Examining the role culture and ethnicity play in the individual's ability to learn mathematics leads to new expectations in seeking a more diverse population of doctorates in mathematics.

Friday, 13 May 2005
9:00 a.m. in Math 211

During the past twenty years, physicists have been using "chip-firing games" to model events in such naturally occurring systems as the earth's crust, snowfields, and the human brain. These chip-firing game models have been studied primarily via computer simulation. It is the purpose of my dissertation to investigate a chip-firing game using a strictly mathematical approach, thereby producing a method for analytically determining the probability distribution of the length of a game on a given connected graph. In this talk, I will introduce the audience to chip-firing games and contrast the simulation approach with the new analytical methods.

Thursday, 5 May 2005
4:10 p.m. in Math 109

Tibetan counting numbers were used before and after the creation of the United Tibetan Writing System. Mathematics, and especially geometry, has had a great impact on the unique Tibetan architecture (stupas) and craft (thangkas). Mathematics has played an important role in the creation of the Tibetan astronomy calendar, involving calculation of celestial phenomena. The Tibetan Astronomical Science - elementary (Byung-rTsis), and advanced (Skar-rTsis) - has drawn from the knowledge of neighboring countries, such as India and China, and also from Buddhists teachings to establish a unique Tibetan system.

Thursday, 28 April 2005
4:10 p.m. in Skaggs 117

How does gravity, the weakest of the fundamental forces, control the universe of planets, stars, and galaxies? It really boils down to the enormous mass contained in astronomical objects. I will describe some important effects of gravity, ranging from the orbits of asteroids, to the interactions of galaxies, to the fate of the universe itself. I will also illustrate how gravity can be used as a tool that enables astronomers to detect, among other things, planets outside the solar system, black holes, and dark matter.

Thursday, 21 April 2005
4:10 p.m. in Jour 304

Previous attempts at defining other correlation measures mostly tried to generalize the inner product definition used in Pearson's correlation coefficient. This does not allow for certain useful correlations, like the Greatest Deviation, or Gini's. In this work the idea in Gideon and Hollister (1987) of seeing correlation as the difference between distance from perfect negative and perfect positive correlation will be used to bring together a general setting. Pearson, Spearman, and Kendall correlation coefficients are then seen as special cases where a linear restriction holds. It will also be seen how to define a wide variety of correlation coefficients. Simple linear regression with these correlations will be discussed in order to illustrate an introduction to statistical estimation with correlation coefficients. The general focus of this paper is simply to outline notation and concepts necessary for using correlation coefficients as estimating functions.

Thursday, 14 April 2005
4:10 p.m. in Math 109

F unctional data analysis methods including functional cluster analysis and functional linear modeling are discussed. The methods are used to describe and compare the shape of elevation cross-profiles taken from three Himalayan valleys. Typical methods for the analysis of these profiles are discussed in a nonlinear regression framework along with the use of model selection criteria. Curve registration is used to align important features in the profiles. Functional cluster analysis is used to group profiles by shape, with the shape based on the estimated curvature of each profile. Functional linear models are then used to explain the variability in the observed shapes of the profiles.

No particular mathematical or negotiation skills are required. This talk is open to everyone.

Thursday, 7 April 2005
4:10 p.m. in Math 109

I describe a dynamic game-theoretic model of the competitive harvesting of a marine fish stock, whose demographics are affected by an erratic and poorly understood climatic regime. I trace the historical development of this model genre, sketch its analysis, and comment on its implications for conservation of the biological resources of the high seas.

Thursday, 31 March 2005
4:10 p.m. in Math 109

Can birthday cakes lead to "B" horror films or world peace? Are strong negotiating skills required to share a pie or is it best to avoid all communication? How about a bundt cake? How about the Middle East? Here we will consider these questions and others and, as the icing on the cake, we'll answer some.

No particular mathematical or negotiation skills are required. This talk is open to everyone.

Tuesday, 29 March 2005
4:10 p.m. in Skaggs 114

The Virasoro algebra is an interesting Lie algebra from a variety of perspectives. A common way to study Lie algebras is through their modules (or representations), that is, ways that the Lie algebra acts on vectors spaces.

I will give an introduction to the Virasoro algebra and its representation theory, comparing it to the classical Lie algebra sl₂(ℂ). In particular, I will discuss simple modules L and Verma modules M, two fundamental types of modules, and their connections to blocks. My own research has focused on the Virasoro algebra module M⊕L. I'll present some of my results on the decomposition of this module by blocks and the potential connections to Hecke algebras and Kazhdan-Lusztig polynomials for the Viraso algebra, concepts which capture the geometric structure of blocks.

Thursday, 17 March 2005
4:10 p.m. in Math 109

Chen, Goldenfeld, and Oono (1996) suggested that their renormalization group method could be the universal technique for asymptotic analysis. Our work suggests that the amplitude equation approach may be even more central. Others might prefer averaging.

Thursday, 10 March 2005
4:10 p.m. in Math 109

Forests are very dynamic, yet they can exhibit behavior in predictable patterns. When an infectious agent is introduced in a forest site these patterns are typically affected and the evolutionary course of the site is altered. We seek to explore these events and how the forest adjusts.

In this paper we develop a mathematical model to characterize these patterns and effects of disease on a forest site. The model is developed for a natural forest (i.e. no artificial planting of trees and/or harvesting/thinning) with a single species of tree and a single pathogenic component. The model consists of a set of integro-differential equations. Under certain assumptions we reduce it to a corresponding set of ordinary differential equations.

A perturbation in the forest system may evolve into a steady state or lead to oscillatory behavior. With a better understanding of these events, forest management decisions can be more effective at realizing a healthy and productive forest system.

Thursday, 3 March 2005
4:10 p.m. in Math 109

The National Council of Teachers of Mathematics has recommended the inclusion of algebraic reasoning as an integrated part of the K-12 curriculum beginning in the elementary grades. With this proposed change comes the need to further understand how algebraic generalization is conceptualized and developed in the minds of both students and teachers at the primary level. To this end, this presentation details three bodies of work. The first study focuses on the strategies used by 5th grade students as they attempt to generalize numeric tasks and puts forth a model describing the factors that influence students' use of these strategies. The second part involves a curriculum development project that uses middle school curriculum and student thinking as a catalyst for the mathematical preparation of teachers. The final study describes the aspects of a teacher's knowledge of mathematics, student thinking, and curricular tasks that are utilized when making instructional decisions concerning algebraic generalization. 

Monday, 28 February 2005
4:10 p.m. in Math 109

Developing an understanding of function is an important component of students' successful participation in mathematics. Several frameworks that describe students' conceptions of mathematical ideas in terms of "processes" and "objects" have come to play a pivotal role in studying student cognition at the undergraduate level. Recently, mathematics educators have begun to focus on the role discourse and communication play in learning and doing mathematics. In this talk, I present an overview of a framework that integrates ideas from a process-object model with a discursive model to describe students' conceptions of functions and show how it can be used to analyze students' discourse. 

Thursday, 24 February 2005
4:10 p.m. in Math 109

Since the publication of Curriculum and Evaluation Standards for School Mathematics in 1989 and the Principles and Standards in School Mathematics in 2000, several curricular reform efforts have been undertaken. However, little is known about the instructional impact of mathematics education reform in high school algebra. This talk will present the results of an investigation into algebra reform practices. The primary goal of the study was to examine the effects of teacher beliefs, school-level environment, and classroom goal structure on mathematics education reform practices in high school algebra. 

Monday, 14 February 2005
4:10 p.m. in Skaggs117

We begin by considering the following question: Given an open set V in ℂⁿ , is it possible to find a larger open set U containing V such that every holomorphic function on V extends to a holomorphic function on U? If , n=1 this is never possible. But in several variables this can sometimes be done, and characterizing those sets for which extension is possible is a difficult problem. Perhaps even more surprising is the fact that, under suitable hypotheses on a hypersurface in ℂⁿ, there may exist a common open set in ℂⁿ to which every sufficiently smooth solution to a certain system of partial differential equations extends holomorphically. Furthermore, such an extension phenomenon may even be observed for sets of higher codimension if they retain some of the complex structure of the ambient space. These sets are the CR manifolds. The associated partial differential equations are referred to as the Cauchy-Riemann equations, and their solutions are the CR functions. The problem of CR extension, then, is to understand under what conditions there exists a common open set in ℂⁿ to which every CR function on a CR manifold extends holomorphically, and, when CR extension is possible, to describe this set.

In this talk, I will discuss the properties we expect of the regions for CR extension. I will describe work on a model class of manifolds that illustrates the limitations of earlier descriptions of CR extension and develops an alternative meeting the proposed criteria. 

Thursday, 10 February 2005
4:10 p.m. in Skaggs117

Whether you know it or not, you have crossed paths with matroids in your mathematical journey. We introduce the main ideas from Matroid Theory and discuss some of my favorite matroid gems (i.e. theorems). We will also see some of the open problems in matroid theory and briefly discuss the major roadblocks to solving these problems.

Tuesday, 8 February 2005
4:10 p.m. in Skaggs117

Beginning with a review of linear algebra, we will explore different types of operators which are "near" normal, and their generalizations to several variables. 

Thursday, 3 February 2005
4:10 p.m. in Math 109

In large-scale population networks the emergence of global behavioral patterns is driven by self-organization of local groups into synchronously functioning ensembles. However, the laws governing such macrobehavior are poorly understood. Here, we propose an extension of the Wilson-Cowan-Amari system which models the behavior of populations of excitatory and inhibitory neurons. We have two goals in this talk: the first is to explain how self-organization of local populations arises in the model in the form of self-sustained synchronous oscillations both in one and two space dimensions. In addition, we show how organization in one spatial region promotes or inhibits organization in another. Theoretical predictions are confirmed by comparison with human electrocorticographic recordings. The second goal is to show how rotating waves arise in the model and how they were used to predict, and subsequently confirm, the existence of rotating waves in rat brain experiments.

All results will be illustrated by videos.

Thursday, 8 December 2005
4:10 p.m. in Math 109

A review of continuum models for production flows involving a large number of items and a large number of production stages is presented. The basic heuristic model is based on mass conservation and state equations for the relationship between the cycle time and the amount of work in progress in a factory. Heuristic extensions lead to advection diffusion equations and to capacity limited fluxes. Comparisons between discrete event simulations and numerical solutions of the heuristic PDEs are made. First principle models based on the Boltzman equation for a probability density of a production lot, evolving in time and production stages are developed. It is shown how the basic heuristic model constitute the zero order approximation of a moment expansion of the probability density. Similarly, the advection diffusion equation can be derived as the first order Chapman-Enskog expansion assuming a stochastically varying throughput time. It is shown how dispatch policies can be modeled by including an attribute in the probability density whose time evolution is governed by the interaction between the dispatch policy and the capacity constraints of the system. The resulting zero order moment expansion reproduces the heuristic capacity constraint model whereas a first order moment will lead to multiphase solutions representing multilane fluxes and overtaking of production lots. A discussion on the similarities and differences of industrial production networks and biological networks is also presented.

Thursday, 1 December 2005
4:10 p.m. in Math 109

My talk will be about a book that is currently in press, which was edited by myself, Eric Hamilton, and Jim Kaput. The title of the book is the same as the title of my talk - above. The project that gave rise to this book was born due to a variety of related concerns of the editors. One was that so much of curriculum reform in mathematics education has attempted only to make incremental improvements in the traditional curriculum that we have inherited, and has largely ignored the voices of some very important stake holders - notably people in professions that are heavy users of mathematics, science, and technology. What might be possible, we asked, if we erased the board of past curriculum goals and began afresh to re-conceive "the 3Rs" for the twenty-first century? And, what new possibilities might arise if we quit ignoring the views of those who are not school people or professors of mathematics?

The general questions that we addressed also were stimulated by the following kinds of observations. In fields ranging from aeronautical engineering to agriculture, and from biotechnologies to business administration, outside advisors to future-oriented university programs increasingly emphasize the fact that, beyond school, the nature of problem solving activities has changed dramatically during the past twenty years. For example, powerful tools for computation, conceptualization, and communication have led to fundamental changes in the levels and types of mathematical understandings and abilities that are needed for success in such fields. … The following questions arise.

What is the nature of typical problem-solving situations where elementary-but-powerful mathematical constructs and conceptual systems are needed for success in a technology-based age of information? What kind of "mathematical thinking" is emphasized in these situations? What does it mean to "understand" the most important of these ideas and abilities? How do these competencies develop? What can be done to facilitate development? How can we document and assess the most important (deeper, higher-order, more powerful) achievements that are needed: (i) for informed citizenship, or (ii) for successful participation in the increasingly wide range of professions that are becoming heavy users of mathematics, science, and technology?

Authors in this book agreed that such questions should be investigated through research - not simply resolved through political processes (such as those that are used in the development of curriculum standards or tests). We also agreed that researchers with broad and deep expertise in mathematics and science should play significant roles in such research - and that input should be sought, not just from creators of mathematics (i.e., "pure" mathematicians), but also heavy users of mathematics (e.g., "applied" mathematicians and scientists). This is because the questions listed above are about the changing nature of mathematics and situations where mathematics is used; they are not simply questions about the nature of students, human minds, human information processing capabilities, or human development.

We also ask:

Why do students who score well on traditional standardized tests often perform poorly in more complex "real life" situations where mathematical thinking is needed? Why do students who have poor records of performance in school often perform exceptionally well in relevant "real life situations?

These latter questions emerged because many participants in our work shared the experience of encountering our former mathematics students when they appeared several years later in courses or jobs where the mathematics that we tried to teach them would have been useful. In some cases, we have been discouraged by how little was left from what we thought we had taught. On the other hand, we often were equally impressed that some students whose classroom performances were unimpressive went on to develop a great deal from seeds that we apparently helped to plant.

Upon further reflection and research about the preceding issues, most of us gradually developed the opinion that, for most topics that we have tried to teach, the kind of mathematical understandings and abilities that are emphasized in mathematics textbooks and tests tend to represent only a shallow, narrow, and often non-central subset of those that are needed for success when the relevant ideas should be useful in "real life" situations. For example, in projects such as Purdue University's Gender Equity in Engineering Project, when students' abilities and achievements were assessed using tasks that were designed to be simulations of "real life" problem solving situations, the majority of understandings and abilities that emerged as being critical for success included were not among those emphasized in traditional textbooks or tests. Consequently, when we recognized the importance of a broader range of deeper understandings and abilities, a broader range of students naturally emerged as having extraordinary potential. Furthermore, many of these students came from populations that are highly under represented in fields that emphasize mathematics, science, and technology; and this was true precisely because their abilities were previously unrecognized. … Such observations return us to the following fundamental question:

What kind of understandings and abilities should be emphasized to decrease mismatches between: (i) the narrow band of mathematical understandings and abilities that are emphasized in mathematics classrooms and tests, and (ii) those that are needed for success beyond school in the 21st century?

Many people assume that students simply need more practice with ideas and abilities that have been considered to be "basics" in the past. Others assume that old conceptions of "basics" should be replaced by completely new topics and ideas (such as those associated with complexity theory, discrete mathematics, systems theory, or computational modeling). Still others assume that new levels and type of understanding are needed for both old and new ideas. Examples include understandings that emphasize graphics-based or computation-based representational media. … My own perspectives lean toward this third option - without denying the legitimacy of the other two. But, this is what I intend to talk about during my visit to the University of Montana. So. no attempt will be made to resolve such issues here. Let me simply point out that, when issues of this type were discussed by authors in our book, three levels of students were given special attention.

• Undergraduate students preparing for leadership positions in fields, such as engineering, where mathematical and scientific thinking tend to be emphasized.
• Middle-school students who, with proper educational opportunities, could have the potential to succeed in universities such as Purdue or Indiana University, which specialize in a variety of fields that are increasingly heavy users of mathematics, science, and technology.
• Teachers (as well as professors and teaching assistants) of the preceding students.

For K-12 students and teachers, questions about the changing nature of mathematics (and mathematically thinking beyond school) might be rephrased to ask: If attention focuses on preparation for success in fields that are increasingly heavy users of mathematics, science, and technology, how should traditional conceptions of the 3R's (Reading, wRiting, and aRithmetic) be extended or reconceived to prepare students for success beyond school in the 21st century? 

Monday, 21 November 2005
4:10 p.m. in NULH

In the early 1900's, Hadamard defined a problem as "ill-posed" if the solution of the problem either does not exist, is not unique, or if it is not a continuous function of the data. Such problems are extremely sensitive to perturbations (noise) in the data; that is, small perturbations of the data can lead to arbitrarily large perturbations in the solution. Contrary to Hadamard's belief, ill-posed problems arise naturally in many areas of science and engineering; one important example is image restoration, which is the process of minimizing or removing degradation (such as blur) from an observed image.

Many algorithms have been developed to compute approximate solutions of ill-posed problems, but they may differ in a variety of ways. For example, there are several different regularization methods one could use, and for each of these, various methods for choosing a regularization parameter.

An important tool in the development of regularization methods is the singular value decomposition (SVD). The difficulty in image processing applications is that the matrices are often very large. In this talk we illustrate the important role played by the SVD when analyzing and solving certain linear systems arising in image restoration. In addition, we consider some approaches that can be used to efficiently (with respect to speed and storage) compute the SVD of structured matrices that arise in these applications.

Friday, 18 November 2005
4:10 p.m. in NULH

Mathematical finance has emerged as a trans-disciplinary area of research. This is characteristic of a new trend in science that categorizes research problems by their level of complexity rather than by traditional disciplinary boundaries. This talk will provide an overview of some of the theoretical issues in derivatives risk management. Topics covered will include stochastic processes and econophysics.

Vasilios ("Bill") Koures has a Ph.D. in theoretical high-energy physics. Shortly after receiving his Ph.D., he worked in the oil industry in exploration geophysics. He then returned to academia, where he taught at the University of Utah and performed research in theoretical and computational physics. After a stint in the defense industry, Bill moved to New York and began working in derivatives research. He worked on various derivatives risk management projects, including interest rates, credit risk, equities, FX, energies, and other commodities. His more recent positions include VP in derivatives research on the Global Commodities desk at J.P. Morgan and Executive Director and Head of Quantitative Research at Mitsui Energy Risk Management, Ltd. At present, Bill lives in Montana, where he founded the Intermountain Institute for Science and Applied Mathematics (IISAM). This institute performs cross-disciplinary research in applied math along with consulting and educational outreach programs. Bill also runs his own trading company: Quantitative Trading LLC.

Thursday, 17 November 2005
4:10 p.m. in Jour 304

The f-vector enumerates the number of faces of a convex polytope according to dimension. The flag f-vector is a refinement of the f-vector since it enumerates face incidences of the polytope. To classify the set of flag f-vectors of polytopes is an open problem in discrete geometry. This was settled for 3-dimensional polytopes by Steinitz a century ago. However, already in dimension 4 the problem is open.

We will discuss the known linear inequalities for the flag f-vector of polytopes. These inequalities include the non-negativity of the toric g-vector, that the simplex minimizes the cd-index, and the Kalai convolution of inequalities.

We will introduce a method of lifting inequalities from lower dimensional polytopes to higher dimensions. As a result we obtain two new inequalities for 6-dimensional polytopes.

The talk will be accessible to a general audience.

Thursday, 3 November 2005
4:10 p.m. in Jour 304

A force constant matrix F (consisted from second derivatives of the molecular potential with respect to nucleus coordinates in the equilibrium configuration) is one of the most important information about the intramolecular dynamics and defines vibrational properties (including infrared and Raman spectra, vibration-rotational spectra, etc.).

There are two main sources for the molecular force field determination. The first way is solving the inverse problem using an experimental data on molecular spectra and electron diffraction measurement. Both (vibrational or generalized structural) problems belong to the class of nonlinear ill-posed problems [1]. Other way is to estimate the molecular force field by carrying out quantum mechanical calculations with a goal to obtain the theoretical equilibrium configuration and force constants.

We have proposed to join these two approaches in the unique statement based on joint treatment of experimental and quantum mechanical data. On this base the concept of regularized quantum mechanical force field (RQMFF) was proposed, and new formulations of inverse problems were given. Stable numerical methods for the solving corresponding inverse problems have been developed. New regularizing algorithms allow us to carry out a special modeling of matrix F based on the different constraints which take into account the relative order of intramolecular forces. Force fields of extended molecular systems (clusters, polymers etc.) are constructed on a base of synthesis of separate blocks of force constants. For the estimation of intermolecular force constants we use the regularizing algorithm based on the joint use of empirical data on the second virial coefficients and results of quantum mechanical calculations.

The next scheme for the calculations of vibrational spectra of the large size molecules such as polymers, nanostructures, biological systems, etc. can be proposed:

1. quantum mechanical analysis of moderate size molecules chosen as key or model molecules which are the fragments of large molecular systems;
2. joint treatment of ab initio and experimental data on vibrational spectra, ED and MW data for model molecules with stable numerical methods with a goal to estimate the accurate molecular force field;
3. organizing a database on structural data and force field parameters transferable in a series of related compounds;
4. synthesis (construction) of a large molecular system from separate fragments included in the database and calculation of its vibrational spectra and thermodynamical functions.

Thursday, 20 October 2005
4:10 p.m. in Math 109

Three dimensional bin packing is well known NP Hard problem. While it might be interesting to study theoretical approximations to this problem, real customers need real solutions to this problem. Dr. Henry has produced a real solution to this problem that produces solutions with container volume efficiencies of 80-90%. This solution is currently being used by the DOD to plan logistical support of troops deployed overseas.

This talk presents the problem both from a mathematical and practical viewpoint, then applies some rules of thumb people use everyday to pack containers, and finally overviews the solution. The strengths and weaknesses of the solution are covered. This will be a fun talk with some interesting challenges and solutions discussed.

Thursday, 13 October 2005
4:10 p.m. in Jour 304

Dynamical processes taking place on networks have received much attention in recent years, especially on various models of random graphs (including "small world" and "scale free" networks). They model a variety of phenomena, including the spread of information on the Internet; the outbreak of epidemics in a spatially structured population; and communication between randomly dispersed processors in an ad hoc wireless network. Typically, research has concentrated on the existence and size of a large connected component (representing, say, the size of the epidemic) in a percolation model, or uses differential equations to study the dynamics using a mean-field approximation in an infinite graph. Here we investigate the time taken for information to propagate from a single source through a finite network, as a function of the number of nodes and the network topology. We assume that time is discrete, and that nodes attempt to transmit to their neighbors in parallel, with a given probability of success. We solve this problem exactly for several specific topologies, and use a large-deviation theorem to derive general asymptotic bounds, which we use, for example, to show that a scale-free network has propagation time logarithmic in the number of nodes, and inversely proportional to the transmission probability.

Friday, 23 September 2005
4:10 p.m. in Jour 304

Carl Pixley was my topology professor at U Texas, Austin. His class was taught by the "Moore method"; students discover and present the proofs on their own. Carl would inspire us to persevere by giving advice in the form of "Pixley Principles". One of my favorites was "...if there is a problem you can't do, there is a simpler one you can't do; do that one first!"

This talk gives a gentle introduction to active sonar. Taking Carl's advice, the simple problem to be done first is: estimate the range and velocity of an object from the echo of a signal it returns. Assume no noise and no clutter.

Thursday, 22 September 2005
4:10 p.m. in Skaggs 114

There is limited use in teaching the performance of mathematical methods to students who will never use any. If, however, the stress is on teaching the reasons behind the methods - why and how they come about - this happens to be the most effective way to teach multi-staged, purposeful, accurate reasoning, something everyone does need. Discovering that the reasons can be made readily understandable does also reduce the anxiety about the subject and thus could increase the number of mathematics users - something the economy could hardly survive without.

Also for students who will work in mathematics and science, dwelling on the processes that lead to the methods is equally important, but for another reason: Their main task will consist not just of applying already known methods, but of the development of new ones. For this purpose it should be useful to have considered the methods which they were taught from a view point of having had to deduce them on their own.

The colloquium will demonstrate such a reasoning-approach to a wide range of topics - 'from C to C' (counting to calculus…) - where familiar methods are often found to be backed by very little understanding about their "why's".

Thursday, 22 September 2005
1:10 p.m. in Math 109