2008-09 Colloquia talks

Abstract: The connection between coding theory and invariant theory was revealed 1970 in the ICM talk by Andrew Gleason. Since then Gleason's Theorem was generalized to many situations. With Eric Rains and Neil Sloane we introduce the formal notion of a Type of self-dual codes. Associated to the Type there is a finite complex matrix group (with explicit generators) such that the invariant ring is spanned by the code polynomials of self-dual codes of this Type.

Abstract: This talk introduces the Ising and q-state Potts model partition functions of statistical mechanics. These models play important roles in the theory of phase transitions and critical phenomena in physics, and have applications as widely varied as tumor migration, foam behaviors, and social demographics. The Potts model is constructed on various lattices, and when these lattices are viewed as graphs (i.e. networks of nodes and edges), then, remarkably, the Potts model is also equivalent to one of the most renown graph invariants, the Tutte polynomial. The Tutte polynomial is the universal object of its type, in that any invariant that obeys a certain deletion/contraction relation (or equivalently a particular state-model formulation), must be an evaluation of it. The emphasis will be on how the Potts model and Tutte polynomial are related and how research into the one has informed the theory of the other, and vice versa. This includes locations of zeros corresponding to phase transitions and graph coloring. There is brief mention of computational complexity analysis and Monte Carlo simulations.

Abstract: Factorization of almost periodic (AP) matrix functions arises naturally in a variety of problems, both theoretical and applied, and for many of them the matrix in question is 2-by-2 and triangular. Even in this setting the factorability properties remain a mystery, in striking difference with both the scalar almost periodic case and with the purely periodic matrix case. We will give a survey of currently available existential results regarding the factorization of general AP matrix functions, as well as available approaches to constructive factorization of the specific matrices mentioned above.

Abstract: Measurement errors occur often in many areas such as biometry, epidemiology and economics. Conventional density estimators and nonparametric regression estimators that ignore measurement errors can be misleading. There have been two lines of attack to correct for measurement errors; see, for example, the Fourier deconvolution method by Fan (1991); Fan & Truong (1993) and Simulation-extrapolation (SIMEX) by Carroll et al. (1999). In this talk, we review the traditional methods and introduce our new nonparametric procedure for estimating the density when data are contaminated with errors and estimating the regression function when there are errors in predictors. The resulting estimators are stable and easy to compute ñ there are no Fourier transformations needed in the calculation and there is no simulation model to assume as it is in SIMEX. They can be also used in the case that measurement errors are non-homogeneous. The form of our new estimators has some similarity to the Shannon Sampling Procedure and is hence named Shannon Weighted Average Procedure (SWAP). Further, the SWAP estimators have faster convergence rates than those of Fourier type estimators. Some simulation studies and data applications will also be presented.

Abstract: The Dehn function of a finitely presented group can be thought of as an optimal isoperimetric inequality for a geometric model of the group. For example, every closed loop of length at most x in the Cayley complex of the group can be null-homotoped using at most delta(x) 2-cells.

We shall discuss the Dehn function and its higher dimensional analogues. The k-dimensional Dehn function delta^k of a group measures how efficiently maps of k-spheres (into suitable spaces on which the group acts) can be extended to maps of (k+1)-balls. delta^k(x) provides an upper bound on the (k+1)-volume of the extension, expressed as a function of the k-volume of the map of the sphere.

We shall discuss some properties of k-dimensional Dehn functions, and look at some examples.

Abstract: Genetic data collected during the second phase of the Third National Health and Nutrition Examination Survey (NHANES III) enable us to investigate the association of a wide variety of health factors with regard to genetic variation. The classic question when looking into the genetic variations in a population is whether the population is in the state of Hardy-Weinberg Equilibrium (HWE). HWE is valid only when certain underlying assumptions have been met. Our interest is to use family data from complex surveys such as NHANES III and develop test procedures not only for overall departure from HWE but also for the departure from random mating. We develop six Pearson Chi^2 based test procedures and six quasi-score test procedures for a diallelic locus of autosomal genes. The finite sample properties of proposed test procedures are evaluated via Monte Carlo simulation studies. We propose to use the score based Rao-Scott first order corrected statistic. Test procedures were applied to three genes from NHANES III genetic data bases, i.e. ADRB2, TGFB1, and VDR.

Abstract: The theory of Azumaya algebras developed parallel to the theory of central simple algebras. However the latter are algebras over fields whereas the former are algebras over rings. One wonders how the K-theory of these objects compare to each other. We look at higher K-theory and reduced K-theory of these objects. In Part one, we present some results and some questions. (In the second part, on Friday, October 31, 2008, in the Algebra Seminar we will present sketch of proofs of some of the results.)

Abstract: Two-stage procedures are frequently used in statistical experiments and analysis relevant to biomedical research to minimize loss due to wrong assumptions, to reduce study cost, and to reduce data dimensions, etc. In this talk I will introduce the two-stage concepts in a statistical decision-making perspective and exemplify the strategy with evaluation of the measurement error of a biomarker.

Abstract: An important tool for studying a (finite) group G is to consider its representations, i.e., homomorphisms from G to GL(n,F) from G into groups of invertible nxn-matrices over various fields F. The n is called the degree of the representation.

In this talk we consider as groups G finite groups of Lie type - these will be introduced informally. They are closely related to many of the finite simple groups which were classified in the 1980's. An example are the general linear groups GL(k,q) of invertible kxk-matrices over a finite field with q elements.

We will consider the following question: For a given such group G and a given (algebraically closed) field F, what is the smallest degree of a non-trivial representation of G over F?

The known answers to this question rely on quite deep mathematics, and the theoretical background is very different depending on the characteristic of the field F: F is the complex numbers, or of prime characteristic l dividing the order of G, in the latter case the defining characteristic (l divides q in the example above) and non-defining characteristic must be considered separately.

Abstract: Cancer remains one of the leading cause of disease death for Americans. Moreover the overall effectiveness of therapeutic treatments is only approximately 50%. Therefore the development of prognostic tools could have immediate impact on the lives of millions of cancer patients. We have developed an integrated, cell-based modeling framework that includes a cellular model for cell dynamics (cell growth, division, death, migration and adhesion), an intracellular regulatory network for cell cycle control and a signaling network for cell decision-making, and a partial differential equation system for extracellular chemical dynamics. This model has produced avascular tumor growth dynamics that agree with tumor spheroid experiments; it has generated realistic vessel sprout morphogenesis in tumor-induced angiogenesis; it has also shown potential for comparing chemotherapeutic strategies for vascular tumor. In particular, we investigate the mechanisms for tumor growth saturation and the roles of VEGF and ECM in tumor angiogenesis. Given the biological realism and flexibility of the model, we believe that it can facilitate a deeper understanding of the cellular and molecular interactions associated with cancer progression and treatment.

Abstract: Dynkin diagrams are graphs that arise naturally in many classification theorems. Just a few examples of their appearances in algebra are: finite groups of rotations in 3-space, reflection groups, Lie algebras, and quiver representations (collections of vector spaces and maps between them). In this talk we will introduce Dynkin diagrams and discuss some of their historical uses in algebra, including theorems of Dynkin and of Gabriel. Then we will introduce Hopf algebras and describe a new appearance of Dynkin diagrams in a recent classification, by Andruskiewitsch and Schneider, of certain types of finite dimensional Hopf algebras.