**Emily Stone**

Email

# 2008 Colloquia

## Spring 2008

### May 16 - Noncommutative Uniform Algebras

**Noncommutative Uniform Algebras**A uniform algebra A is a Banach algebra such that ||*f*^{2}||= ||f||^{2}, for all f ∈ A. It is well known that any *complex* uniform Banach algebra is automatically commutative and is isometrically isomorphic with a subalgebra of C_{C}(X) [Hirschfeld and Żelazko, 1968]; such algebras are the most classical and well studied ones. Commutative, real uniform algebras have also been studied for years. Any such algebra *A* is isometrically isomorphic with a real subalgebra of C_{C}(X) for some compact set *X*; furthermore in most cases *X* can be just divided into three parts X_{1}, X_{2} and X_{3} such that A_{|x1} is a complex uniform algebra, A_{|x2} consists of complex conjugates of the functions from A_{|x1}, and A_{|x3} is equal to C** _{H}**(

*X*

_{3)}. In case of real uniform algebras the condition ||

*f*

^{2}||= ||f||

^{2}no longer implies commutativity - the algebra of quaternions serves as the simples counterexample. On the other hand there have been very little study of non commutative real uniform algebras. We show that any such algebra is isometrically isomorphic with a real subalgebra of C

**(**

_{H}*X*) - the algebra of all continuous functions defined on a compact set

*X*and taking values in the field

**H**of quaternions. We will also produce a general structural description of such algebras and address the question whether there are non trivial example of such algebras other then the direct sum of the entire algebra C

**(**

_{H}*X*) and a commutative real uniform algebra.

Friday, 16 May 2008

2:10 p.m. in 103

1:30 p.m. Refreshments in Math Lounge 109

### May 9 - Classroom Voting in Contemporary Mathematics

Presentation of Master’s Project

**Classroom Voting in Contemporary Mathematics**“I hate math,” “I’ve never been good at math,” “Thank goodness this is the only math class that I have to take.” These are the common comments that instructors of lower level math courses hear from their students when asked to respond to the question “how do you feel about math.” With this in mind, an alternative instruction method seemed appropriate to try in Contemporary Mathematics, Math 107.

With my overarching goal of providing a positive learning environment and fostering a more positive attitude about math, my next step was to determine the best way to utilize a new instruction method: classroom voting. In this way, my project became two-fold: developing a test bank of voting questions and supporting math achievement through a more positive attitude towards math. Challenges included the proper implementation of the question/answer instruction process, designing the multiple-choice voting questions and their corresponding answers, and collecting the necessary quantitative and qualitative data to support my findings.

Friday, May 9, 2008

2:10 pm in Math 108

### May 7 - Topos-like Properties in Two Categories of Graphs and Graph-like Features in an Abstract Category

Presentation of Master’s Thesis

**Topos-like Properties in Two Categories of Graphs**

**and Graph-like Features in an Abstract Category**Often in the study of graph theory, the usual notion of a graph is that of a simple graph with at most one edge between vertices, and at most one loop on any vertex (some say no loops). The usual notion of a graph homomorphism is a mapping of graphs that sends vertices to vertices, edges to edges, and preserves incidence of the mapped vertices and edges. A more general view is to create a category of objects and morphisms that allows the graphs (i.e. the objects) to have multiple edges between two vertices and multiple loops at a vertex, coupled with more general graph homomorphism (i.e. the morphisms) that allows edges to be mapped to vertices as long as that map still preserves incidence, and of course, still maps vertices to vertices. Here this more general category of graphs is named theCategory of Conceptual Graphs.

We investigate three topos defining properties of two subcategories of the Category of Conceptual Graphs. The first subcategory is the Category of Simple Loopless Graphs with Strict Morphisms in which the graphs are simple and loopless and the morphisms are restricted to only sending edges to edges (i.e. strictly), and the second subcategory is the Category of Simple Graphs with Strict Morphisms where at most one loop is allowed on a vertex. We find that these two graph categories have only a few topos-like properties. We also define some small graph-like objects in an abstract category that are their graph counterparts when viewed in any of the concrete categories of graphs. We also study these graph-like objects in some other familiar (concrete) categories, e.g. the Category of Abelian Groups and Homomorphisms and the Category of Sets and Functions.

May 5, 2008

3:10 pm in Math 108

### May 5 - Examining the Skew-Normal Distribution

Presentation of Master’s Project

**Examining the Skew-Normal Distribution**We wish to examine the skew-normal distribution, its properties, and provide proofs. We then wish to extend our exploration from this vantage point. Along the way, we will explore variance-mean mixture models, the variance-gamma (VG) distribution, and extend to a skew-variance-gamma (SVG) distribution. To accomplish this, we will first analyze and defend much of the results in Azzalini (1985, A Class of Distributions which Includes the Normal Ones. *Scandinavian Journal of Statistics*, 12, 171-178).

May 5, 2008

4:10 pm in Math 311

### May 2 - “Spectral Preserver Problems in Uniform Algebras”

Doctoral Dissertation Defense

**“Spectral Preserver Problems in Uniform Algebras”**There has been much interest in characterizing maps between Banach algebras that preserve a certain equation or family of elements. There is a rich history in such problems that assume the map to be linear, so called linear preserver problems. More recently, there has been an interest in not assuming the map is linear a priori and instead to assume it preserves some equation involving the spectrum, a portion of the spectrum, or the norm.

After a brief introduction to uniform algebras, we give a rigorous development of the theory of boundaries. This includes a new alternative proof of the famous Shilov Theorem. Also a generalization of Bishop's Lemma is given and proved. Two spectral preserver problems are introduced and solved for the class of uniform algebras. One of these problems is given in terms of a portion of the spectrum called the peripheral spectrum. The other is given by a norm condition.

The first spectral preserver problem concerns weakly-peripherally multiplicative maps between uniform algebras. These are maps T : *A*→*B* such that σ* _{∏}*(TƒT

*g*)∩σ

_{∏}(ƒ

*g*)≠Ø for all ƒ ,

*g*∈

*A*where σ

_{∏}(ƒ) π σ f is the peripherial spectrum of ƒ. It is proven that if

*T*is a weakly-peripherally multiplicative map (not necessarily linear) that preserves the family of peak functions then it is an isometric algebra isomorphism.

The second of these preserver problems shows that if *T* : *A*→*B* is a map (not necessarily linear) between uniform algebras such that ∥Tƒ*Tg* +1∥ = ∥ƒ*g* +1∥ for aƒ ,g∈*A* then *T* is a weighted composition operator composed with a conjugation operator. In particular, if *T* (1) =1 and *T* (*i*) = *i* then *T* also is an isometric algebra isomorphism.

**Committee**: T. Tonev (Chair), J. Halfpap, K. Stroethoff, K. Yale, and E. Uchimoto (Physics and Astronomy)

Friday, May 2, 2008

3:10 – 5:00 pm in Math 103

### May 1 - The Peripheral Spectrum and Function Algebra Isomorphisms

**The Peripheral Spectrum and Function Algebra Isomorphisms**There is a long history of studying linear maps between Banach algebras that preserve some non-algebraic structure, such as invertibility or the norms of elements, and determining whether this necessarily implies the map must also have further algebraic structure, such as being multiplicative or a weighted composition operator. In recent years, researchers have begun to ask similar questions about maps that are not assumed to be linear. Here we will present results demonstrating that maps between uniform algebras or Lipschitz algebras that satisfy certain preservation properties with respect to subsets of the spectrum must be isometric algebra isormorphisms or weighted composition operators, and this holds regardless of the fact that we do not assume the linearity of the maps.

Thursday, 1 May 2008

4:10 p.m. in 103

3:30 p.m. Refreshments in Math Lounge 109

### May 1 - “Regularization Methods for Ill-Posed Poisson Imaging Problems: Mathematical Theory”

Doctoral Dissertation Defense

**“Regularization Methods for Ill-Posed Poisson Imaging Problems: Mathematical Theory”**The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative logarithm of the Poisson likelihood functional in place of the ubiquitous least squares fit-to-data functional. However, if the underlying mathematical model is assumed to have the form *z* = *Au*, where *A* is a linear, compact operator, the problem of minimizing the negative-log of the Poisson likelihood functional is ill-posed, and hence some form of regularization is required. For us, this involves solving a variational problem of the form

def

u_{∝} = arg min ℓ (*Au*;*z*) = ∝*J*(*u*)

u≥0

where ℓ is the negative-log of the Poisson likelihood functional, and *J* is a regularization functional. The main result of this thesis is a theoretical analysis of this variational problem for three different regularization functionals. However, we also present an efficient computational method for its solution and we demonstrate the effectiveness of the approach in practice by applying the algorithm to simulated astronomical imaging data corrupted by typical CCD camera noise.

Thursday, May 1, 2008

12:30 – 2:00 pm in Math 108

### April 30 - Departmental Awards Ceremony

4:10 p.m. - Dell Brown Room

Reception at 3:30

### April 28 - A Comparison of Analytical Methods for DNA Microarrays

Presentation of Master’s Project

**A Comparison of Analytical Methods for DNA Microarrays**The ability to measure gene expression and accurately determine whether a gene or a set of genes is differentially expressed has become a popular area of investigation. This is partly a result of the fact that the differential expression of genes can be indicative of the health or changing health of an animal. The use of DNA Microarrays, in particular the Affymetrix oligonucleotide gene chip, to obtain gene expression measurements has become commonplace, but with such technology arises potential roadblocks. One such roadblock is analyzing the overwhelming amount of data that is acquired with any experiment implementing the use of DNA Microarrays. The focus of this paper is, therefore, to develop and compare two methods for analyzing gene expression measurements obtained from oligonucleotide gene chips.

The first of these methods, which has become a standard approach, is a parametric empirical Bayes method in which a mixture model is constructed and then used to estimate the mixing probabilities that correspond to the possible patterns of differential expression. The second method, which is of our design, is a permutation test implementing a multivariate test statistic that is calculated using the probe-level data. Using a range of posterior probabilities for the empirical Bayes method and variety of false discovery rates for our method, the error rates of the two approaches are then compared using a benchmark data set.

April 28, 2008

4:10 pm in Math 211

### April 24 - Regularization Methods for Ill-Posed Poisson Imaging Problems: An Introduction and Overview

**Regularization Methods for Ill-Posed Poisson Imaging Problems: An Introduction and Overview**In this talk, we discuss the following deblurring method: solve

def

u_{∝} = arg min ℓ (*Au*;*z*) = ∝*J*(*u*)

u≥0

where *z* is blurred, noisy data, *A* is a compact operator,ℓ is the negative-log of the Poisson likelihood function, α > 0 is the regularization parameter, and *J* is the regularization functional. We will discuss the notions of ill-posedness, regularization, and also how the choice of the functional *J* effects the properties of the deblurred image *u _{α}*.

Lastly, we will present the computational technique used in practice to obtain *u _{α}* and some numerical results with three different regularization functions.

Our goal will be to present the main ideas of our work in a way that is accessible to a broad audience.

Thursday, 24 April 2008

4:10 p.m. in 103

3:30 p.m. Refreshments in Math Lounge 109

### April 21 - Traps and Pitfalls in Statistical Decision-Making

**Traps and Pitfalls in Statistical Decision-Making**Abraham Maslow is credited with the quote "When the only tool you have is a hammer, then every problem begins to look like a nail." Academic researchers frequently become comfortable with the mechanics of research design and statistics in their respective fields of inquiry without considering whether these methods are most efficient for answering their research question. Research is a multi-faceted system comprised of numerous elements, in which the modification of any one element will affect the entire system. Despite the delicate interplay of research design elements and statistical analysis, often little attention is paid a priori to the issue of statistical conclusion validity. For example, it is not uncommon for individuals to focus primarily on whether results are "statistically significant" while ignoring the question of "whether this statistical decision is valid, or if so, what substantive interpretation is justified by it?" Anticipating common traps and pitfalls that threaten the validity of the statistical decisions is beneficial for all members of the research team. This presentation, drawn from many years of advising graduate students and faculty on research design and statistical analysis in basic and applied sciences, will address strategies for maximizing efficiency in design and reducing the potential for invalid statistical decisions.

**About the presenter**: Dr. Williams is a Professor of Dental Public Health and Behavioral Sciences at the Clinical Research Center Director at University of Missouri – Kansas City. Dr. Williams has been involved in the design, assessment of validity and reliability, and analysis of numerous clinical, patient outcome and psychometric studies over the past 18 years. As Director of the Clinical Research Center, she is integral in the design, implementation and statistical analysis of a variety of federal- and industry-funded clinical trials. Dr. Williams actively consults in the research endeavors of other faculty and has served as the primary methodologist and statistician for more than thirty clinical research projects. Dr. Williams' expertise in the area of psychometrics has been pivotal in the development of numerous survey instruments for assessing critical thinking ability, knowledge, attitudes, and behaviors of health professionals. She is the lead biostatistician on an AIDS medication trial (R01 MH68197) and the primary methodologist for a HRSA funded health promotion project aimed at empowering individuals with severe mental illness.

Monday, 21 April 2008

4:10 p.m. in 103

3:30 p.m. Refreshments in Math Lounge 109

Sponsored by UM PACE, PArtnerhsip for Comprehensive Equity

Hosted by Solomon Harrar (Mathematical Sciences)

& Kari Harris (Public and Community Health Sciences)

### April 16 - Size-Structured Population Models: Sensitivity & Probabilistic/Stochastic Formulations for Inclusion of Uncertainty/Variability in Growth

**Size-Structured Population Models:**

**Sensitivity & Probabilistic/Stochastic Formulations for Inclusion of Uncertainty/Variability in Growth**Motivated by current efforts in modeling growth and mortality in shrimp populations to be used in vaccine production systems, we consider the classical Sinko-Streifer size-structured population model and derive sensitivity partial differential equations for the sensitivities of solutions with respect to initial conditions, growth rate, mortality rate and fecundity rate. Sample numerical results to illustrate use of these equations are also presented. Furthermore, we compare two approaches for inclusion of uncertainty/variability in modeling growth in these size-structured population models. One entails imposing a probabilistic structure on growth rates in the population while the other involves formulating growth as a stochastic Markov diffusion process. We present a theoretical analysis that allows one to include comparable levels of uncertainty in the two distinct formulations in making comparisons of the two approaches.

Wednesday, 16 April 2008

4:10 p.m. in 103

3:30 p.m. Refreshments in Math Lounge 109

### April 15 - Math film Festival

### April 14 - Is fair voting and apportionment possible? A mathematical look

The Department of Mathematical Sciences is pleased to present a special Colloquium talk for Math Awareness Week:

**Is fair voting and apportionment possible? A mathematical look**Michael de Villiers is co-author of *Is Democracy Fair?*, and was inspired to draft this book by the first democratic elections in 1994 in his home country of South Africa, where he teaches mathematics at the University of KwaZulu-Natal. As he watched his country emerge from apartheid and, in its formative democratic stage, struggle with deciding on the "best" systems of voting and apportionment, de Villiers examined the many options that exist and the role mathematics plays in political representation. The talk will have three objectives:

- To demonstrate how mathematics applies to the analysis of problems in the seemingly non-mathematical areas of social and political science
- To challenge the stereotype that mathematics is of value only in certain applied sciences, such as physics, chemistry, and computer science
- To raise voter education awareness by exposing them to a variety of election decision procedures and methods of apportionment and their strengths and weaknesses

Monday, 14 April 2008

4:10 p.m. in NULH

3:30 p.m. Refreshments in Math Lounge 109

Michael de Villiers' visit is supported by a gift from Dr. Frank Gilfeather '64.

### April 3 - The Proof-Writing Process: A Look at Novice Mathematics Students

**The Proof-Writing Process: A Look at Novice Mathematics Students**Proof writing is a serious concern for all mathematicians. Educators must also be concerned with how to instruct students in learning to write mathematical proofs. Math education research can lend answers to how students write proofs, what strategies they are using in the process, the products of proof writing, and how to teach students to write proofs. This talk is based on my recently completed study designed to describe the detailed processes and strategies used during the proof-writing process in order to more completely understand this process.

Specifically, this study is designed to answer the following questions:

- What are the proof writing strategies of a novice mathematics proof writer?
- What strategies are in use during a successful proof writing attempt?
- In what specific ways do novice mathematics proof writers use heuristics or strategies when working through a proof, which go beyond the application of standard problem-solving heuristics?
- Do the strategies used by individuals remain constant across multiple questions or do the questions affect the choice of strategies?

In this talk, we will take an in-depth look at a few key participants in the study and investigate what can be learned from their work. I will describe their work, tendencies, errors, and successes, and give some ideas of what this data could tell us about proof-writing strategies. This talk is designed to give a flavor of math education research and to gain an appreciation of what can be learned through such research. I will also discuss further research opportunities for how we can continue to study the proof-writing process.

Thursday, 3 April 2008

4:10 p.m. in 103

3:30 p.m. Refreshments in Math Lounge 109

### April 3 - “Proof Processes of Novice Mathematics Proof Writers”

Doctoral Dissertation Defense

**“Proof Processes of Novice Mathematics Proof Writers”**Previous studies have shown that a large portion of undergraduate mathematics students have difficulties constructing, understanding, and validating proofs. However, proofs are the foundation of mathematics; it is therefore essential that every university mathematics student be able to step through the proof writing process. Even at the secondary education level, students should be able to “produce logical arguments and present formal proofs that effectively explain their reasoning” (NCTM, 2000).

Previous research has sought to describe the strategies involved in the process of mathematical problem solving. It has covered both the larger process, based on Polya (1973), of read the problem, devise a plan, implement the plan, and verify the results, as well as the individual strategies which would be included within these larger categories.

This study seeks to describe those distinct strategies specifically related to the process of proof writing. The framework for this study was based on the three large categories of proof-writing processes defined by Weber (2004): procedural, syntactic, and semantic processes. This study expands on Weber’s definitions by adding to the description of the individual strategies involved in each of these types of proof productions. That is, it seeks to describe the detailed processes and strategies used during the proof-writing process in order to more completely understand this process.

Specifically, this study is designed to answer the following questions:

- What are the proof writing strategies of a novice mathematics proof writer?
- What strategies are in use during a successful proof writing attempt?
- In what specific ways do novice mathematics proof writers use heuristics or strategies
- when working through a proof, which go beyond the application of standard problemsolving
- heuristics?
- Do the strategies used by individuals remain constant across multiple questions or do the
- questions affect the choice of strategies?

Thursday, April 3, 2008

9:10 a.m. in Jeanette Rankin Hall 205

### March 20 - Teaching Quantitative Literacy

**Teaching Quantitative Literacy**As individuals and as a society in the 21st century, we face challenges and choices that could affect our survival as a species. Understanding those issues and making informed decisions require fundamental quantitative skills that all college and university students should possess. In courses, careers, and life, students will face decisions about personal finance, voting issues, food, lotteries and gambling, risky behavior, and taxes. They ought to have the skills to understand the federal debt, the mathematics of pollution and deforestation, and health care issues. As teachers and students, we are all called to higher levels of quantitative literacy in order to be more effective citizens.

The grand challenge is to infuse quantitative literacy (QL) throughout the undergraduate curriculum. This talk will have a more modest theme: implementing a single QL course for liberal arts students. I will begin by briefly surveying the rationale for developing a QL course and justifying such a course as a legitimate alternative to college algebra for liberal arts students. Most of the talk will consist of examples of activities, problems, and projects used in an existing QL course. Discussion is encouraged and can focus on the implementation of a QL course or on the grand challenge of QL across the curriculum.

Thursday, 20 March 2008

4:10 p.m. in Math 103

3:30 p.m. Refreshments in Math Lounge 109

Sponsored by the Department of Mathematical Sciences & the Office of the Provost with support from Pearson Education Math & Science

### March 4 - Surfaces, 3-manifolds, and character varieties

**Surfaces, 3-manifolds, and character varieties**The geometry and topology of a surface can be understood by cutting it along essential curves. The success of this approach to surfaces has motivated a similar approach in dimension 3. In this setting, instead of looking for essential curves in a 2-dimensional space, we are interested in essential surfaces in a 3-dimensional space.

Essential surfaces are difficult to find - many manifolds do not contain any - and finding them is a fundamental difficulty in 3-manifold topology. The character variety and machinery of Culler and Shalen provides one of the best ways to find these surfaces. Their work was critical in many of the most important achievements in 3-manifold topology over the last 35 years.

I will introduce their techniques and talk about two related questions that I have answered in my research.

Monday, 4 March 2008

4:10 p.m. in Math 103

3:30 p.m. Refreshments in Math Lounge 109

### February 14 - Division algebras over fields of characteristic p

**Division algebras over fields of characteristic p**The real Hamilton quaternions form a four dimensional real vector space with an associative multiplication and an identity element such that every nonzero element has a multiplicative inverse. In fact, this is the only such non-trivial finite dimensional division algebra that exists over the real numbers. The Hamiltonians are an example of a cyclic algebra. If you ask which division algebras occur over the rational numbers you get a plethora of different division algebras, yet they are all still cyclic algebras.

The situation changes slightly when you look for division algebras with dimension *p ^{n}* over a field of characteristic

*p*. In this talk I will discuss characterizations of cyclic algebras along with examples of non-cyclic algebras over a field of characteristic

*p*which remain non-cyclic after any prime to p extension.

Thursday, 14 February 2008

4:10 p.m. in Math 103

3:30 p.m. Refreshments in Math Lounge 109

### Feb 12 - Combinatorics of Trig Functions

**Combinatorics of Trig Functions**We will discuss some historical results on the combinatorics of trig functions. We will show that extensions of these results give generating functions for certain statistics on permutations with given patterns. Some specific tools and techniques for generating functions will be outlined.

Tuesday, 12 February 2008

4:10 p.m. in Math 103

3:30 p.m. Refreshments in Math Lounge 109

### February 4 - Spatial problems in mathematical ecology

**Spatial problems in mathematical ecology**In this talk, I will introduce two spatial problems in theoretical ecology together with their mathematical solutions.

The first part of the talk concerns competition between plants for sunlight. In it, I use a mechanistic Kolmogorov-type competition model to connect plant population vertical leaf profiles (or VLPs) to the asymptotic behavior of the resulting dynamical system. For different VLPs, conditions can be obtained for either competitive exclusion to occur or stable coexistence at one or more equilibrium points.

The second part of the talk concerns the spatial spread of infectious diseases. Here, I use a family of SI-type models to examine the ability of a disease, such as rabies, to invade or persist in a spatially heterogeneous habitat. I will discuss properties of the disease-free equilibrium and the behavior of the endemic equilibrium as the mobility of healthy individuals becomes very small relative to that of infecteds. The family of disease models consists variously of systems of difference equations (which I will emphasize), ODEs, and reaction-diffusion equations.

Monday, 4 February 2008

4:10 p.m. in Math 103

3:30 p.m. Refreshments in Math Lounge 109

### January 31 - Knowledge Assisted Data Management and Retrieval in Multimedia Database Systems

**Knowledge Assisted Data Management and Retrieval in Multimedia Database Systems**The proliferation of multimedia data (e.g., graphics, images, animation, video, audio, text, etc.) and ever-increasing demand for multimedia applications signify the pertinent need for efficient and effective multimedia data indexing, storage and retrieval mechanisms. Due to the special characteristics of multimedia data, the Multimedia Database Management Systems (MMDBMSs) have emerged and attracted immense research attention in recent years.

Though many research efforts have been devoted to this area, it is still far from maturity and numerous open issues need to be addressed. In this talk, a systematic and integrated framework is presented, with the focus on two essential challenges in developing the MMDBMS, i.e., semantic gap and data organization. The techniques and methodologies developed for this framework can be applied to many practical application domains. In this talk, video event/concept mining is used as an example to demonstrate the potential of this framework.

Thursday, 31 January 2008

4:10 p.m. in Math 103

3:30 p.m. Refreshments in Math Lounge 109

## Fall 2008

### December 10 - Reconciling Non-malleability with Homomorphic Encryption

**Reconciling Non-malleability with Homomorphic Encryption**Jomomorphic encryption schemes allow for very simple "computations" on encrypted data, and are therefore convenient tools in the design of conceptually simple cryptographic protocols. Non-malleability is a property of encryption schemes that is necessary when making security guarantees against malicious adversaries. However, these two desirable properties of encryption are mutually exclusive.

In this talk, I will discuss how these two opposing demands can be reconciled into a powerful notion for public-key encryption. An interesting application of these new results is an anonymous data-collection protocol, which is efficient, conceptually simple, yet secure in the strongest known sense. I will also discuss some consequences regarding automated analysis of cryptographic protocols. This is joint work with Manoj Prabhakaran.

Wednesday, 10 December 2008

4:10 p.m. in Social Science 362

Refreshments in same room from 3:30 to 4:00

### November 24 - Statistical Learning in Online Advertising

**Statistical Learning in Online Advertising**We’ll start by discussing some of the data that is created by online advertising and the role that data plays in what is now a $20 billion industry. Then I’ll talk about a few interesting open problems that I’m working on for my dissertation.

Fundamentally, researchers in online advertising are trying to understand how people respond to ads and how to make advertising more effective. I’ll begin by shocking you with the unsophisticated “state-of-the-art” in conversion attribution. Then, having thoroughly destroyed that straw man, we’ll build up a much more sensible mathematical model. After that I’ll touch on another accessible problem or two from the same arena.

By the end of this talk you should have some sense for the kinds of problems that people with mathematical talents are working on at Microsoft, you’ll see some examples of the kinds of data that are being created by the Internet, and you’ll learn a little bit about how statistics is being applied to these problems. If you’re reading this, then this talk is probably accessible to you.

Monday, 24 November 2008

3:10 p.m. in Math 103

4:00 p.m. Refreshments in Math Lounge 109

### November 17 - Cleaning Graphs with a Greedy Algorithm

**Cleaning Graphs with a Greedy Algorithm**Following the decontamination metaphor for searching a graph, suppose every vertex and edge of a graph is initially contaminated, or 'dirty'. Brushes are placed on some vertices and at each step, a vertex is 'cleaned', whereupon it sends one brush along each dirty incident edge (cleaning those edges). Brushes may not traverse clean edges. The model presented is one where the edges and vertices are continually recontaminated, say by algae, so that cleaning is regarded as an on-going process. Ideally, the final configuration of brushes, after all vertices and edges have been cleaned, should be a viable starting configuration to clean the graph again. The minimum number of brushes needed to continually clean a graph is called the brush number and is NP-hard to determine in general. Thus, the main results of this talk will focus on determining the (asymptotic) brush number of random regular graphs using a greedy algorithm.

Monday, 17 November 2008

3:10 p.m. in Math 103

4:00 p.m. Refreshments in Math Lounge 109

### November 10 - A Research Problem that Just Won’t Quit

**A Research Problem that Just Won’t Quit**In this talk, I will present an overview of a research topic that I have been pursuing, off and on, for seven years. In that time, I have approached the problem from a number of angles—theoretically, computationally, and statistically—and it has borne, and continues to bear, nice fruit.

My main goal will be to keep the discussion accessible to a broad audience for as long as possible, always motivating the various explorations from a point of view grounded in intuition, pragmatism, and/or an appreciation for beauty that seems to me to distinguish the mathematician among scientists. In light of the former two, I will motivate the problem from its primary application in image processing.

Monday, 10 November 2008

3:10 p.m. in Math 103

4:00 p.m. Refreshments in Math Lounge 109

### October 6 - Mathematics and the Thirty Meter

*Mathematics and the Thirty Meter*

In this talk we will review the basic principles underlying TMT, a proposed ground-based optical telescope with a thirty meter primary mirror; see the Thirty Meter Telescope website for background information. We will focus on interesting mathematical problems that arise in the implementation of a multiconjugate adaptive optics system for TMT. These include the estimation of the atmospheric turbulence profile from sensor measurements and the control of deformable mirrors to correct for turbulence aberrations that degrade optical images.

Monday, 6 October 2008

3:10 p.m. in Math 103

4:00 p.m. Refreshments in Math Lounge 109

### September 8 - Inference for Multivariate Data

**Inference for Multivariate Data**Multivariate data appear naturally in the life sciences because usually more than one response variable is of interest ("multiple endpoints"). We look at different ways to analyze multivariate data. Methodological advancements in recent years have made it possible to analyze many multivariate data sets using completely nonparametric methods. We will provide an overview of current statistical research on these methods, illustrated by examples. In particular, we will focus on inference methods that can be used when the data have a mix of ordinal and quantitative response variables.

Monday, 8 September 2008

3:10 p.m. in Math 103

4:00 p.m. Refreshments in Math Lounge 109

### August 26 - Mathematica 6 in Education and Research

**Mathematica 6 in Education and Research**

We'll be discussing a wide variety of practical and theoretical applications. Attendees will not only see the new features of Version 6, but will also receive examples of this functionality to begin using immediately. No Mathematica experience is required, and students are encouraged to attend. Please feel free to send me any specific topics you'd like to cover, and I'll do my best to address them in the talk.

Tuesday, 26 August 2008

4:10—6:00 p.m. including Q&A in Math 103

### August 25 - Paced Algebra to Calculus Electronically

**Paced Algebra to Calculus Electronically**At Colorado State University, the Department of Mathematics has for decades offered a five-credit pre-calculus program to its students using a mastery system that has proven effective for preparing them for the various calculus courses we offer. More recently we have improved that program, launching an innovative new online pre-calculus program: Paced Algebra to Calculus electronically (PACe). This new program has demonstrated an increase in student success rates (based on grades A, B, and C) from 57% to 66% . That 66% is high compared with other institutions offering pre-calculus, particularly when one considers our students’ opportunities to return and complete their courses in a subsequent semester. Not only have we shown an increase in overall success rates but the percentage of A’s has increased from 24% to 35%, better preparing students for their calculus courses where we have shown a 10% increase in retention in the subsequent Business Calculus Course.

The PACe Curriculum consists of five one-hour semester credit courses, combinations of which are prerequisite to one of three calculus courses at CSU. PACe was designed based on the best-practices research base for online mathematics instruction and is distinct in many regards. First of all, it is designed from assessment up approach. Instruction focuses on both concept and skill using a multiple representations approach—Numerical, Verbal, Graphical, and Symbolic—throughout the program. A full complement of instructional videos accompanies each unit. Assessment is frequent, and formal proctored assessment includes the application of an innovative new approach to using the TI calculator—we are the development site for using Texas Instruments calculator emulator software in an assessment environment.

Monday, 25 August 2008

3:10 p.m. in Math 103

4:00 p.m. Refreshments in Math Lounge 109