# 2006 Colloquia

## Spring 2006

Representations of Quivers
Paul Smith
University of Washington

A quiver is another name for a directed graph, usually with a finite number of vertices and arrows. A representation of a quiver consists of a vector space at each vertex and a linear map for each arrow.

The fundamental problem is to classify such representations. This is an impossible problem in general. Nevertheless one may in simple cases classify the representations: for example, one vertex and one arrow, a loop from that vertex to itself, leads to the Jordan Normal Form classification of matrices. If one takes a Dynkin diagram of type A, D, or E , and puts an arrow on the edge one gets a quiver and these are exactly the quivers with the following property: there is only a finite number of representations that can not be written as direct sums of smaller representations (an indecomposable representation) and every representation is a direct sum of copies of these indecomposable ones. Moreover, in that case, the indecomposables are in bijection with the positive roots of the root system corresponding to the Dynkin diagram.

For more complicated quivers there are infinitely many indecomposable representations and these usually come in families that are parameterized by interesting geometric objects. We will give some examples.

Representation theory of quivers interacts with a wide range of other branches of mathematics (even string theory). There are a number of reasons for this but one fundamental reason is that in many areas of mathematics one is interested in collections of objects and maps between them: one then gets a quiver by assigning a vertex to each object and an arrow to each map.

This talk will be a gentle introduction to the subject.

Friday, 21 April 2006
4:10 p.m. in Math 109

Understandings of First-year Mathematics Majors about Differential Calculus — One Episode on Differentiability
Antonio Olimpio Junior
Math Education Candidate

The transition to undergraduate mathematics has been reported by researchers in Mathematics Education as being a source of many difficulties for many freshmen. In this talk, I will present, from my doctoral research in Mathematics Education, an episode with a pair of students focusing on the concept of differentiability. In the research, I investigated understandings about the concepts of function, limit, continuity and derivative produced by full-time first-year students from UNESP - The State University of Sao Paulo at Rio Claro, Brazil.

Thursday, 20 April 2006
4:10 p.m. in Math 109

Multivariate Factorial Designs when The Number of Factor Levels is Large
Solomon Harrar
Statistics Candidate

We obtain the asymptotic distributions of different multivariate parametric and nonparametric tests for the situation where the number of replications is limited, whereas the number of treatments goes to infinity (large k, small n case). For the parametric case, we consider the Dempster's ANOVA-type, Wilks Lambda, Lawley-Hotelling and Bartlett-Nanda-Pillai Statistics. In the nonparametric case, we propose the rank-analogs of the Dempster's ANOVA-type, Lawley-Hotelling and Bartlett-Nanda-Pillai statistics. The tests are based on separate rankings for the different variables.

We provide a finite sample approximation procedure in both the parametric and nonparametric cases. The finite performance of the tests is investigated through simulations. It turns out that the proposed nonparametric tests perform very well as compared to their parametric competitors, especially in the presence of outliers.

An example illustrates the application.

Thursday, 6 April 2006
4:10 p.m. in Math 109

Looking at Learning from a Complex Systems Point of View
Andrew Hurford
Math/C&I Candidate

Educational research and theory building are increasingly being influenced by the sciences of complexity, systems theories, and non-linear mathematics. This talk will consider several models of learning from dynamical systems perspectives and will provide an introduction to classroom-based technological tools that have significant potential for research on and teaching about learning as a complex system. Although complex systems views of learning are very new, it will be argued that they have much in common with more established theories of learning and that they can be extended beyond traditional individualistic views of learning toward enabling educators to leverage the learning potential of groups. Qualitative and quantitative data relative to the proposed arguments will be presented.

Tuesday, 21 March 2006
4:10 p.m. in LA 11

"Robustifying" Parametric Models via Mixtures of Polya Tree Priors
Tim Hanson
University of Minnesota

Mixtures of Polya trees models are straightforward to code and provide a highly flexible alternative when a parametric model may only hold approximately. In this talk, I provide computational strategies for obtaining semiparametric inference for mixtures of finite Polya trees models given a standard parameterization, including models that would be difficult to fit using Dirichlet process mixtures. Recommendations are put forth on choosing the level of a finite Polya tree and model comparison is discussed. Several examples demonstrate the utility of Polya tree modeling including data on bivariate (CD4,CD8) counts fit to a semiparametric linear mixed model; the classic V.A. lung cancer study data fit to proportional hazards, proportional odds, and accelerated failure time models; and serology scores modeled with a stochastic order constraint.

Friday, 17 March 2006
4:10 p.m. in Math 109

A Three-dimensional Variational Approach to Video Segmentation
Aaron Luttman
University of Montana

In order to engage in photosynthesis, leaves use pores on their surface - called stomata - to absorb CO2. The opening of these pores results in the evaporation of H2O, which is a detriment to leaf function. Thus a leaf is faced with the global optimization problem of maximizing CO2 uptake for a fixed amount of H2O loss. In solving this problem, stomata in spatially homogeneous patches often synchronize their apertures, even though this does not result in optimal local CO2 uptake. In order to visualize these patches, a dye is injected into a leaf so that it fluoresces when closing its stomata. Understanding how synchronized patches of stomata results in an optimal CO2 uptake for the entire leaf requires a thorough analysis of these fluorescence patterns. Using an experimental background model to drive video segmentation, we use a variational level-set approach for extracting the spatially synchronized stomatal patches from video taken of the leaf fluorescence. Methods of two-dimensional pattern analysis can then be used to analyze the dynamics of the stomatal patches.

Thursday, 9 March 2006
4:10 p.m. in Math 109

Multivariate Factorial Designs when The Number of Factor Levels is Large
Solomon Harrar
Statistics Candidate

We obtain the asymptotic distributions of different multivariate parametric and nonparametric tests for the situation where the number of replications is limited, whereas the number of treatments goes to infinity (large k, small n case). For the parametric case, we consider the Dempster's ANOVA-type, Wilks Lambda, Lawley-Hotelling and Bartlett-Nanda-Pillai Statistics. In the nonparametric case, we propose the rank-analogs of the Dempster's ANOVA-type, Lawley-Hotelling and Bartlett-Nanda-Pillai statistics. The tests are based on separate rankings for the different variables.

We provide a finite sample approximation procedure in both the parametric and nonparametric cases. The finite performance of the tests is investigated through simulations. It turns out that the proposed nonparametric tests perform very well as compared to their parametric competitors, especially in the presence of outliers.

An example illustrates the application.

Thursday, 6 April 2006
4:10 p.m. in Math 109

Marginal Mixture Analysis of Correlated Bounded-Response Data
with an Application to Ultrasound Risk Assessment
Yan Yang
Statistics Candidate

Data with bounded responses are common in many areas of application. Often the data are bounded below by zero with excess zero observations. Essentially continuous responses may contain a substantial portion of zeros, either because no effects occur or due to limits of detection. Discrete data may exhibit a significantly higher percentage of zeros than expected under idealized models. In these settings ordinary generalized linear models fail. Three methods in the literature for modeling zero-inflated data are left-censored regression models, two-part models, and latent mixture models. We introduce a general class of zero-inflated mixture (ZIM) models that unifies and generalizes these three classes of models. In particular, we develop unified estimation procedures, large sample inferences and general computational algorithms. Novel diagnostics are proposed for assessing the adequacy of a ZIM model. We extend ZIM models to correlated data with excess zeros using the theory of generalized estimating equations. Risk threshold estimates are also provided for the incidence and magnitude of correlated adverse outcomes. We illustrate the issues and methodology in the context of an ultrasound safety study of the occurrence and extent of lung hemorrhage due to focused ultrasound exposure in laboratory animals. 1

Tuesday, 28 February 2006
4:10 p.m. in Math 109

Pulse interactions and self-replication in activator-inhibitor type reaction-diffusion systems.
Tasso Kaper
Boston University

This phenomenon of self-replicating spots and pulses was discovered in the Gray-Scott model and in a series of gel reactor experiments. In this talk, I will present some analytical and numerical results for pulse dynamics and self-replication in one and two space dimensions in the Gray-Scott model and a class of activator-inhibitor systems. The results have been obtained in collaboration with Arjen Doelman, Wiktor Eckhaus, Rob Gardner, Dave Morgan, and Bert Peletier.

Friday, 24 February 2006
4:10 p.m. in Math 109

Factors in Statistics Learning:
Developing a dispositional attribution model to describe differences in the development of statistical proficiency.
Jennifer Kaplan
Mathematics Education Candidate

This talk will describe initial results of a study designed to classify differences in the development of proficiency in hypothesis testing by introductory students based on dispositions exhibited by the students. The talk will define five components that comprise statistical proficiency as well as the general concept of a psychological disposition and two particular dispositions. Findings from the psychology literature on human reasoning as they relate to statistics learning will be discussed briefly. Results of a quantitative study linking cognitive dispositions to statistics learning will be presented and a qualitative study designed to further understand the relationship will be described.

Thursday, 23 February 2006
4:10 p.m. in Math 109

Non-linear process specifications in hierarchical spatio-temporal models.
Mevin Hooten
Statistics Candidate

Natural systems exhibiting complex and highly non-linear behavior can often be characterized by scientifically-based deterministic models. However, such models are only approximations to the real process of interest and contain uncertainties in parameters and representativeness. Thus, statistical inference (i.e., parameter estimation and process prediction) for such natural processes can be achieved through the hierarchical incorporation of conventional deterministic spatio-temporal models (e.g., differential and integral equation models). For example, when discretized for implementation in a computational setting, many such models suggest a first order Markovian specification (termed "matrix models" in the ecological literature). Parameterizations motivated by partial differential equations and integro-difference models are effective but can be awkward in non-Gaussian settings. More intuitive, and thus, more accessible specifications are possible by parameterizing the process model directly based on scientifically meaningful dynamical components. When considered in this context, such specifications imply a very general class of models capable of accommodating many different types of spatio-temporal processes. The utility of these hierarchical "matrix" models in an ecological setting is illustrated with an application focusing on characterizing the spread of invasive species in the presence of sampling uncertainty..

Tuesday, 21 February 2006
4:10 p.m. in Math 109

The Process and Impact of Implementing Standards-Based Mathematics Curriculum
and the Georgia Performance Standards.
Samuel Obara
Mathematics Education Candidate

The goal of this study was to investigate a middle school coordinated implementation of Standards-Based Curriculum-Connected Mathematics Project materials (CMP), new Georgia Performance Standards and the accountability system. The state of Georgia has replaced its Quality Core Curriculum Standards with the Georgia Performance Standards (GPS), which are statements of what is expected at each grade level. The CMP is judged to be a middle school mathematics curriculum consistent with the GPS. To ensure that schools and school systems meet their goals, some school districts have implemented an accountability system.

Three case studies were constructed of 6th grade teachers and their mathematics coach in one school in Georgia during the coordinated implementation of the Georgia Performance Standards (GPS), the standard-based curriculum (CMP), and the accountability system. They attended a one-week institute in summer planning for implementing CMP materials in their classes. They had previously participated in staff development for implementing GPS, and they began teaching CMP materials in their classes in fall 2005. Multiple interviews were held with the teachers and their mathematics coach about GPS plans, the institute, CMP use in their classes and the accountability system. The teachers were also observed teaching materials from the CMP.

The study identifies both common and individual concerns and issues with respect to the ongoing implementation of these three influences. It underscored the importance of teamwork and collaboration in bringing about a new vision of the school mathematics program.

Tuesday, 14 February 2006
4:10 p.m. in Math 109

Teaching Innovations for Related Rates Problems in First Semester Calculus
Nicole Engelke
Mathematics Education Candidate

Related rates problems in first semester calculus are a source of difficulty for many students. These problems require students to be able to visualize the problem situation, attend to the nature of the changing quantities, reconceptualize the variables in a geometric formula as functions of time, and relate these functions either parametrically or through function composition. Being able to successfully solve a related rates problem by engaging in the problem solving behaviors described above relies on the students' ability to engage in transformational/covariational reasoning. I have developed a sequence of teaching activities which employs a computer program designed to foster the students' exploration of related rates problems in a covariational context. I am investigating the impact of these activities on students' abilities to solve related rates problems.

Thursday, 9 February 2006
4:10 p.m. in Math 109

Hierarchical Team Learning using Fuzzy Perceptron Algorithms
Hideyuki Nojiri
Kumamoto Gakuen University

An approach to the problems of learning in team theory and perceptrons is presented. Economic theory of teams was initiated by Marschak and developed by Radner. The Marschak-Radner models of static teams are groups of individuals with a common goal but with individual information structures and decision rules. In the previous works, we studied fuzzy information structures and fuzzy decision rules and presented various models of team decision-making under fuzziness. Team theory enables one to decide analytically the optimal decision rules only in few cases. This drawback and the need for a computational distributed algorithm lead us to approximate the functional optimal team decision problem to a parametric one. Here, we discuss the formal relationship between perceptrons and team models, and introduce the various learning concepts of perceptrons and fuzzy sets to the extended team models because there is yet no fully developed theory of team learning. Hierarchical learning team models using fuzzy perceptron algorithms are also proposed. These models use the learning rules to adjust a weight matrix interpreted as the intensity of the team member's informal human relations expressed by the ideas of fuzzy relations.

Thursday, 26 January 2006
4:10 p.m. in Math 109

## Fall 2006

Weakly Peripherally-Multiplicative Operators Between Uniform Algebras
Scott Lambert
University of Montana

Let X be a compact Hausdorff space. A uniform algebra is a sub-algebra of continuous functions, ƒ : X→ ℂ with the uniform norm. In this case the spectrum is the range i.e. σ(ƒ)=ƒ(X) and σg(ƒ):{λ ∈σ(ƒ):∣λ∣ = ∥ƒ∥. The mere existence of this norm in the definition places restrictions on the algebraic structure so that a mapping between uniform algebras that preserves certain analytic conditions is necessarily an algebraic isomorphism. There are many results with this theme. For example Aaron Luttman and Thomas Tonev have produced the following:

If T:AB is a surjective, unital mapping between uniform algebras that is peripherally multiplicative (i.e. for all ƒ, gAgg)=σg (TƒTg)) the T is an algebraic isomorphism.

In this talk we prove a stronger result: If  T:AB is a mapping between uniform algebras that preserves the peaking functions and is weakly-peripherally multiplicative (σgg)∩σg(TƒTg)≠Ø for all ƒ, g∊A) the T is an algebraic isomorphism. (A function hA is a peaking function if σg(h)={1})

Also some extensions of this theorem will be discussed.

Thursday, 30 November 2006
4:10 p.m. in Math 109

Empirical Likelihood
Arne Bathke
University of Kentucky

Empirical likelihood (EL) is a nonparametric inference method with asymptotic properties that are in general similar to the parametric maximum likelihood. For example, there are EL versions of likelihood ratio tests and of Wilks' theorem. As its parametric equivalent, the EL approach often provides an efficient and practical inference tool in situations where other inferential methods do not succeed. In this presentation, I will first give a general introduction to empirical likelihood. Then, I plan to demonstrate a new procedure for combining multiple tests in samples of right-censored observations. The new method is based on multiple constraint censored empirical likelihood where the constraints are formulated as linear functionals of the cumulative hazard functions. A useful application of the proposed method is examining the survival experience of one or more populations by combining different weighted log-rank tests. Real data examples are given using the log-rank and Gehan-Wilcoxon tests. Simulation results demonstrate that, in addition to its computational simplicity, the combined test performs comparably to, and in some situations more reliably than previously developed procedures.

Thursday, 30 November 2006
4:10 p.m. in Math 109

Conjectures on Circuits, Clones, and Connectivity in Matroids
Talmage James Reid
University of Mississippi

Two elements of a matroid are clones if the map that interchanges the two elements and fixes all other elements is an automorphism. Clones are important in the study of matroid representability. We give results on the number and size of clone sets in representable matroids.

Smith conjectured in 1979 that two distinct cycles in a k-connected graph meet in at least k vertices when k >= 2. Matroid extensions of this conjecture are considered. We also give a result that characterizes the connected binary matroids with two different circuit sizes.

Finally, we consider conjectures of Wu on deletable and contractible elements in 4-connected matroids. This is the result of work with many others such as Cotwright, Lemos, McMurray, Robbins, Sheppardson, Wei, Wu, and Zhou.

Friday, 17 November 2006
2:10 p.m. in Math 211

Part of the 2006 Montana Matroid Workshop

On the spectrum of matroids
Talmage James Reid
University of Mississippi

Murty characterized the connected binary matroids with a single circuit size in 1971. Matroids with this property include the duals of projective and affine geometries. Here we provide a partial characterization of connected binary matroids with two circuits sizes c and d under the condition that c < d and d is odd. This is joint work with Lemos and Wu.

Thursday, 16 November 2006
4:10 p.m. in Math 109

Part of the 2006 Montana Matroid Workshop

A Partially Anecdotal History of Matroid Theory
Thomas Brylawski
University of North Carolina

From its birthplace in the atmosphere of Harvard junior fellows in the thirties (subsequently renowned mathematicians in several fields obsessed with a proof of the four-color theorem) to Waterloo in the early sixties and then back to Cambridge at M.I.T. in the late sixties, the early years of matroid theory (aka "combinatorial geometries") are evoked. The birth and recent results of the Tutte polynomials are given, all with various memories of my and my students personal story.

Thursday, 16 November 2006
10:10 a.m. in Math 109

Part of the 2006 Montana Matroid Workshop

A Panel Discussion: Using Projects throughout the Undergraduate Mathematics Curriculum
Montana State University

Although many of us are convinced of the pedagogical value of having our students do projects or larger writing assignments, we often find ourselves faced with practical concerns:

Where can I find problem and project ideas?

What makes a problem a good group problem?

How can I structure the assignment to encourage students to make progress throughout the semester?

How should such projects be evaluated?

In this panel discussion, four instructors share their experiences and ideas concerning using projects in mathematics courses.

Thursday, 9 November 2006
4:10 p.m. in Math 109

What's Really Being Taught
vs.
What's Really Being Tested
vs.
What's Being Said
Alan Zollman
Northern Illinois University

A study of the mathematics content correlations of states' high-stakes tests compared to the state's learning standards, and compared to the curriculum actually taught in school districts. The results identify why money and effort have not changed test scores.

Thursday, 26 October 2006
4:10 p.m. in Math 109

Color-Permuting Automorphisms of Cayley Graphs
Liz McMahon
Lafayette College

Given a group G with generators Δ, it is well-known that the set of color-preserving automorphisms of the Cayley color digraph CayΔ(G) is isomorphic to G. Many people have studied the question of finding graphs (including Cayley graphs) with a given automorphism group G, the graphical regular representation problem. This talk asks a different question: how much larger than G can the full (digraph) automorphism group of a given Cayley graph for G be? The question doesnt have a complete answer yet, so we will survey results known so far. All of these concepts will be defined, and many lovely (and colorful) pictures will be shown.

Thursday, 12 October 2006
4:10 p.m. in Math 109

Teaching Excellence Series

Evaluating Student Writing for Mathematicians
Nancy Mattina
Director, The Writing Center

Faculty in all disciplines ask students to create texts as a way to show what they are learning in a course. But many faculty have no recent, formal training in designing and evaluating writing assignments. This colloquium will address how to evaluate the effectiveness of the writing assignments you make and the quality of the students' written work you receive. We will look at sample grading rubrics and assignments in mathematics courses and consider how they relate to grading, plagiarism, general education, and faculty workload at UM.

Thursday, 5 October 2006
4:10 p.m. in Math 109

Book Presentation:

Shift-Invariant Uniform Algebras on Groups,
Birkhauser, 2006
Thomas Tonev
University of Montana

T he book is of interest to mathematicians interested in analytic functions and commutative Banach algebras; researchers, graduate and post-graduate students familiar only with the fundamentals of complex and functional analysis. Its central subject - the theory of shift-invariant algebras - is an outgrowth of the established theory of generalized analytic functions. Associated subalgebras of almost periodic functions of real variables and of bounded analytic functions on the unit disc are carried along within the general framework. There are given characterizations of semigroups such that classical theorems of complex analysis hold on the associated shift-invariant algebras. Bourgain algebras, orthogonal measures, and primary ideals of big disc algebras are described. The notion of a harmonic function is extended on compact abelian groups, and corresponding Fatou-type theorems are proven. Important classes of inductive limits of standard uniform algebras, including Blaschke algebras, are introduced and studied. In particular, it is shown that algebras of hyper-analytic functions, associated with families of inner functions, do not have a big-disc-corona.

Thursday, 28 September 2006
4:10 p.m. in Math 109

Scissors, Glue, and Hilbert, Too.
Hilbert's Third Problem
Seth Braver
University of Montana

Your seventh grade teacher may have taught you how to cut a triangle into pieces and glue them together to form a rectangle. She did not teach you the analogous trick in three dimensions -- and with good reason. In 1900, Hilbert conjectured (and Max Dehn proved) that it is impossible to cut a tetrahedron into a finite number of pieces and glue them together to form a rectangular box.

I shall discuss the history of this problem, which extends back to Euclid, and present a modified form of Dehn's proof, accessible to anyone who knows what a group is.

Thursday, 14 September 2006
4:10 p.m. in Math 109

On Extensions of Holomorphic Motions
Zbigniew Slodkowski
University of Illinois at Chicago

Holomorphic motions are isotopies f_z(w), holomorphic in the parameter z, of maps that are themselves (usually) not holomorphic but quasiconformal. They were introduced by Mane, Sad and Sullivan in the context of their work on the topological dynamics of rational selfmaps of the Riemann sphere. The talk will start with the outline of this background and then will discuss the solution of the problem of existence of extensions of holomorphic motion of a set to holomorphic motions of the whole Riemann sphere (posed by Sullivan and Thurston. Some applications will be sketched.

Tuesday, 5 September 2006
4:10 p.m. in Math 109