I will talk about new programs at the National Science Foundation. Most of the discussion will be on programs in the Division of Mathematical Sciences and the Directorate of Education and Human Resources. However, I will discuss some Foundation-wide programs such as CAREER (a program for junior investigators), POWRE (a program for women), and KDI (Knowledge and Distributed Intelligence). I will also talk about what its like to work as a program director at NSF.

Thursday, February 12, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

Cyanobacteria .(blue-green algae) are the oldest inhabitants on the Earth. These procaryotes can change their buoyancy depending on solar irradiation, and adjust their position to the optimal phototrophic regime.

Blue-green algae may release toxins in water, some of the toxins being more toxic than the cobra toxin. If exposed to the optimal conditions, cells grow rapidly and usually lose their vertical stability in a water column. As a result of such catastrophic behavior, blue-greens form the dense bloom which follows the surface scum that ends the cycle of population development.

The system of non-linear differential and integral-differential equations is proposed to describe this phenomenon. Mathematical analysis of this set of equations was performed, and the particular analytical solutions are found.

Thursday, February 19, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

Wildfire managers are obligated to meet ecosystem management objectives such as cost minimization and efficiency (Williams et al. 1993). This, however, is difficult because each objective is dependent upon wildfire controllability and wildfire behavior. Currently, there is no functional form that defines the relationship between wildfire behavior and controllability and therefore, there is no physical basis for efficient economic analysis. In the first section I generate a fire data set and test it as a cusp catastrophe. Results suggest that catastrophe theory may be an effective tool to model wildfire controllability as measured by the modeled change in fireline intensity, windspeed, initial fuel moisture and fuel loading. The resulting production function relationship presents three management possibilities. The model may be used to; identify environmental factors that systematically predict wildfire controllability and the range over which sudden changes in fire behavior occur; to quantify uncertainty of fire behavior in terms of environmental factors, and; to define a manageable environmental variable that can be used to determine marginal costs and benefits of wildfire management activities. The second section enhances the C+NVC theory by including the physical effects of wildfire behavior and environmental conditions as developed in the first section. To facilitate development of the model I use two factors of production, suppression, which is variable throughout all time periods, and presuppression, which is fixed in the short run and variable in all other instances. I show that the annual C+NVC curve is the economic envelope to the seasonal C+NVC curves. Next, I characterize net value change as a monotonically increasing function of fireline intensity. Because the fire model relates physical effects to fireline intensity, the C+NVC model is enhanced in three ways; the introduction of environmental variables will enable fire managers to assess fire management programs for severe and average expected fire behavior and identify conditions that lead to management uncertainty; fire behavior is defined in terms of intensity and controllability, therefore, the cusp model generates different estimates of suppression efficiency depending upon environmental conditions and expected fire behavior, and; the NVC manifold provides a method by which to evaluate the marginal effectiveness of presuppression.

Thursday, February 26, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge

The Ramsey number R(3,t) has asymptotic order of magnitude t²/log t
Jeong Han Kim
Microsoft Research

The Ramsey number R(s,t) is the minimum integer n for which every red-blue coloring of the edges of a complete n-vertex graph induces either a red complete graph of order s or a blue complete graph of order t. In this talk, we describe a dynamic probabilistic method which is used to settle an old problem on R(3,t). Namely, R(3,t) has asymptotic order of magnitude t²/logt.

Thursday, March 5, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

We discuss some ideas useful for developing a better than linearly convergent algorithm for minimizing a non-smooth convex function of n variables. For the single variable case, we describe a fully implementable algorithm which has global and superlinear convergence. The method appropriately combines polyhedral (V-shaped) approximation and quadratic (U-shaped) approximation.

Thursday, March 12, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

A basic problem in algebra is to classify, for a given field K, the finite-dimensional division algebras having center K. The theorems of Frobenius and Wedderburn solve this problem in case K is the real field or a finite field. The classification for the rational field, or more generally an algebraic number field, was completed in the 1930s using deep results from algebraic number theory.

A finer study of the subfield structure of those algebras was initiated in the 1960s. Beginning with a polynomial f(x) with coefficients in K, one may ask whether there is a K-division algebra containing a root of f(x). The answer is known to depend on delicate properties of f(x) and K. In this talk I describe a few examples, and then discuss open questions and recent work concerning the Galois groups which arise in this manner.

Thursday, March 26, 1998
>4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

Formal mathematical definitions are among the most important tools in mathematics. The purpose of this talk is to discuss the results of a study that investigated undergraduate mathematics majors' understanding and use of formal definitions in real analysis.

The results showed that even good students did not necessarily fully appreciate the role of formal definitions in mathematics and that this factor as well as students' sometimes incomplete conceptual understandings influenced their ability to understand and use formal definitions in a mathematically acceptable way.

Thursday, April 9, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

Extremal questions are studied in many branches of mathematics. For instance, in graph theory we might ask: How many edges can a graph on n vertices contain if it has no subgraph isomorphic to the complete graph K_m? In projective geometry we might ask: How many points can a subset of a projective plane of order q contain if no k of the points are colinear?

Matroid theory is an abstraction of graph theory, projective and affine geometry, and several other branches of mathematics. Questions of the type posed above can be asked about matroids. We will address several such questions and present some intriguing problems and conjectures in this area.

This talk will include enough background on matroid theory, starting with the definition and motivating examples, to be widely accessible.

Thursday, April 16, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

Distributed computing deals with problems that arise when a set of autonomous computers interact. Among the most basic of these problems are those of coordination: ensuring that computers act in a mutually consistent manner and do not interfere with each other. It turns out that, under certain reasonable assumptions, many basic coordination problems are provably impossible to solve.

In this talk, I'll discuss an approach to such results via elementary topology, that reveals a surprising connection between these impossibility results and the classical Brouwer fixed point theorem for Euclidean space.

This work is joint with Fotis Zaharoglou. The talk will not assume any prior knowledge of distributed computing.

Thursday, April 23, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

Powerful telescopes on the surface of the earth are capable of imaging very faint objects. However, the clarity of these images is no better than that which can be obtained from an amateur's telescope with a 6 inch mirror---hence, the Hubble Space Telescope. This blurring of astronomical images is due to distortions of the wave fronts of light caused by atmospheric temperature variations. One way to correct these distortions is to physically "undistort" the wave fronts with deformable mirrors. This is called Adaptive Optics (AO). Recently, an alternative, software-based approach called Phase Diversity (PD) has been developed. In this talk, I will present an overview of AO and PD, touching on a variety of topics from Physics, Mathematics, and Computer Science. These topics include:

• Stochastic Processes (used to model atmospheric turbulence caused by wind).
• Control Theory (used in the control of deformable mirrors in AO).
• Mathematical Modeling (used to quantify the relationship between the noisy, blurred, recorded image and the "true" image).
• Inverse Problems (used to estimate the true image from the recorded images in PD).
• Computational Mathematics (used to formulate algorithms to efficiently solve the PD estimation problem).
• Computer Science (used to implement these algorithms on parallel computers).

Thursday, April 30, 1998
4:10 p.m. in Forestry 305
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

Come Celebrate Math Awareness Week
April 27th - May 1st, 1998
Theme: Mathematics and Imaging

The NCTM Standards make the following three recommendations regarding computer technology:

1. A computer should be available in every classroom for demonstration purposes.
2. Every student should have access to a computer for individual and group work.
3. Students should learn to use the computer as a tool for processing information and performing calculations to investigate and solve problems.

Whether we support these standards or not, parents, society, business, industry, etc. are expecting teachers to try to incorporate some form of technology into the classroom. Research suggests that unless preservice teachers have observed technology being used in the content classroom, they will be slow to implement it into their own classrooms. How we as mathematicians and teacher educators model the use of technology in our classes becomes a great influence on how it will be used by prospective teachers.

The types of computer activities that have been and are being used in the classroom as well as the research into their effectiveness will be discussed. Also a variety of computer demonstrations that have been used in college classrooms will be shown to give an idea of how this modeling could take place

Thursday, May 7, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

When operator algebras were first invented, there was a close link between their theory and developments that were taking place in finite dimensional algebra about the same time. Since then, the two subjects, operator algebra and finite dimensional algebra, i.e., ring theory, have grown apart, following very different paths.

Recent advances in the structure of operator spaces, the theory of which some call Quantized Functional Analysis, have provided new opportunities for interaction between operator algebra and ring theory.

My objective in this colloquium is to describe some of these developments and to illustrate some of the latest technology in this area with concrete, finite dimensional examples.

* Partially funded by the MONTS Speaker Program

Thursday, June 4, 1998
3:10 p.m. in MA 211
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

* Big Sky Conference on Discrete Mathematics

In a certain sense, the art of juggling is a physical realization of many of the principles that mathematicians and computer scientists know and love. These include the search for patterns, the design and analysis of appropriate algorithms, and the prospect of facing problems of unbounded difficulty. In particular, juggling is typically a very discrete activity, and as such, is governed by a rich family of combinatorial constraints. Recently, a new and unexpectedly simple way of describing juggling patterns has been discovered. This has led to a bewildering array of previously unknown patterns, as well as several new combinatorial theorems relating linear extensions of partially-ordered sets to chromatic polynomials of associated graphs. In this talk we will describe these developments, and attempt to demonstrate some of these new tricks.

Thursday, September 10, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

* This talk is sponsored by and presented as an element of The University of Montana President's Lecture Series. The Big Sky Conference is also sponsored by the National Science Foundation and the Department of Mathematical Sciences.

 * Big Sky Conference on Discrete Mathematics

Around 1975, A.M. Hobbs asked whether every 2-connected multigraph G with minimum degree at least four contains a circuit C such that G-E(C) is still 2-connected. A positive answer to this question would imply the Cycle Double Cover Conjecture. The answer is "no" in general due to an example involving Petersen's graph and multiple edges. The answer is "yes" for simple graphs (Mader, Jackson) and planar multigraphs (Fleischner, Jackson). In joint work with Jan van den Heuvel and Sean McGuinness, we show:

A 2-connected multigraph G with minimum degree at least four and containing no Petersen graph as a minor, contains two edge-disjoint circuits Csuch that G-E(C) is 2-connected.

An obvious question is "How far can the above result be generalised to matroids?" In particular:

What is the largest (minor-closed) class of matroids such that every connected matroid M in this class having cogirth at least four has a circuit C such that M \C is connected?

For cographic matroids, the question above is equivalent to:

Which 2-connected graphs with girth at least four have a bond (a minimal edge-cut) B such that the graph resulting from contracting all edges in B is still 2-connected?

With Jan van den Heuvel, we show that all graphs which are not contractible to K₅ have this property. There are interesting issues regarding the complexity of finding such a removable circuit.

Friday, September 11, 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

* The Big Sky Conference is sponsored by the National Science Foundation and the Department of Mathematical Sciences

Imagine a long, straight pipe filled with gas. The gas is ignited, and a traveling wave of combustion forms and moves down the pipe. The gas initially consists of two chemical species (one of which is possibly oxygen), and the combustion process causes the formation of a third, inert gaseous product.

This talk will present the governing equations for this system. Then, by converting to coordinates that move with the combustion wave, we can show that the whole problem reduces essentially to a system of two ordinary differential equations in a phase plane. For small amplitudes, an approximate solution will be presented, that describes the combustion wave in terms of a classical "soliton" (solitary wave). For moderate amplitude waves, very accurate numerical solutions can be obtained using a type of shooting method. Large amplitude waves ultimately form a detonation shock, and this can also be accounted for in the numerical method. This is a joint work with Bill Derrick in the Math Department.

Thursday, 24 September 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

In addition, Dr. Forbes will present a Chemistry lecture, "Burning Down the House", Monday, September 21st at 4:10 p.m. in CP102.

We examine the topological study of manifolds, particularly from the late 50's and 60's, some important milestones, and how the subject has undergone abrupt changes of direction, due to new discoveries or rediscoveries.

Thursday, 15 October 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

Professor Browder has been at Princeton since 1963. He has served as an editor for the Annals of Mathematics. He is a member of the National Academy of sciences, and a participant in many activities of the National Research Council. He was Chairman of the Briefing Panel on Mathematics for the Office of Science and Technology Policy of the White House in 1983. He has been elected to many offices in the American Mathematical Society, including President (1989-91). The Proceedings of a conference - convened at Princeton in 1994 in his honor - lists 28 direct mathematical descendants, 78 mathematical grandchildren, and 15 mathematical great- grandchildren.

Hopf algebras are algebraic objects which combine a beautiful structure with many important applications in mathematics and physics.

I will begin my talk with a number of motivating examples from a variety of mathematical fields. Then I will give an elementary introduction to the structure of Hopf algebras, and give some examples of ways in which Hopf algebras have proven useful in (seemingly) very different areas of mathematics and physics.

This talk will be accessible to graduate students.

Our profession is in desperate trouble - immediate and present danger. The absolute numbers and the trends are clear. If something is not done soon, we will see mathematics department faculties decimated and an already dismal job market completely collapse. Simply put, we are losing our students.

-From the Garfunkel-Young Notices of AMS article "The Sky is Falling"

I will argue that undergraduate math education has been frozen for so long that it is unable to meet modern demands expected of it by its students and by client departments. Not only has course material become weighted down by what Peter Lax calls "inert" material, but the ways of teaching mathematics have also become bogged down by the lecture method.

This talk will suggest new ways of injecting technology into math courses so that students play and get the feel of a new idea before the jargon goes on. It will also talk about good use of technology versus poor uses of technology.

It will also deal with some moral dilemmas that face math instructors such as:

• Confusion about the role of Algebra: Is algebra prerequisite to everything?
• Confusion about how much can be done without formal proofs.
• Confusion about whether students have the right to know about advanced mathematics.
• Confusion about the nature of mathematics.

Thursday, 12 November 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

The educational system in Japan is quite different from that in the USA. Recently there is a lot of talk in Japan on the need to reform the educational system. This might sound surprising for the country that enjoyed steady economic growth in recent years that was certainly supported by the current educational system. What has changed in the last couple of years? What are the possible directions of the reforms? How can one tell whether the reforms will lead to progress or regress in the future development of the society?

These questions, as well as general comments on the exchange visit to Toyo University, will be discussed.

Thursday, 19 November 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

An encore video presentation *

When operator algebras were first invented, there was a close link between their theory and developments that were taking place in finite dimensional algebra about the same time. Since then, the two subjects, operator algebra and finite dimensional algebra, i.e., ring theory, have grown apart, following very different paths.

Recent advances in the structure of operator spaces, the theory of which some call Quantized Functional Analysis, have provided new opportunities for interaction between operator algebra and ring theory.

My objective in this colloquium is to describe some of these developments and to illustrate some of the latest technology in this area with concrete, finite dimensional examples.

Thursday, 3 December 1998
4:10 p.m. in MA 109
Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

* Originally delivered on June 4, 1998

Andrew Wiles' proof of Fermat's Last Theorem relies in turn on his proof that a wide class of elliptic curves are "modular" (this is the so-called Taniyama-Shimura conjecture). I will begin by describing what this conjecture says in concrete, friendly terms.

But elliptic curves are not the only objects thought to be modular. In recent work, Ken Ono and I have devised a new method which proves the modularity of a certain "Calabi-Yau threefold". As a result we can prove Beukers' conjectured "supercongruence" for the Apery numbers (these are combinatorial sums introduced by Apery to establish the irrationality of certain values of the Riemann zeta function).