UIUC Dept. of Mathematics Seminar Calendar

      March 2003             April 2003              May 2003      
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Thursday, April 24, 2003

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Thursday, April 24, 2003

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Nicolae Popescu (Institute of Mathematics of the Romanian Academy)
Galois action on Plane Compacts
Abstract: Let Q be the field of rational numbers and C the field of complex numbers. Let A be the algebraic closure of Q in C and G:=Gal(A/Q), endowed with the Krull topology. We characterize all the subsets M of A, such that G acts transitively and continuously on M. More precisely, we prove that G acts transitively and continuously on a subset M of A if and only if there exists a (topological) homeomorphism T: G/H->M, where H is a closed subgroup of G, G/H is endowed with the quotient topology and M with the topology induced from that of C. The well-known Cantor sets give generic examples of such infinite sets M.

Group Theory Seminar
1:00 pm in Altgeld Hall, room 347, Thursday, April 24, 2003

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Ilya Kapovich (UIUC)
Genericity for Whitehead's algorithm, for stabilizers in Aut(F) and for one-relator groups, part II
Abstract: We use a combination of algebraic and probabilistic methods (in particular Large Deviation Theory) to show that the classical algorithm of Whitehead for solving the automorphic equivalence problem in a finitely generated free group F works in linear time. on an exponentially generic set of inputs. Moreover, it turns out that a generic element of F has trivial stabilizer in Out(F). Our results also imply that generic one-relator groups are complete, that is, they have trivial centers and trivial outer automorphism groups. Even the fact that complete one-relator groups exist is new, and we establish their existence by a classical probabilistic argument, that is, by proving that a randomly chosen one-relator group is complete with "positive probability". Moreover, we are able to compute the precise asymptotics of the number of isomorphism types of k-generator one-relator groups when the length of the defining relator goes to infinity.

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, April 24, 2003

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Ruhan Zhao (University of Toledo)
Weighted Composition Operators on the Bergman Space
Abstract: Let D be the open unit disk in the complex plane. Let u be an analytic function on the unit disk D and let $\varphi$ be an analytic self-map of D. The weighted composition operator $u C_&ob;\varphi&cb;$ is defined as follows: for an analytic function f on D, $(u C_&ob;\varphi&cb;)f(z)=u(z)f(\varphi(z))$. These operators can be considered as a combination of a multiplication operator (when $\varphi$ is the identity map) and a composition operator (when $u\equiv 1$). Weighted composition operators appear naturally. It is known that the isometries between the Hardy spaces $H^p$, $1\le p<\infty$, $p\neq 2$, are weighted composition operators. A similar result holds for the isometries on the Bergman space $L^p_a$. Both composition operators and multiplication operators have been extensively studied in recent decades. However, the study on weighted composition operators is still a new territory. Recently, M. Contreras and A. Hernadez-Diaz characterized bounded and compact weighted composition operators on Hardy spaces by using the Carleson measure. In this talk, we are going to characterize bounded, compact and Schatten class weighted composition operators on the Bergman space $L^2_a$ by using generalized Berezin transforms. An estimate of the essential norms of weighted composition operators on the Bergman space is also given. Most of our results remain true for the Hardy space and weighted Bergman spaces. This is a joint work with Zeljko Cuckovic.

Probability and Statistics Seminar
4:00 pm in 2 Illini Hall, Thursday, April 24, 2003

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Edward. F. Vonesh, Ph.D. (Senior Baxter Research Scientist, Applied Statistics Center, Baxter Healthcare Corporation)
Approximation Techniques in Nonlinear Mixed Models: Strengths, Weaknesses and Implementation
Abstract: Generalized linear and nonlinear mixed models are used extensively in such fields as population pharmacokinetics (PK), population pharmacodynamics (PD), bioassay, studies of biological or agricultural growth, and epidemiology. As these models are typically nonlinear in the random effects, maximum likelihood estimation (MLE) requires maximizing an integrated likelihood function (i.e., marginal likelihood) that generally has no closed form expression. To circumvent the computational challenges associated with maximizing an integrated likelihood, a number of estimation techniques have been proposed based on various Taylor series approximations. We will review the strengths and weaknesses of several estimation techniques including those based on the nonlinear mixed effects (NLME) algorithm of Lindstrom and Bates (1990), the penalized quasi-likelihood (PQL) algorithm of Breslow and Clayton (1993), the conditional second-order generalized estimating equations (CGEE2) algorithm of Vonesh et. al. (2002) and a Laplacian-based maximum likelihood (LMLE) algorithm similar to that described by Vonesh (1996) and Wolfinger and Lin (1997). We do so by briefly examining the theoretical basis for and limitations of these approximations, and by investigating their asymptotic properties, both theoretically and via simulation. We will also discuss some common difficulties encountered when implementing these techniques (e.g., starting values for variance-covariance parameters) and methods for potentially overcoming these difficulties. The methods will be illustrated using both simulated and real data with all analyses carried out using SAS-based software.