21 January 2010
Quantum K-Theory of Grassmannians and the Geometry of Spaces of Curves
Leonardo Mihalcea
Department of Mathematics
Baylor University

The 3-point, genus 0, Gromov-Witten invariants of a Grassmannian count rational curves of degree d satisfying certain incidence conditions - if the number of curves is expected to be finite. Recently, Givental and Lee defined the K-theoretic Gromov-Witten invariants, which associate an integer to spaces of rational curves in question, regardless of whether it is finite or not. The resulting quantum cohomology theory - the quantum K-theory - encodes the associativity relations satisfied by the K-theoretic Gromov-Witten invariants. I will show how the Gromov-Witten invariants and quantum K-theory algebra can be computed explicitly. The key is a "quantum=classical" phenomenon: the K-theoretic Gromov-Witten invariants for Grassmannians are equal to structure constants of the ordinary K-theory of certain two-step flag manifolds.

11 February 2010
GasDay: Forecasting Customer Demand for Natural Gas
George F. Corliss
Electrical and Computer Engineering
Marquette University
Milwaukee, Wi

Natural gas for heating takes 2 - 3 days to get to Milwaukee from Louisiana. If our local utility does not order enough, they buy it on the spot market at premium prices. If they order too much, the pipeline companies charge them penalties. The GasDay lab at Marquette University licenses to local natural gas utilities software to forecast the gas demands of their customers hours, days, and months in the future. We license to 23 utilities across the US. Each day, software written by our students helps forecast 20% of the gas used by residential, commercial, and industrial customers in the US. Our customers tell us our forecasts save their customers tens of millions of dollars each winter. We also offer analysis consulting services including peak day studies and data cleaning. We will discuss some of the mathematical modeling challenges, as well as the entrepreneurial challenges of running a $500K business with students within the university.

18 February 2010
Max-Plus Algebra and the Computation of Nonlinear Controls
Ben G. Fitzpatrick
Loyola Marymount University

Mathematical methods for linear systems have had enormous impact on engineering applications, especially in control systems. Nonlinear approaches, however, suffer from many drawbacks, not the least of which are computational difficulties. The recent successes of max-plus and more general idempotent structures for attacking nonlinear control problems offer the potential for revolutionary improvements in our ability to design and implement nonlinear controls for real applications. In this talk, we describe the basic concepts behind max-plus methods. The key observation is that Bellman's dynamic programming principle yields a linear equation in the max-plus arithmetic. We will outline the max-plus approach to nonlinear control and provide some preliminary results in the extension from deterministic to stochastic problems. We will discuss numerical approximation schemes and examples problems in counterinsurgency and UAV sensing planning.

16 March 2010 (TUESDAY)
Leavitt path algebras and graph C*-algebras: at the crossroads of algebra and functional analysis
Mark Tomforde
Department of Mathematics
University of Houston

In the past 15 years Functional Analysts have considered a method for constructing C*-algebras from directed graphs. These graph C*-algebras include many previously considered classes of C*-algebras, and provide a unified framework in which to consider many examples. Furthermore, numerous properties of the graph C*-algebra can be translated into properties of the graph, allowing for a very satisfying theory in which C*-algebraic questions can be reformulated as graph questions. In the past 5 years Algebraists have used similar methods to construct a K-algebra from a directed graph. These K- algebras are called Leavitt path algebras, and they generalize the Leavitt algebras (which are fundamental examples of algebras without the Invariant Basis Number property). To many researchers' surprise, it has been found that when properties of a graph correspond to properties of the associated C*-algebra, these same graph properties correspond to the algebraic properties of the associated Leavitt path algebra. What is even more astonishing is that neither the graph C*- algebra results nor the Leavitt path algebra results are obviously logical consequences of the other. Often different methods are used in the proofs, and moreover, neither set of results can easily be seen to imply the other. This has left many researchers wondering exactly what the relationship is between the Leavitt path algebras and the graph C*- algebras. In this talk I will describe the many similarities (and a few of the differences) between these two classes.

18 March 2010
Global stability in a multi-species periodic Leslie-Gower model
Robert J. Sacker
University of Southern California
Los Angeles, California

The d-species Leslie--Gower competition model is studied in which all the parameters are p-periodic. It is shown that whenever the coupling parameters are small, there is a positive p-periodic state that is exponentially asymptotically stable and globally attracts all initial states having positive coordinates. The theory extends to systems with delays and to certain systems having fractional quadratic form.

22 April 2010
A Skew-Normal Approximation to the Distribution of Aggregate Claim
Kumer P. Das
Department of Mathematics
Lamar University
Beaumont, Texas

In collective risk model, an insurance portfolio is regarded as a process that produces claims over time. The size of these claims are taken to be independent, identically distributed random variables, independent also of the number of claims generated. This makes the total claims the sum of a random number of iid individual claim amounts. The third central moment of aggregate claims under both the compound Poisson and compound negative binomial distribution is not zero. In fact, for positive claim amount distributions, the third central moment of aggregate claims is positive in each case. This skewed nature of the distribution of aggregate claims assures that a normal approximation may not be the best approximation to the aggregate claims distribution. In this study, a more general approximation to the distribution of aggregate claims using the skew normal distribution has been sought. In many occasions the obtained results are sharper than that of translated gamma approximation technique.

29 April 2010
Stabilization of hydrodynamic instabilities in Hele-Shaw flows
Prabir Daripa
Department of Mathematics
Texas A&M University

Motivated by applications and numerical simulation of fronts, multi-layered Hele-Shaw flow will be considered as a way to control growth of instabilities. Upper bound results on the growth rates will be discussed in cases of constant viscosity layers and variable viscosity layers. The upper bound provides a way to assess cumulative effects of many layers and many interfaces on the growth rates of unstable waves. As an application of the bound, we obtain some necessary conditions for suppressing instability of two-layer flows by introducing arbitrary number of constant viscosity fluid layers in between. This necessary condition has very practical relevance because it narrows the choice of internal layer fluids based on surface tensions of all interfaces and viscosity of fluids. Importance of this condition which has been hitherto unknown is also discussed. Other consequences of these upper bounds and necessary conditions are discussed. The case of internal fluid layers having unstable viscous profiles will be treated. The effect of diffusion on the instability will also be discussed.