# Colloquia

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## Kimberly Ayers – Carroll College

### A Skew Product Model for Hybrid Dynamical Systems

A hybrid system is a dynamical system that exhibits both continuous and discrete behavior.  Think of a bouncing ball: when the ball is in the air, its velocity and position are changing smoothly in time.  However, when the ball hits the ground, it instantaneous reverses directions, exhibiting a discrete change in its velocity.  In this talk, we’ll examine a particular type of hybrid system from the vantage point of skew products.  We will begin by isolating the discrete behavior and examining its dynamics, exploring ideas such as chain recurrence and chaos.  We’ll then examine the behavior of the product space, and explore different classical recurrence concepts within the context of the dynamics on the product space and projections.

## Maria C. Quintana – University Carlos III of Madrid

### On the Structure of Linearizations for Rational Approximations of Nonlinear Eigenvalue Problems

Given a nonlinear matrix-valued function $$F(\lambda):\mathbb{C} \longrightarrow \mathbb{C}^{m\times m},$$ the Non-Linear Eigenvalue Problem (NLEP) consists in computing numbers $$\lambda \in \mathbb{C}$$ (eigenvalues) and non-zero vectors $$v \in \mathbb{C}^{m}$$ (eigenvectors) such that $$F(\lambda) v = 0,$$ under the regularity assumption $$\det(F(z)) \not\equiv 0$$. NLEPs arise in a variety of applications in Physics and Engineering. Nowadays, a useful approach to tackle them is based on Rational Approximation (RA), which leads to rational eigenvalue problems (REPs). Then, for solving REPs, linearizations of rational matrices are used, which is one of the most competitive methods for this task. In this talk we present the notion of local linearizations of rational matrices. A local linearization of a rational matrix $$R(\lambda)$$ preserves the zeros and poles of $$R(\lambda)$$ locally, that is, in subsets of $$\mathbb{C}$$ and/or at infinity. By using this new notion of linearization, we study the structure of linearizations constructed in the literature for RAs of NLEPs on a target set. Moreover, we provide very simple criteria to determine when a linear polynomial matrix is one of these linearizations.

## Andrew Gilbert – Pacific Northwest National Lab

### Inverse methods for material quantification using neutron and X-ray radiography

The need for methods to complete non-destructive quantitative inspections remains an active one for a variety of applications, e.g., baggage and cargo scanning. Recently, there is a renewed interest in inspection of objects with a beam of neutrons. The way that neutrons travel through an object is unique to X-rays, potentially offering useful information beyond what would be available using a typical X-ray radiograph. We will discuss recent work at the Pacific Northwest National Lab on quantification of material composition of an object using radiography as well as the inverse algorithms that underpin the method. New work involving combining data from a complex neutron interrogation system, a so-called neutron associated particle imager, will also be discussed. This system presents interesting possibilities for developing new methods to combine multiple observables of multiple particles into a cohesive and meaningful output.

## Jake Downs – University of Montana

### Inferring Holocene precipitation in west central Greenland using the Unscented Transform

We investigate changing precipitation patterns in Greenland during a period of elevated  temperatures called the Holocene thermal maximum (~10,000 - 6,000 years ago), exploiting a new chronology of ice sheet extent through the Holocene and an inverse modeling approach based on the unscented transform (UT) . The UT is applied to estimate changes in annual precipitation in order to reduce the misfit between modeled and observed ice sheet margin positions. We discuss the basic theory of the UT and show how it can be applied to the problem of time dependent data assimilation. Our results indicate that Holocene warming coincided with elevated precipitation, without which modeled retreat in west Greenland is more rapid than suggested by observations. This result highlights the important role that changing precipitation patterns had in controlling ice sheet extent during the Holocene.

## Neal Bushaw – Virginia Commonwealth University

### Small Percolating Sets

Bootstrap percolation is a simple monotone cellular automaton which was originally introduced by Chalupa, Leath and Reich as a model of ferromagnetism in the late 1970s.   In this model, we think of some vertices of a graph as being initially *infected*.  Even worse, this infection can spread -- an *uninfected* vertex with many infected neighbors will itself become infected.  Does the infection spread to the entire graph?  Will it stop?  Can it be efficiently quarantined?

In this talk, we give an introduction to bootstrap percolation and its history, highlighting a few major breakthroughs, classic problems, and important variants.  Then, we'll proceed to a simple sounding extremal problem -- which graphs have a small set of vertices whose infection will eventually spread to the entire graph?  This question was the topic of this summer's Graph Brain Project; we will describe several results which came out of the summer's work, as well as that workshop's somewhat unusual characteristics.

No background knowledge will be assumed -- the aim of this talk is to introduce you to the area and its problems, rather than to show complicated proofs.

## Vilma Mesa – University of Michigan

### Algebra Instruction at Community Colleges: Investigating the relationship with student outcomes

This project seeks to assess the connection between quality instruction in community college algebra courses and students' outcomes in these courses. It is a work in progress, so I will describe the context of the project, the research questions, and what we have found so far, and some connections to teaching undergraduate mathematics.

## Rick Brown – University of MontanaPhD Candidate

### Semivariogram Methods for Modeling Whittle-Matérn Priors in Bayesian Inverse Problems

In this talk, I will briefly present a mathematical description of the 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) in the isotropic case. I will show that this connection breaks down when the domain is finite due to the effect of boundary conditions and that it can be re-established using extended domains. I will then introduce the semivariogram method for obtaining point estimates of the Whittle-Matérn covariance parameters, which completely specifies the Gaussian prior needed for stabilizing the inverse problem. I will extend these results to the anisotropic case, where the correlation length in one direction is larger than another. Finally, I will consider the case where the the correction length is spatially dependent. Two-dimensional image examples will be presented throughout the talk.

## Diego Martinez – University Carlos III of Madrid

### Coarse Geometry and Inverse Semigroups

In this talk we will discuss, mainly, two seemingly disconnected notions in mathematics: coarse geometry and inverse semigroups.

Geometry often studies certain objects (such as sets or manifolds) equipped with a distance function. For instance, one classical problem would be to classify every compact manifold up to diffeomorphism. Coarse geometry shifts the point of view, and defines two sets to be coarse equivalent if they look the same from far away. In this way, for instance, a point and a sphere are indistinguishable from each other. Coarse geometry then studies properties that remain invariant under this weak equivalence relation, that is, properties of the space that only appear at infinity.

On the other hand, an inverse semigroup is a natural generalization of the notion of group, and is closely related to the idea of groupoid. Starting with one of these objects we will introduce how to construct a metric space, in the same fashion as the Cayley graph construction in the context of groups. We will then study its coarse structure, in particular its property A and its amenability. Time permitting, we will also relate these properties to analogue properties in some operator algebras.

## Jingjing Sun – University of MontanaDepartment 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

Available Dates:

January 13, 27
February 3, 10, 24
March 2, 9, 23, 30
April 6, 13, 20