Doctoral Dissertation Defense

“The Szegö Kernel for Non-Pseudoconvex Domains in ℂ² .”
Michael Gilliam
Dept of Mathematical Sciences

Dissertation Committee
Jen Halfpap, Chair (Mathematical Sciences), Eric Chesebro (Mathematical Sciences),
Michael Schneider (Physics and Astronomy), Karel Stroethoff (Mathematical Sciences),
Thomas Tonev (Mathematical Sciences)

Thursday, May 12, 2011
1:10 pm in Math 103

Doctoral Dissertation Defense

Wednesday, May 11, 2011
1:10 pm in Math 103

In 1959 Paul Erds (Canad. J. Math. 11 (1959), 34-38) famously proved, nonconstructively, that there exist graphs that have both arbitrarily large girth and arbitrarily large chromatic number. This result, along with its proof, has had a number of descendants that have extended and generalized the result while strengthening the techniques used to achieve it. We follow the lead of Xuding Zhu (J. Graph Theory 23 (1996), 33-41) who proved that, for a suitable graph H, there exist graphs of arbitrarily large girth that are uniquely H-colorable (a homomorphism property generalizing coloring). We establish an analogue of Zhu's result in a digraph setting with a certain type of homomorphism.

Let C and D be digraphs. A mapping ƒ : V (D) → V (C) is a C-coloring if for every arc uv of D, either ƒ(u)ƒ(v) is an arc of C or ƒ(u) = ƒ(v), and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colorable if it admits a C-coloring and that D is uniquely C-colorable if it is surjectively C-colorable and any two C-colorings of D differ by an automorphism of C. In this dissertation, we prove that if D is a digraph that is not C-colorable, then there exist graphs of arbitrarily large girth that are D-colorable but not C-colorable. Moreover, for every digraph D that is uniquely D-colorable, there exists a uniquely D-colorable digraph of arbitrarily large girth. In this talk, we sketch the proof of the former result, taking care to stress the main techniques over the fine details.

Dissertation Committee:
P. Mark Kayll, Chair (Mathematical Sciences), Min Chen (Computer Science),
Solomon Harrar (Mathematical Sciences), Jennifer McNulty (Mathematical Sciences),
George McRae (Mathematical Sciences)

Singular integrals operators (SIOs) are very important in the study of partial differential equations, operator theory, abstract harmonic analysis, and many other fields. For example, in Fourier analysis, SIOs have connections with some very important problems regarding the convergence of Fourier series. On the real line, the prototypical SIO is the Hilbert transform. We will use the Hilbert transform to highlight common techniques that are used to establish mapping properties of integral operators, while, in the process, addressing common issues that arise while computing specific singular integrals. 

Monday, 9 May 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Doctoral Dissertation Defense

Thursday, May 5, 2011
2:00 pm in Native American Center 105

The most common category considered in (undirected) graph theory is a category where graphs are defined as having at most one edge incident to any two vertices and at most one loop incident to any vertex. The morphisms are usually described as a pair of functions between the vertex sets and edge sets that respect edge incidence. We will relax these conditions to allow multiple edges to be incident to any two vertices, multiple loops to be incident to any vertex, and morphisms will be allowed to map edges to vertices, but they must still preserve edge incidence. With combinations of the three restrictions and relaxations of the three restrictions we define and study five categories of graphs.

One asks when can an abstract system of objects and their morphisms be represented by a familiar system of sets with structure and their structure preserving functions. We answer this question for the categories of graphs giving a characterization of five categories of graphs and their morphisms. We follow the lead and spirit of F. W. Lawvere's groundbreaking categorical characterization of the Category of Sets and Functions (Proc. Nat. Acad. Sci. U.S.A., 52 (1964), pp.1506-1511) and D. Schlomiuk's characterization of the Category of Topological Spaces and Continuous Functions (Trans. Amer. Math. Soc., 149 (1970), pp.259-278).

In both characterizations of the Category of Sets and Functions and the Category of Topological Spaces and Continuous Functions, a list of elementary (or first order) axioms are provided so that when combined with a second order axiom (there exists "small" products and coproducts) a functor equivalence between the axiomatically defined category and the concrete category is formed. We provide such an elementary theory for the five categories of graphs.

Dissertation Committee:
George McRae, Chair (Mathematical Sciences),
Peter Golubstov (Mathematical Sciences),
Kelly McKinnie (Mathematical Sciences),
Nikolaus Vonessen (Mathematical Sciences),
Joel Henry (Computer Science)

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

The use of observations and models to predict the future state of a system is a hallmark of the scientific method that often has practical application. As a result, estimation and prediction are central pursuits across a vast range of disciplines, including the physical, biological, and social sciences; engineering; and finance. In many cases the system of interest is composed of a large number of interacting components that render estimation and prediction difficult. This challenge motivates this talk in which I will review essential aspects of, and the basic theory for, estimating and predicting complex systems. One such system, Earth's atmosphere, will be used to illustrate techniques that deal with complexity. The success of these methods for reducing uncertainty in weather forecasts will be contrasted against a failure to reduce uncertainty in climate-change forecasts. This contrast motivates a mathematically based reconsideration of model formulation and calibration for complex systems.

Monday, 25 April 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Sponsored by a gift from Dr. Frank Gilfeather '64.

In 1966 S. Hedetniemi conjectured that the chromatic number of the categorial product of two graphs with finite chromatic number would be the minimum of the chromatic number of the two graphs. In 1985 A. Hajnal found two graphs with an uncountable chromatic number that when taken together in the categorial product, produce a graph with a countable chromatic number. However Hedetniemi's conjecture remains open today.

In this talk, we will explore a portion of the results and techniques research into this conjecture has created as well as my own research into a new technique that provides a special case of the conjecture.

Thursday, 21 April 2011
3:10 p.m. in Math 211
4:00 p.m. Refreshments in Math Lounge 109

Doctoral Dissertation Defense

Monday, April 18, 2011
3.00 pm in Native American Center 105

Despite the importance of teaching proof in any undergraduate mathematics program, current mathematics education research has documented student difficulties with proof (Dreyfus, 1999; Harel & Sowder, 2003; Selden & Selden, 2003; Weber, 2004). In this qualitative case study, conducted over the course of one academic year, nine undergraduate students were each interviewed once every two weeks. Based on a pre-determined set of problems, in each interview, students were asked to complete, evaluate or discuss mathematical proofs. The results of these interviews were then qualitatively analyzed using two different frameworks. The first focused on proof type, which refers to what kind of proof is created and how it comes about. The second framework addressed identifying each student’s proof scheme, which “constitutes ascertaining and persuading for that person” (Harel & Sowder, 1998). Using these structures as a guide, the question researched was: What, if any, identifiable paths do students go through while learning to prove? Even though, no paths were found common to all participants, the results of the study indicate that as students become more comfortable with proof, they are inclined towards a certain proof type, semantic, and this coincides with becoming more successful with proof writing in general.

Dissertation Committee:
Bharath Sriraman, Chair (Mathematical Sciences),
James Hirstein (Mathematical Sciences),
Thomas Tonev (Mathematical Sciences),
David Erickson (School of Education),
Libby Knott (Washington State University, Mathematical Sciences)

In this presentation I will discuss a mathematical structure suitable for studying systems which somehow deal with information. I will show that the appropriate structure is a category which satisfies some additional axioms. Within such category we will be able to introduce problems of decision making (or information processing) and address a question: what does it mean that one source of information is more informative than the other. We will see that many such categories can be constructed as monoidal Kleisli categories. The key ingredients for this construction are: a base category with products ("deterministic" morphisms), a functor, producing objects of "distributions" and a natural transformation, representing "independent product of distributions". Several examples of such categories will be discussed.

Monday, 28 March 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

A sheaf on a Heyting algebra H is a functor F : H^op → SET that satisfies the Gluing axiom. In this talk we explore sheaves from a semicategorical perspective and show that a sheaf is an idempotent semifunctor F : H^co → REL. The key difference in the outlook is that in the usual sheaf theoretical perspective H is interpreted as a multi object category while the semicategory setting views H as an one object supremum enriched semicategory. Using this latter perspective we expand on Higg's Q-valued set version of sheaf theory to construct a semicategory theory of sheaves in terms of semifunctors. 

Tuesday, 22 March 2011
1:10 p.m. in Math 211
4:00 p.m. Refreshments in Math Lounge 109 

This talk will review the basics of the game of Nim and look at the Game of Simplicial Nim. It is well known in combinatorial game theory that any impartial game is equivalent to a game of Nim. This equivalence is called the Grundy value of a game. After looking at how this equivalence works we next look at how we can extend these ideas to simplical sets. Finally we will explore some variations of simplicial nim that I know nothing about, but I am convinced there is more to the story.

Monday, 21 March 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge

Doctoral Dissertation Defense

Thursday, March 17, 2011
12:30 pm in Native American Center 105

Recent educational reforms have called for greater incorporation of informal (i.e. creative, investigative and generative) approaches to learning mathematics in K-12 classrooms (NCTM, 1991; NCTM, 2000). In spite of such reforms, mathematics education research indicates the continuance of formal (i.e. rule-driven, algorithmic) approaches to the subject in elementary school teacher candidates. Consequently, the literature has called for teacher educators to challenge pre- service teachers’ formal notions of mathematics (i.e. Seaman, et al., 2005). This dissertation study reveals that informal mathematical activity coupled with personal reflection has a transformative effect on the formal beliefs of elementary school teachers. Contrasting results were obtained for beliefs about mathematics as opposed to beliefs about mathematics instruction. The transformation of beliefs was also found to depend upon the nature of the informal activity. These differential results have prompted a “critical zone” theory for the use of such activities as agents of educational reform in elementary school teacher preparation.

Dissertation Committee:
Bharath Sriraman, Chair (Mathematical Sciences),
Albert Borgmann (Philosophy),
Jon Graham (Mathematical Sciences),
James Hirstein (Mathematical Sciences),
Ke Norman (Mathematical Sciences)

An historic graph theoretic result is Paul Erdős' 1959 theorem that there exist graphs that have both arbitrarily large girth and arbitrarily large chromatic number. This was interesting not only because it was a surprising achievement, but also because of the proof's elegant use of probabilistic arguments. In this talk, we will explore the history of this result and its descendants and briefly describe how my own research fits into this body of work. 

Monday, 14 March 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109  

Recently attempts have been made to dissolve the measurement problem in standard quantum mechanics by arguing that the state of a quantum mechanical system is to be interpreted as a credence function of a rational agent instead of a representation of the underlying properties of a quantum system. Towards that end, there has been an attempt to prove that under certain rationality constraints, and agent must set their degrees of belief equal to quantum probability values. In this talk I criticize one of the assumptions of the proof, and ultimately argue that even if one accepts that assumption that there is a crucial flaw in the derivation that cannot be fixed. The result is that this approach to resolving the measurement problem in standard quantum mechanics is unacceptable. 

Thursday, 3 March 2011
3:10 p.m. in Math 211
4:00 p.m. Refreshments in Math Lounge 109

Effective inquiry-oriented teaching demands sensitivity toward students’ common ways of thinking about content, as does the design of tasks to support that teaching. In the research literature, little is known about the variety of ways in which students think about matrix multiplication, which is a fundamental part of any introductory undergraduate linear algebra course. In my talk, I will discuss a framework I have developed for understanding student thinking about matrix multiplication. The data for my study comes from the first in a series of classroom teaching experiments (CTEs) in inquiry-oriented undergraduate linear algebra. The framework for student thinking about matrix multiplication was developed based on data which came from a set of semi-structured clinical interviews conducted with students about halfway through the semester of the aforementioned initial CTE. In addition to presenting this framework for student thinking about matrix multiplication, I will discuss the ways in which the analysis of the interview data informed the revision of an instructional sequence designed to support students’ learning of span, linear dependence and independence. 

Monday, 12 December 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

It is widely accepted that textbooks play a prominent role in the teaching of mathematics in K-12 schools in the U.S. The particular textbook a teacher uses, and how the teacher implements this curriculum, can influence not only what students learn, but also how they learn it. This study examined the effect of two types of mathematics content organization on high school students’ mathematics learning. The study involved more than 2000 students and 33 teachers in 5 states. In this presentation, we will discuss the main results of the first year of this longitudinal study and the development of the assessment instruments employed.

Wednesday, 7 December 2011
4:10 p.m. in Math 103
3:30 p.m. Refreshments in Math Lounge 109

Presentation of Master’s Project

Plant ecologists are interested in estimating survival and other demographic rates for dormant plants. When dormant, plants may be alive but unobservable for one or more years. When individuals are alive but unobservable, separating mortality from dormancy can be difficult. Recently, researchers have proposed using multistate mark-recapture models to estimate survival and transition rates of plants that have dormant states. We used a simulation approach to explore the ability of a multistate mark-recapture model and two restricted models to estimate survival and transition rates for three plant species with frequent dormancy. All of the methods we considered produced biased estimates for survival and transition probabilities. Furthermore, for survival estimates, the pattern of survival across states was consistent for all three species regardless of the pattern in true survival. Our results suggest that estimates from current methods should be interpreted with caution and additional data on individual fates may be required.

Tuesday, December 6, 2011
3:10 pm in Math 211

Master’s Committee:
Solomon Harrar, Chair (Mathematical Sciences),
David Patterson (Mathematical Sciences),

Paul Lukacs (Wildlife Biology)

Measuring mutation rates are very important today for a variety of reasons including investigating multiple antibiotic resistance, and measuring disease/infection progression (Cystic Fibrosis). Mutation is also thought to be the sole mechanism by which natural selection acts upon. Current methods of estimating mutation rates of unicellular organisms are somewhat more complicated than they could be and/or expensive. This presentation will cover current methods and propose and test a new method that results from analysis of a mathematical model for mutation. Results will be presented and an application of the mathematical model will be discussed.

Monday, 5 December 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

We briefly describe the ideas and results which have led to finding very large prime numbers, and to finding efficient ways of distinguishing prime numbers from composite numbers. Our story begins with Ancient Greeks and perfect numbers, it includes the ideas and contributions of Fermat in the 17th century, Euler in the 18th, Lucas in the 19th, the huge amount of progress in the area of primality tests during the 20th century, and the celebrated result of Agrawal, Kayal, and Saxena, which in 2002 surprised the international scientific community with the discovery of the first general polynomial-time primality test. We also describe the so-called "practical version" of AKS, in which we played a part. We end by discussing some contributions on the subject obtained by our research group. 

Friday, 2 December 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Engineering design and scientific inquiry are approaches used by STEM professional in their work as they explore the natural world and invent solutions that transform our lives. These approaches are also ideal for engaging students in STEM learning by providing the conditions that allow them to explore and invent too. In this session we will explore the similarities and differences between inquiry and design. Using a couple of hands-on activities as context we will discuss how design and inquiry can be used to teach a wide range of STEM content.

Monday, 14 November 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Craig Bauling

Wolfram Research

Wednesday, October 12
4:00 — 6:00 pm, Math 306

"Short course in Mathematica" video preview

Colloquium

On-Base Percentage — the Moneyball Statistic 

The book Moneyball talks about the efforts of the Oakland Athletics to build a team using sabermetrics. Oakland learned that it was important and relatively inexpensive to sign players who were effective in getting on-base. We look at on-base percentages in the history of Major League Ball. By fitting a random effects model, we see how the distribution of on-base talent has changed over time. In addition, we find particular players who have unusually high on-base percentage seasons. We focus on the career on-base trajectories of 16 players and discuss how we can effectively simultaneously estimate their true trajectories. 

Monday, 19 September 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109 

Evening talk

Looking at Spacings to Assess Streakiness 

There is much interest in studying the hot hand behavior of individuals and teams in sports. We collect success/failure data for each of the hitters in a particular baseball season. We develop a Bayes factor statistic on the spacings to see if players are truly streaky. By looking at the ensemble of test statistics over all players, we see if the pattern of streakiness is more than one would predict from a consistent hitting model. We see that the pattern of streakiness depends on how one defines a success in a plate appearance. 

7:00 p.m. in Math 103 

Tuesday, September 6

12:10 — 1:30, UC 330-331
Student Forum: "Math, the language that connects mankind."

3:10 — 4:00 pm, Math 103

Colloquium & Undergraduate Mathematics Seminar: "Mathematics changed my life" (See below)

Wednesday, September 7

12:10 — 1:30, UC 330-331
Chair, Staff & Administrator Workshop: "Retaining Minority Students and Increasing the Number of Majors"

3:00 — 4:00 pm, UH 004
SACNAS Informational Meeting

It almost killed me at first. In my first semester in college I was a chemical engineering major and enrolled in calculus. I could not understand my calculus class and dropped it, falling back to college algebra and trig. That first semester I earned nine units of Ds (It is hard to believe but I do have a Ph.D. in mathematics and hold the post of University Distinguished Professor).

I made a momentous decision in my second semester in college. I dropped engineering but re-took calculus. I did OK. In my fourth semester I decided I was going to earn a PhD in mathematics or physics. I was hooked, I was fascinated. It changed my life. The fact that I made that decision to continue on in mathematics provided me with the tools to address a complex array of problems. Understanding mathematics has been fun, applying it to solve problems dealing with military communication systems has been exciting, introducing students to world-shaking mathematical ideas continues to be exhilarating. This is the message that I try to convey to students.

My first semester experience in mathematics is all too common for first year students. My second semester experience is all too rare. Yet mathematics is now more important to our citizenry than ever before. Mathematical thinking, and the tools that mathematicians have developed, now permeate the fabric of modern life. Purchasing groceries (error-detecting codes), withdrawing money from an ATM (cryptography) or looking at the weather maps on the evening news (massive mathematical models describing the behavior of the air and moisture) are all part of a sophisticated mathematical infrastructure.

If our society is to survive and prosper, we have to produce a mathematically literate society. I have decided to dedicate the final years of my academic life to convince our students that mathematics is relevant, that it is fascinating, and that students arriving on campus should increase the mathematical content of their course work. It is important that parents, K-12 teachers and university mathematicians understand how critical mathematics is for our students and for our communities.

In this talk I will describe some of the problems in number theory that have fascinated me and how that very theoretical preparation, and my own view that mathematics is applicable, led a number theorist to play the role of applied mathematician. I will also relate how this background prepared me for my current position, where I try to convince everyone to become a math major. 

Tuesday, 6 September 2011
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109 

Sponsored by the Diversity Council, the Office of the President, the Office of the Provost, the Office of Student Affairs & the Department of Mathematical Sciences. 

Cancelled