1) to employ mathematical models that are so general and flexible that they can account for many possible mechanisms, both known and unknown, of biomedical processes of interest; (2) to find those model parameters whose optimal values are independent of observations. These universal parameter values may reveal general regularities in biomedical processes (that can be called natural laws). Existence of such universal parameters presupposes that the model does not meet the conditions required for the consistency of the maximum likelihood estimator.

I illustrate this approach with the discovery of a natural law governing cancer metastasis. Specifically, I will show that under minimal mathematical and biological assumptions the likelihood-maximizing scenario of metastatic cancer progression is always the same: complete suppression of metastatic growth before primary tumor resection followed by an abrupt growth acceleration after surgery. This scenario is widely observed in clinical practice, represents a common knowledge among veterinarians, and is supported by a wealth of experimental studies on animals and clinical observations accumulated over the last 115 years. Furthermore, several biological mechanisms, both hypothetical and experimentally verified, have been proposed that could explain this natural law. The above scenario does not preclude other possibilities that are also observed in clinical practice. In particular, metastases may surface before surgery or may remain dormant thereafter. 

September 27, 2021 at 3:00 p.m. in Math 305

Leonid Kalachev – University of Montana

Classical infectious disease modeling paradigms shifted by the SARS-CoV-2 pandemic 

Classical infectious disease models during epidemics have widespread usage, from predicting the probability of new infections to developing vaccination plans for informing policy decisions and public health responses. However, it is important to correctly classify reported data and understand how this impacts estimation of model parameters. The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic has provided an abundant amount of data that allows for thorough testing of disease modeling assumptions, as well as how we think about classical infectious disease modeling assumptions. We use simulations to demonstrate the minimal data (infected, active, quarantined, and recovered) needed for collection and reporting that are sufficient for reliable model parameter identification and prediction accuracy. Using a classical example of influenza epidemics in an England boarding school, we show that the Susceptible-Infected-Quarantined-Recovered model is more appropriate than the commonly employed Susceptible-Infected-Recovered model. We demonstrate the role of misclassification and the importance of correctly classifying reported data to the proper compartment in a COVID-19 disease model and implications of using “right” data in the “wrong” model. The role of misclassification and the importance of correctly classifying reported data will have downstream impacts on predictions of number of infections, as well as minimal vaccination requirements.

October 11, 2021 at 3:00 p.m. in Math 305

Aaron Luttman – Senior Technical Advisor, Pacific Northwest National Laboratory

Mathematics Research for Security and Science in the US National Laboratories 

The US Department of Energy (DOE) maintains 17 national laboratories, which employ hundreds of mathematicians, working on research from modeling of coastal ecosystems, to advanced energy solutions, to nuclear security. We build computational codes for modeling and simulation; we develop algorithms for analyzing scientific data; and we’re beginning to play an essential role in the advance of machine learning and artificial intelligence (AI/ML). In this presentation, we’ll highlight a few mathematical research programs that are currently underway in the DOE complex, including modeling of nuclear fusion devices, design of AI/ML models for characterizing the structure and performance of materials like uranium, and some of the graduate level mathematics that underwrites why some deep neural networks actually work. Each of these topics has important open problems, and we’ll also discuss how to get involved in research and get jobs in our community, as well as some of the details of the national lab ecosystem and work environments. 

October 18, 2021 at 3:00 p.m. in Math 305

Faculty Evaluation Committee meeting, 3:00 p.m. in Math 103

Chad Topaz – Williams College, co-founder of the Institute for the Quantitative Study of Inclusion, Diversity, and Equity

Mathematical and Computational Approaches to Social Justice 

Civil rights leader, educator, and investigative journalist Ida B. Wells said that "the way to right wrongs is to shine the light of truth upon them." This talk will demonstrate how mathematical and computational approaches can shine a light on social injustices and help build solutions to remedy them. We will present quantitative social justice projects on topics ranging from diversity in art museums to equity in criminal sentencing to affirmative action, health care access, and other fields. The tools engaged include crowdsourcing, clustering, hypothesis testing, statistical modeling, Markov chains, data visualization, and more. I hope that this talk leaves you informed about the breadth of social justice applications that one can tackle using quantitative tools in careful collaboration with other scholars and activists.

November 1, 2021 at 3:00 p.m. in Math 305

Katie Oliveras – Seattle University

Measuring Water Waves: Using Pressure to Reconstruct Wave Profiles 

How does one measure waves in the ocean?  And how accurate are these methods at capturing the height and shape of the wave?  In this talk, I will discuss an inverse problem related to measuring water-waves using pressure sensors placed inside the fluid. We will begin by introducing the partial differential equations that describe fluid motion.  Then, using a non-local formulation of the water-wave problem, we will see how to directly determine the pressure below both traveling-wave and time-dependent solutions of Euler's equations.

This method requires the numerical solution of a nonlinear, nonlocal equation relating the pressure and the surface elevation which is obtained without approximation. From this formulation, a variety of different asymptotic formulas are derived and are compared with both numerical data and physical experiments. 

November 8, 2021 at 3:00 p.m. in Math 305

Ryan Grady –  Montana State University

Persistence in Data, Cosheaves, and K-Theory 

TDA is a family of techniques which uses topological structures to analyze data. I'll begin by introducing some aspects of TDA; in particular, I'll discuss persistence modules. Next, I'll describe a reformulation of persistence in terms of cosheaves on a stratification of parameter spaces. Finally, I'll indicate the utility of the aforementioned translation by computing persistent invariants via cosheaves, e.g., in terms of algebraic K-theory. 

November 15, 2021 at 3:00 p.m. in Math 305

Anna Halfpap – University of Montana

Positive co-degree problems for 3-graphs 

How many lines can you place between n points before you are guaranteed to find a set of 4 points between which all 6 possible lines are present? How many 3-sets can you take from the first n integers before you are guaranteed to find a set of 4 integers among which all 4 possible 3-sets have been chosen? The first question is a basic problem in extremal graph theory. The second is also an extremal question -- this time to do with 3-graphs, a generalization of "normal" graphs in which edges contain 3 points instead of 2.

Extremal problems for hypergraphs (of which 3-graphs are a special type) are rich, interesting, and often very difficult. In this talk, we will introduce a new type of extremal hypergraph problem, that of maximizing the positive co-degree of a hypergraph subject to some forbidden sub-hypergraph. We will describe the connections between this question and other extremal questions on hypergraphs, and will present some exact results. Joint work with Cory Palmer and Nathan Lemons. 

November 22, 2021 at 3:00 p.m. in Math 305

Chad Topaz – Williams College, co-founder of the Institute for the Quantitative Study of Inclusion, Diversity, and Equity

Mathematical and Computational Approaches to Social Justice, part 2

Continued from Nov. 1^st, a viewing of the recorded talk by Chad Topez.  Expect to have Zoom working this time, so attend in Math 103 or by Zoom.  Discussion with colleagues to follow.

November 29, 2021 at 3:00 p.m. in Math 103

Jasper Weinburd – Harvey Mudd College

Collective Behavior in Locust Swarms from Differential Equations to Data 

Locusts are devastating pests that infest and destroy crops. Locusts forage and migrate in large swarms which exhibit distinctive shapes that improve efficiency on the group level, a phenomenon known as collective behavior. One of the difficulties in understanding and preventing these collective behaviors has been a lack of biological data for individual interactions between locusts. In this talk, I’ll first describe mathematical models for these phenomena on both the collective and individual levels. I’ll then discuss a collaboration with students at Harvey Mudd College using field data derived from video footage of locust swarms. We digitized nearly 20,000 locust trajectories and revealed individual behaviors that depend on a locust’s motion and the relative position of its nearby neighbors. Finally, I will illustrate the challenges and potential benefits of incorporating these field observations into our models of locust swarms.

December 6, 2021 at 3:00 p.m. in Math 305