Colloquia 2008-2009

Current Talks:

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: TBA

Abstract: TBA

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.

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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 – 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.

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Past Talks:

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.

Department of Mathematics and Statistics
ILB 325
University of South Alabama
Mobile, AL 36688
phone: (251) 460-6264 (voice), (251) 460-7969 (fax)
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