2018 Colloquia

Ricela Feliciano-SemideiUniversity of Montana – PhD Candidate

Understanding Conditional Probability with the Monty Hall Problem

Conditional probability is an important concept that is widely used in social sciences, natural sciences, health sciences and business. There is a high need to understand how students learn conditional probability and to develop effective teaching interventions to improve their learning experiences. This study developed a teaching intervention that incorporated the Monty Hall problem into a game learning model. Through the implementation of the teaching intervention in a statistics introductory course at college level, researchers investigated students’ conceptions on conditional probability. The impact of the teaching module on students’ learning was examined through pretest and posttest. Researchers identified the Law of Large Numbers as a key concept required prior to learning conditional probability. Findings from the study had implications to improve the teaching and learning of conditional probability.

Michael Dorff – Brigham Young University

Analytic functions, harmonic functions, and minimal surfaces

Complex-valued harmonic mappings can be regarded as generalizations of analytic functions and are related to minimal surfaces which are beautiful geometric shapes with intriguing properties. In this talk we will provide background material about these harmonic mappings, discuss the relationship between them and minimal surfaces, present some new results, and pose a few open problems.

Vanni Noferini – University of Essex

Localization results for indefinite eigenvalue problems

Sylvester's law of inertia states that the number of positive, zero or negative eigenvalues of a matrix is invariant under congruence, and the same is true for pencils when at least one matrix is definite (and both are allowed to undergo independent congruences). Nothing was known thus far for indefinite pencils, and almost nothing for nonlinear problems. I will present new results of ours in this area, including inertia-based lower and upper bounds for the number of eigenvalues in a real interval. This talk is based on joint work with Yuji Nakatsukasa (University of Oxford).

Matthias Chung – Department of Mathematics at Virginia Tech.

Computational Challenges of Inverse Problems

Inverse problems are omnipresent in many scientific fields such as systems biology, engineering, medical imaging, and geophysics. The main challenges toward obtaining meaningful real-time solutions to large, data-intensive inverse problems are ill-posedness of the problem, large parameter dimensions, and/or complex model constraints. This talk discusses computational challenges of inverse problems by exploiting a combination of tools from applied linear algebra, parameter estimation and optimization, and statistics. For instance, for large scale ill-posed inverse problems, approximate solutions are computed using a regularization method that solves a nearby well-posed problem.  Oftentimes, the selection of a proper regularization parameter is the most critical and computationally intensive task and may hinder real-time computations of the solution. We present a new framework for solving ill-posed inverse problems by computing optimal regularized inverse matrices. We further discuss randomized Newton and randomized quasi-Newton approaches to efficiently solve large linear least-squares problems, where the very large data sets present a significant computational burden (e.g., the size may exceed computer memory or data are collected in real-time). In this framework, randomness is introduced as a means to overcome computational limitations, and probability distributions that can exploit structure and/or sparsity are considered. We will present numerical examples, from deblurring, tomography, and machine learning to illustrate the challenges and our proposed methods.

Zhuang Niu – University of Wyoming

The classification of C*-algebras

A C*-algebra is an algebra of bounded linear operators acting on a Hilbert space, closed under adjoint operation and closed under the norm topology. Prototype examples include the algebra of n by n matrices and the algebra of continuous function on a compact Hausdorff space, and in general, C*-algebras arise naturally in the studies of dynamical systems, mathematical physics, group theory, and representation theory, etc. In this talk, I will give an overview of the recent progress on the classification of C*-algebras using the K-theory information.

Dr. Edray H Goins – Purdue UniversityColloquium & Reception honoring Dr. Gloria Hewitt

Yes, Even You Can Bend It Like Beckham

In the 2002 film by Gurinder Chadha, character Jesminder 'Jess' Bhamra states "No one can cross a ball or bend it like Beckham'' in a reference to the international soccer star's ability to cause the ball to swerve. In 2010, French researchers Guillaume Dupeux, Anne Le Goff, David Quere and Christophe Clanet published a paper in the New Journal of Physics detailing both experimental and mathematical analyses of a spinning ball in a fluid to show that it must follow a spiral.

In this talk, we give an overview of their discussion by reviewing the Navier-Stokes equation in a Serret-Frenet coordinate system. This talk is dedicated to the memory of Angela Grant and her love of mathematics in sports.

Wend Werner – University of Muenster in Germany

Algebraic composition with more than two ingredients

Are we biologically biased towards thinking that products, sums and the like always require the contribution of exactly two agents in order to spawn a third one? We will survey some sample theories where ternary (and higher) algebraic operations show up and see that sometimes, there is a binary structure governing these somewhat exotic structures and that on other occasions, this is less true.

Teacher Learning in a Professional Learning Community (PLC) in The Netherlands

In 2013, the Dutch government started funding for professional learning communities (PLC’s) in The Netherlands. Aim of this funding was to do research about ways to improve mathematics teaching by teachers involved in these PLC’s. Radboud Teachers Academy started a PLC with 12 mathematics teachers from 11 different middle schools and high schools, starting from the idea of Lesson Study, a Japanese method for teacher development. As a result, teachers reported that they were inspired and that their teaching skills increased. However, the intended shift from instruction centred teaching to attention on student learning proved hard to realize.

Nhan NguyenUniversity of Montana – PhD Candidate

Kummer subspaces of generic abelian crossed products(And other things I learned during a PhD here)

An element $$x$$ of a central simple algebra is called $$\textit{p-central}$$ if $$x$$ is central or $$x^p$$ is central but $$x^{p'}$$ is not for any $$1\leq p'<p$$. $$p$$-central elements play an important role in the structure and presentations of central simple algebras. In this talk we discuss abelian crossed products (to be defined), their $$p$$-central subspaces, and present some open problems in regarding them.

Atish Mitra – Montana Tech

Extension Theory: Large Scale vs Small Scale

Classical extension theory deals with extensions of continuous functions between topological spaces. In this talk we will discuss progress made in the extension theory of morphisms in various large scale categories, and will compare the results and techniques with that of classical extension theory. We will also outline an axiomatic viewpoint that unifies a classical extension result in topology with recent results in the coarse category.

Ellie Bayat MokhtariUniversity of Montana – PhD Candidate

Information Processing in Hippocampal Interneuron Synapses

Understanding of the brain as an extremely sophisticated information processing system has gained tremendous momentum in the past few decades and it continues to advance.

Information theory  proposed by Shannon in 1948 provides primary tools to uncover how the central nervous system (CNS) acquires, transforms, stores, and uses information to control the body in a complex environment. In this talk I will provide a brief overview of the basics of information theoretic functionals and describe how these concepts are applied to estimate the information transfer in  both deterministic and stochastic models of hippocampal synapses. The Stochastic model can be used to simulate the main sources of variability in synaptic transmission. In addition, we assume that a synapse serves as a dynamic memory buffer that can store and transfer information. However, it is clear that it cannot carry infinite amount of information about the temporal activity of presynaptic neuron. We address the question of how much further back in time a synapse can store and transmit information; in other words what is the capacity of a particular synapse in the transmission information.

Department Picnic & Softball game

Friday, May 4, 5:00 p.m.
Bonner Park Band Shelter

Henry Riely – Washington State University

The Bellman function technique in harmonic analysis

The Bellman function was developed by applied mathematician Richard Bellman in the 1950s for the field of optimization called dynamic programming. Roughly speaking, the idea is to simplify a problem by breaking it into smaller sub-problems in a recursive fashion.

More recently, starting with the work of Donald Burkholder in the 1980s, the Bellman function has found utility in harmonic analysis, providing new proofs of old results, and even giving some new results for which classical proofs do not exist.

In this talk, we will trace the origins of the Bellman function technique, and walk through an example of a Bellman function proof of a simple fact from harmonic analysis.

Enrique Alvarado – Washington State University

The Steiner problem, and it's variations

The Steiner problem is one of finding the shortest network spanning a finite number of points on the plane. We will start by introducing the initial version of this problem, called Fermat's problem: find a point in the plane where the sum of whose distances from three given points is minimal. This will lead us to the Steiner problem and to some of its variations which will include on going research. Throughout the talk, we will encounter different areas of mathematics such as differential geometry, computational geometry, graph theory, and calculus of variations.

This talk is intended to be for a general mathematical audience, including both undergraduate and graduate students in mathematics and the mathematical sciences.

Bram Mesland – Max Planck Institute for Mathematics Bonn

KK-theory in geometry and physics

Kasparov's KK-theory is a bivariant homology theory for C*-algebras providing a general framework for index theory. Cycles for the theory come from the abstract analogue of a first-order differential operator. I will provide an exposition of the theory for non-specialists, based on the Atiyah-Singer index theorem. If time permits I will discuss applications to number theory and physics.

Tien Chih – Montana State University-Billings

Homotopy in the Category of Graphs

Homotopy Theory is the study of bending spaces into each other. While this is an essential part of the study of Topology, it does not translate immediately to discrete settings such as Graphs. However, we can use the language of Categories to abstract ideas from homotopy, and apply them to Graph Theory.

We begin by discussing the basic definitions of graph homotopies first established by Anton Dochtermann in 2008. We then discuss new results in homotopy of graphs, including a way to find a unique representative for each homotopy class of graphs. This work was done in collaboration with Dr. Laura Scull of Fort Lewis College.

Justin Marks – Gonzaga University

Manifold Methods for Averaging Subspaces

Applications of geometric data analysis often involve producing collections of subspaces, such as illumination spaces for digital imagery. For a given collection of subspaces, a natural task is to find the mean of the collection. A robust suite of algorithms has been developed to generate mean representatives for a collection of subspaces of fixed dimension, or equivalently, a collection of points on a particular Grassmann manifold. These representatives include the ag mean, the normal mean, the projection mean, and the Karcher mean. In this talk, we catalogue the types of means and present comparative heuristics for the suite of mean representatives. We respond to, and at times, challenge, the conclusions of a recent paper outlining various means built via tangent-bundle maps on the Grassmann manifold.

Monday, October 15, 2018 at 3:00 p.m. in Math 103 Refreshments at 4:00 p.m. in Math Lounge 109

October 22 – FEC meeting

Sean English – Ryerson University

Recent Problems in Hypergraph Saturation

Extremal graph theory is the branch of mathematics concerned with maximizing or minimizing some parameter across a restricted set of graphs. The most studied problem in extremal graph theory involves maximizing the number of edges over all (simple) graphs on a fixed number of vertices that avoid a certain substructure. For example, the seminal problem in this field, solved by Mantel in 1907, studies the maximum number of edges over all triangle-free graphs on n vertices. This was later generalized to all complete graphs by Turán in 1941. In this talk, we will give a brief overview of extremal problems for graphs and hypergraphs (graphs where edges may contain more than two vertices), and then talk about some recent advances on the saturation problem, which is a minimization problem, in some sense the dual of the classical extremal question of maximizing the number of edges.

Melody Alsaker – Gonzaga University

Improved D-bar Reconstructions of Human Ventilation from Electrical Impedance Tomography Data

In Electrical Impedance Tomography (EIT), electrical signals applied to human skin are used to reconstruct the internal electrical properties of the body, resulting in an image. The mathematical reconstruction process is an extremely ill-posed inverse problem, which causes challenges in the use of EIT for medical imaging applications. The D-bar reconstruction method for EIT is a mathematical inversion process with its roots in inverse quantum scattering theory.  Recent advances in D-bar methods EIT have resulted in techniques to improve image resolution via the inclusion of spatial a priori data. In this talk, we present an overview of the D-bar method along with some recent results, including EIT images and movies of human ventilation.

Jillian Glassett – Washington State University

Spectrally Arbitrary Patterns

A sign pattern is a matrix whose entries are from the set $$\{0,+,-\}$$, while a zero-nonzero pattern is a matrix whose entries are from the set $$\{0,*\}$$.  The idea behind sign and zero-nonzero patterns came from need to solve problems in economics (and other areas) when only the signs of the entries in a matrix are known. Sign and zero-nonzero patterns are areas of interest in qualitative matrix theory and, due to the connections with graph theory, combinatorial matrix theory. In this talk, we will discuss some background information about patterns before focusing on spectrally arbitrary patterns. We will go through a few techniques on how to determine if a pattern is spectrally arbitrary, including some techniques based on algebraic properties of rings and fields.