2016 Colloquia
Spring
K-theory and twisted groupoid C*-algebras
Elizabeth Gillaspy
Analysis candidate
The goal of this talk will be to explain some of what we know about the K-theory of twisted groupoid C*-algebras, and why we should care.
In the first part of the talk, I will define the objects in the title and explain how the K-theory of groupoid C*-algebras can tell us about other mathematical objects, from dynamical systems to string theory. My hope is that this will be a gentle introduction to the topic(s) at hand; no prior familiarity with groupoids or with C*-algebras will be assumed.
In the second part of the talk, I'll discuss the particular question I've investigated about the K-theory of twisted groupoid C*-algebras, why I chose it, and the progress that I've made so far. Time permitting, I will sketch some of the proofs, so this part of the talk will be more technical than the first part.
Thursday, February 11, 2016 at 2:10 p.m. in Math 103
Refreshments at 3:00 p.m. in Math Lounge 109
A snapshot of holomorphic dynamics in a neighborhood of a fixed point
Sara Lapan
Analysis Candidate
Holomorphic dynamics is a fascinating area of mathematics that lies at the intersection of complex analysis and dynamical systems. In pop-culture, holomorphic dynamics is known for its beautiful fractals, like the Mandelbrot set and Julia sets. This talk will be a gentle introduction to holomorphic dynamics and, along the way, we will see and explore many beautiful fractals.
More specifically, we will focus on holomorphic self-maps that fix a point and explore the dynamics near that fixed point. Are nearby points attracted to (or repelled from) that fixed point when the map is iterated? If so, how? These questions are of great interest in holomorphic dynamics in one (and several) complex variables. We will begin our discussion in dimension one and reveal the local dynamics near a fixed point of a general holomorphic map. We will see how the coefficient of the linear term in the power series expansion of the map near the fixed point can determine how points near that fixed point behave under iteration. We will then focus on a particularly interesting class of maps that are called tangent to the identity. In dimension one, the Leau-Fatou Flower Theorem provides a beautiful description of the behavior of points in a full neighborhood of a fixed point for this class of maps. This theorem from the early 1900s serves as inspiration for the study of maps tangent to the identity in higher dimensions. In higher dimensions, our picture of the dynamics near a fixed point is still being formed. We will briefly introduce the study of maps tangent to the identity in several complex variable and we will establish the foundations for our discussion in the subsequent analysis seminar.
Wednesday, February 17, 2016 at 2:10 p.m. in Math 103
Refreshments at 3:00 p.m. in Math Lounge 109
Nonparametric Regression Method for Agreement Measure Between an Ordinal Measurement and a Continuous Measurement.
AKM Fazlur Rahman, Ph.D.
Department of Biostatistics and Bioinformatics, Emory University
Statistics Candidate
In this talk, I will present a non-parametric regression framework for an agreement measure between an ordinal measurement and a continuous measurement given a covariate. Unlike in the usual non-parametric regression methods, agreement measure is not observed as a random variable for each subject, thus commonly used non-parametric methods such as kernel methods are not directly applicable. Adopting the idea of stratified sampling in the framework of kernel regression, I will present the proposed inferential procedures including regression function estimation, asymptotic properties, and hypothesis testing. Finite sample performance of the proposed method will be illustrated via simulation studies. I will also present an illustration of the proposed method to a recent posttraumatic stress disorder (PTSD) study which reveals an interesting impact of depression severity on the agreement measure between a self-reported symptom instrument score (continuous measure) and clinician diagnosis (ordinal measure) in PSTD patients.
Monday, February 29, 2016 at 3:10 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Hierarchical group testing for multiple infections
Peijie Hou
Statistics Candidate
Group testing, where individuals are tested initially in pools, is widely used to screen a large number of individuals for rare diseases. Triggered by the recent development of assays that detect multiple infections at once, screening programs now involve testing individuals in pools for multiple infections simultaneously. Tebbs, McMahan, and Bilder (2013, Biometrics) recently evaluated the performance of a two-stage hierarchical algorithm used to screen for chlamydia and gonorrhea as part of the Infertility Prevention Project in the United States. In this article, we generalize this work to accommodate a larger number of stages. To derive the operating characteristics of higher-stage hierarchical algorithms with more than one infection, we view the pool decoding process as a time-inhomogeneous, finite-state Markov chain. Taking this conceptualization enables us to derive closed-form expressions for the expected number of tests and classification accuracy rates in terms of transition probability matrices. When applied to chlamydia and gonorrhea testing data from four states (Region X of the United States Department of Health and Human Services), higher-stage hierarchical algorithms provide, on average, an estimated 11 percent reduction in the number of tests when compared to two-stage algorithms. For applications with rarer infections, we show theoretically that this percentage reduction can be much larger.
Monday, March 7, 2016 at 3:10 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Microbiome: Data Normalization and Comparison of Bacterial Vaginosis Species Composition
Katia Smirnova
Statistics Candidate
Human Microbiome Project (HMP) is a large scale nationwide study that utilizes next generation sequencing technology (NGS) to investigate the relationships between the human microbiota composition, diet and health status. We discuss common statistical challenges arising in microbiome data analysis and methods to approach these problems. One particular characteristic of these studies is that the data are often quite sparse but collected on a large number of variables, many of which are possible contaminants. First, we propose a network based data normalization method, which is known in microbiome literature as filtering. This method removes variables from a high dimensional sparse data set that are most likely present due to contamination. Then we discuss Co-Inertia Analysis approach to identify microbial species that contribute to major differences between healthy and bacterial vaginosis data. This disease, caused by excessive bacteria in vagina, affects approximately 30% of reproductive age women. The importance of this disease is that it has a high recurrence rate, and associated with miscarriage, preterm birth, and increased risk of acquiring other sexually transmitted infections.
Monday, March 14, 2016 at 3:10 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Solving polynomial eigenvalue problems via linearizations is backward stable.
Javier Pérez Álvaro
Polynomial eigenvalue problems arise directly from applications in mechanics and control theory, from finite element discretizations of continuous models, or as approximations of nonlinear eigenvalue problems, and are still a challenge for modern eigenvalue methods. The most widely used approach for solving the polynomial eigenvalue problem associated with a matrix polynomial \(P(\lambda)=\sum_{i=0}^d \l^i A_i\) is to linearize to produce a larger order pencil (i.e. a linear matrix polynomial) \(L(\lambda) = \lambda X + Y\), whose eigenstructure is then found by any method for generalized eigenproblems (such as the QR algorithm). Though this may not be the best way to address the polynomial eigenvalue problem, from the point of view of efficiency and storage, it has been used extensively, because of the advantages of the QR algorithm (robustness and stability).
To determine whether this method to solve polynomial eigenvalue problems is backward stable or not has been an open problem until very recently.
The goal of this talk will be to explain our most recent results about the backdward stability of this approach to solve polynomial eigenvalue problems. In the first part of the talk I will introduce polynomial eigenvalue problems and the concept of a matrix polynomial. In the second part of the talk I will introduce the concept of a linearization of a matrix polynomials. In the third part of the talk I will make a gentle introduction to Numerical Linear Algebra, and, in particular, I will introduce the ideas of a backward error analysis and a backward stable numerical algorithm. Finally, I will explain some of the ideas behind our backward error analysis of the polynomial eigenvalue problem solved via linearizations.
Thursday, March 17 at 3:10 p.m. in Math 108
Design Principles for Infusing Active Learning in Undergraduate Calculus
Eric Stade
Department of Mathematics, University of Colorado
David Webb
Freudenthal Institute US, School of Education, University of Colorado
This presentation will provide a brief overview of undergraduate mathematics at the University of Colorado Boulder, and related activities that we have designed and used in the calculus sequence through the support of Helmsley Foundation funding, the APLU Mathematics Teacher Education Partnership, an IBL grant, and (quite frankly) significant personal and departmental commitments to ensure more widespread experience of active learning in calculus courses. Using principles of Active Learning, students are encouraged to conjecture, explore and communicate their reasoning in the process of solving mathematics problems. Underlying this approach is research that has demonstrated how undergraduate students who are involved in active learning techniques can learn more effectively in their classes, resulting in increased achievement and improved dispositions towards mathematics.
Monday, March 21, 2016 at 3:10 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Database for Dynamics: a new approach to model gene regulatory networks
Tomas Gedeon
Montana State University
Experimental data on gene regulation is mostly qualitative, where the only information available about pairwise interactions is the presence of either up-or down- regulation. Quantitative data is often subject to large uncertainty and is mostly in terms of fold differences. Given these realities, it is very difficult to make reliable predictions using mathematical models. The current approach of choosing reasonable parameter values, a few initial conditions and then making predictions based on resulting solutions is severely subsampling both the parameter and phase space. This approach does not produce provable and reliable predictions.
We present a new approach that uses continuous time Boolean networks as a platform for qualitative studies of gene regulation. We compute a Database for Dynamics, which rigorously approximates global dynamics over entire parameter space. The results obtained by this method provably capture the dynamics at a predetermined spatial scale.
Monday, April 11, 2016 at 3:10 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Native American Mathematics
Robert E. Megginson
Arthur F. Thurnau Professor
University of Michigan
In the early 1930s, Will Ryan, Director of Indian Education for the Bureau of Indian Affairs, eliminated algebra and geometry from the Uniform Course of Study in BIA schools. This was done in a well-intentioned but misguided effort to make BIA education more culturally relevant for American Indians, in the belief that mathematics has had no historical or cultural importance for the indigenous peoples of the Western Hemisphere. In fact, examples abound of the importance of mathematics in many Native cultures of the Americas. The well-developed number systems of pre-contact Mesoamerica are probably the most well-known, to the extent that the Mayan place-value system is sometimes taught in our schools as an example of a number system with a different base that a highly developed non-Western culture found useful. For those who have not seen this system, it will be presented along with some of its number-theoretic underpinnings and consequences, as well as the cultural values that led to some of its structure. (It will also be seen why some people have claimed that the European conquest of Mesoamerica was due in part to the Mexica people knowing too much number theory!) An interesting approach to counting among North American tribes of the Algonkian language group (e.g., the Siksika or Blackfeet language) will also be discussed as an example of how not just language, but mathematics, can carry culture, as well as analogs among the Maori of New Zealand.
MAA Strengthening Underrepresented Minority Mathematics Achievement (SUMMA)
Monday, April 18, 2016 at 3:10 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
This talk sponsored in part from a grant from the National Science Foundation:
Montana Supports the Mathematician of Tomorrow
Swarms in bounded domains
Dieter Armbruster
Arizona State University
Swarms of animals, fish, birds, locusts etc. are a common experience and their coherence and dynamics have been the focus of research in mathematics and biology in the last 20 years. A number of different models for the onset of coherent swarming and the shapes of the swarms have been proposed but studies of the dynamics of these swarms and their interaction with other swarms or boundaries are lacking. This talk will present recent research on scattering at boundaries and collisions between two swarms for two swarming models, the Vicsek and the Attraction-Repulsion model. We will show that, while individual particles are specularly reflected at a boundary, the swarm as a whole is reflected inelastically. A fundamental refraction law for a swarm, impacting on a planar boundary is derived. Inelastic collisions lead to alignment with the wall and to damped pulsating oscillations of the swarm as well as new dynamical swarm solutions. Setting two identical flocks on a collision course is studied in a simplified form analytically and for large swarms numerically. Depending on the scattering parameters swarms will diverge, converge or form bound states where the translational kinetic energy is converted into a rotational kinetic energy.
Monday, May 2, 2016 at 3:10 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Enhanced Gibbs sampling for an application to X-ray imaging
Kevin Joyce
PhD candidate
University of Montana
Image deblurring techniques derived from convolution require, a priori, an estimate for the convolution kernel or point spread function (PSF). Standard techniques for estimating the PSF involve imaging a bright point source, but this is not always feasible (e.g. high energy radiography). This work takes a novel non-parametric approach to modeling a radially symmetric PSF, in which an estimate can be obtained from the calibration image of a vertical edge. Moreover, we employ a hierarchical Bayesian model that in addition to providing a method for estimation, also gives a quantification of uncertainty in the estimate. We will present a recently developed improvement to the Gibbs algorithm for simulating samples from the Bayesian posterior of the hierarchical model, referred to as partial collapse. The improved algorithm has been independently derived in several other works, however, it has been shown that partial collapse may be improperly implemented resulting in a sampling algorithm that that no longer converges to the desired posterior. The algorithm we present is proven to satisfy invariance with respect to the target density.
Monday, April 25, 2016 at 3:10 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Doctorial Dissertation Defensse
Point Spread Function Estimation and Uncertainty Quantification
Kevin Joyce
PhD Candidate
University of Montana
An important component of analyzing images quantitatively is modeling image blur due to eects from the system for image capture. When the eect of image blur is assumed to be translation invariant and isotropic, it can be generally modeled as convolution with a radially symmetric kernel, called the point spread function (PSF). Standard techniques for estimating the PSF involve imaging a bright point source, but this is not always feasible (e.g. high energy radiography). This work provides a novel non-parametric approach to estimating the PSF from a calibration image of a vertical edge. Moreover, the approach is within a hierarchical Bayesian framework that in addition to providing a method for estimation, also gives a quant cation of uncertainty in the estimate by Markov Chain Monte Carlo (MCMC) methods.
In the development, we employ a recently developed enhancement to Gibbs sampling, referred to as partial collapse. The improved algorithm has been independently derived in several other works, however, it has been shown that partial collapse may be improperly implemented resulting in a sampling algorithm that that no longer converges to the desired posterior. The algorithm we present is proven to satisfy invariance with respect to the target density.
The other component of this work is mainly theoretical and attempts to develop fromrst principles the requisite functional analysis to make the integration based model derived in therst chapter rigorous. The literature source is from functional analysis related to distribution theory for linear partial dierential equations, and brie y addresses i nite dimensional probability theory for Hilbert space-valued stochastic processes, a burgeoning and very active research area for the analysis of inverse problems. To our knowledge, this provides a new development of a notion of radial symmetry for \(L^2\) based distributions. This work results in d ning an \(L^2\) complete space of radially symmetric distributions, which is an important step toward rigorously placing the PSF estimation problem in the i nite dimensional framework and is part of ongoing work toward that end.
Thursday, May 5, 2016 at 11:10 a.m. in Math 211
Doctorial Dissertation Defensse
The Dynamics of Vector-Borne Relapsing Diseases
Cody Palmer
PhD Candidate
University of Montana
We will begin with a review of the relevant history and theory behind disease modeling, investigating important motivating examples. The concept of the fundamental reproductive ratio of a disease, \(R_0\), is introduced through these examples. The compartmental theory of disease spread and its results are introduced, particularly the next-generation method of computing \(R_0\). We review center manifold theory, as it is critical to the reduction of the dimension of our problems. We review diseases that have a relapsing character and focus in on relapsing diseases that are spread by vectors in a host population. The primary example of such a disease is Tick-Borne Relapsing Fever (TBRF). Motivated by TBRF we establish a general model for the spread of a vector-borne relapsing disease.
With a model in hand we confirm that it meets the required hypotheses for the use of compartmental theory. A technical computation then leads to an explicit form of \(R_0\) that is given in terms of the number of relapses. Further technical computations then allow us to describe the bifurcation at \(R_0 = 1\), finding that it is always transcritical regardless of the number of relapses. We also show the existence of a unique endemic equilibrium for all values of \(R_0\) greater than 1.
Variations of the simple model are explored. Adding in removal to the recovered compartment, in which individuals leave an earlier relapse state and recover, we find how this changes \(R_0\) and show that the bifurcation at \(R_0\) is still transcritical. We investigate the addition of latent infective compartments and describe how they affect \(R_0\). We also find the reproductive ratio when there are two host species that undergo the same number of relapses.
We establish a continuity result between the reproductive ratios of systems with differing numbers of compartments. This allows us to state the reproductive ratio of a smaller system as a limit of the reproductive ratio of a larger system. This result is then used to compute the reproductive ratio for a coupled host-vector system where the hosts undergo a different number of relapses. We close with some conclusions and directions for future work.
Wednesday, May 11, 2016 at 10:10 a.m. in Math 103
Fall
Colloquium
Linkage of Symbol p-Algebras of Prime Degree
Adam Chapman
Michigan State University
Two symbol algebras of the same prime degree p over a field of characteristic p are said to be "left linked" if they share a common cyclic Galois field extension of the center, and "right linked" if they share a common pure inseparable field extension of the center.
We discuss the connection between left linkage and right linkage and present some open problems.
Monday, August 29, 2016 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Colloquium
Escapades in Burn-off Chip-firing on Graphs
Mark Kayll
University of Montana
Start by placing piles of indistinguishable chips on the vertices of a graph. A vertex can fire if it's supercritical; i.e., if its chip count exceeds its valency. When this happens, it sends one chip to each neighbour and annihilates one chip. Initialize a game by firing all possible vertices until no supercriticals remain. Then drop chips one-by-one on randomly selected vertices, at each step firing any supercritical ones. Perhaps surprisingly, this seemingly haphazard process admits analysis. And besides having diverse applications (e.g., in modelling avalanches, earthquakes, traffic jams, and brain activity), chip firing reaches into numerous mathematical crevices. The latter include–alphabetically–algebraic combinatorics, discrepancy theory, enumeration, graph theory, stochastic processes, and the list could go on (to zonotopes(!)). I'll share some joint (old and new) work–with my colleague and former PhD student Dave Perkins–that touches on a few items from this list. The talk'll be accessible to non-specialists. Promise!
Monday, September 19, 2016 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Colloquium
Essential dimension of generic symbols (and other great things I learned on sabbatical)
Kelly McKinnie
University of Montana
The essential dimension of an algebraic object is loosely defined as the minimal number of independent parameters needed to define the object over a base field. The essential dimension of an algebraic object was only formally defined in the late 90's and in that case it was for abelian groups. We will take a look at the original motivations for essential dimension and discuss some recent results. One example we will look closely at is the essential dimension of generic symbol algebras \((x_1,y_1) \otimes \cdots \otimes (x_n,y_n)\) (to be defined in the talk). These algebras have different essential dimensions over \(\mathbb C\) and over a field of characteristic \(p\).
Monday, September 26, 2016 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Q&A
Darrel Choate
2016 Distinguished Alumni Award Recipient and a Boeing Company Technical Fellow
We are pleased to welcome Mr. Choate back to the Math Dept., where he will give a short overview of his extensive career, and take questions from the audience. Choate earned a bachelor's and master's degree in Mathematics in ’65 and ’67. He was part of the Boeing Company’s contribution to the "Star Wars" Program, or Strategic Defense Initiative, which has had an influence on the current U.S. ballistic missile defense architecture. Over his long career he has also worked for the Aerospace Corporation and the Kaman Science Corporation, and has worked in retirement on infrastructure development projects around the world, including a tsunami rebuilding effort in Japan. We look forward to hearing how his training in Mathematics has featured in all these endeavors.
Thursday, September 29, 2016 at 3:00 p.m. in Skaggs 117
Refreshments at 4:00 p.m. in Skaggs
Colloquium
Equation-free techniques for infectious disease data
Josh Proctor
Senior Research Scientist at the Institute for Disease Modeling, Seattle Washington
Equation-free techniques are emerging as a promising analysis and modeling tool for the investigation of complex, dynamical systems. The surge in popularity of these methods stems from the ability to discover models directly from measurement data, making them well suited to describe systems for which the governing equations are either partially known or heuristically posited. In the life sciences, complex systems without a standard set of governing equations include neuroscience, infectious disease spread, metabolic/regulatory networks, and ecological networks. As a motivating example, equation-free methods can be applied to data collected by public health surveillance systems focused around the eradication of infectious diseases. The increased awareness for gathering high-quality data and the advent of new monitoring tools is beginning to generate large sets of data describing the spread of infectious disease. I will focus on how recently developed equation-free methods, such as Koopman operator theory, Dynamic Mode Decomposition (DMD), and Sparse Identification of Nonlinear Dynamics (SINDy), can help both characterize and analyze time-series data collected from complex systems. These techniques offer insight into high-dimensional datasets and complex systems that are nonlinear, parameterized, and multi-scale. This presentation will also include a discussion on how these methods can be used for prediction and nonlinear forecasting. Importantly, these approaches are helping modify infectious disease risk calculations at a sub-national level for countries in Africa helping guide large-scale intervention strategies to help prevent childhood disease.Monday, November 21, 2016 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109
Peter Coffee, VP for Strategic Research, Salesforce
From Mass to Force: Finding Value and Direction in Big Data
Friday, December 9, 2016
University of Montana, Gallagher Business Building, room GBB 123
3:00 pm