Colloquia

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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.

Leonid Kalachev – University of Montana

October 11, 2021 at 3:00 p.m. in Math 103

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

Emily Stone – University of Montana

Data Science meets Public Health Data in Montana: My adventures with seasonal and SARS-CoV-2 flu from Montana counties.

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

Available Dates:
September 20
October 4, 18
November 1, 15, 22
December 6

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.

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.

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.

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.

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.

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.

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.

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.

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.

Ted Owen – University of Montana PhD CandidateDoctoral 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.

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.

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.

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.

Mohsen Tabibian – University of Montana PhD CandidateDoctoral 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.