An Overview of Finite Element Analysis

Review of Projective Schemes

The genomics of vitamin D-mediated prostate cancer preventionIn this talk we will briefly review the evidence that high vitamin D status is protective against prostate cancer. We will then explain how we have used transcriptomics and ChIP-sequencing to identify molecular pathways regulated by vitamin D which may mediate the protection from prostate cancer. We will focus on the use of tools for transcriptomics and ChIP-sequencing from the perspective of the end-user. Associated Reading: Kovalenko, PL et al., (2010) 1,25 dihydroxyvitamin D-mediated orchestration of anticancer, transcript-level effects in the immortalized, non-transformed prostate epithelial cell line, RWPE1. BMC Genomics 11:26 (Available on-line) Ji, H. et al., (2008) An integrated software system for analyzing ChIP-chip and ChIP-seq data. Nature Biotechnology 26:1293Click here for a full schedule of BIOINFORMATICS SEMINARS, past and present.

Part 1Moving-Boundary Approaches for American Security ValuationIn these two talks, we develop computational schemes to solve the American option pricing problem, which is a particular example of an optimal-stopping problem. Solving this problem is non-trivial, even under the simplistic Black-Scholes model. Pricing the option requires solving a free-boundary partial differential equation (PDE) problem, a problem in which along with the option price, a boundary not known a priori (known as the optimal-exercise boundary) needs to be determined. Furthermore, empirical evidence suggests that some assumptions made under the Black-Scholes model do not hold. Thus, allowing for more-complex stochastic models to represent asset price movement is desirable, but complicates the option pricing problem even further.The Black-Scholes model assumes that volatilities of asset prices are constant, and that asset price processes are continuous. Empirical evidence, however, suggests otherwise. In the first talk, we therefore develop moving-boundary approaches that price the option in the presence of stochastic volatility and jumps. These moving-boundary approaches transform the free-boundary problem into a sequence of monotone fixed-boundary problems which are significantly easier to solve. In the second talk, we discuss two possible ways of overcoming the Œcurse of dimensionality inherent in numerical methods, with the aim of extending the moving-boundary approaches to price multidimensional American options.

On The Geometry of Jet Spaces
Abstract: The jet schemes of an algebraic variety bear important information for its singular locus, and are used to define the arc space, but not much is known about their geometry. The relations between the variety and its jet schemes will be shown, some questions and conjectures proposed by Mustata will be formulated, and some results will be discussed.

Traces, Norms, and Discriminants, using SAGE

Cyclic Sets (d'apres Drinfeld)

Thom Spaces and Cobordism

Strichartz Estimates on Asymptotically de Sitter Spaces

Joint with the University of Illinois Urbana Champaign Some Statistical Issues in Computerized Adaptive Testing - A Close Look at Graduate Record ExaminationThe rapid development of technology has brought different modes of testing to the large-scale assessment market. For example, the Graduate Record Examination (GRE) was originally developed as a paper-and-pencil test; a Computerized Adaptive Testing (CAT) mode was adopted in 1993. In 2006, Educational Testing Service (ETS) -- the company that owns the GRE -- introduced a plan to replace the CAT mode with Internet-Based Testing (IBT). However, in 2007 ETS announced that it was canceling the IBT mode after encountering difficulties in its administering. My presentation introduces the basic psychometric elements behind the development of the three modes of testing. I will focus my discussion on some issues of sequential design in large scale computerized adaptive testing.

Onset of Kuppers-Lortz-like Dynamics in Finite Rotating Thermal Convection

Measurement error models with applications to biomedical sciencesData measured with errors occur frequently in many biomedical fields. For example, in low level microarray data from either the cDNA microarray or the Affymetrix GeneChip system, each observation is an original signal coupled with a background noise. In neuroimage studies, observed predictors are often summarized from regions of interest and thereby involve errors-in-variables problems. Statistical analyses that ignore measurement errors can yield biased and inconsistent estimates and thus lead to erroneous conclusions with various degrees in data analysis. In this talk, we present two real medical studies in genetics and neuroscience, where different error problems take place. We address two measurement error models along with solution algorithms to correct for errors, including the Fourier-type deconvolution and the Simulation-extrapolation (SIMEX). Some simulations and real data analysis are demonstrated to illustrate the use of the methods and their relative merits. Associated Reading: Xiao-Feng Wang, Zhaozhi Fan, Bin Wang. 2010. Estimating smooth distribution function in the presence of heteroscedastic measurement errors. Computational Statistics and Data Analysis 54:25-36.Click here for a full schedule of BIOINFORMATICS SEMINARS, past and present.

Part 2Moving-Boundary Approaches for American Security ValuationIn these two talks, we develop computational schemes to solve the American option pricing problem, which is a particular example of an optimal-stopping problem. Solving this problem is non-trivial, even under the simplistic Black-Scholes model. Pricing the option requires solving a free-boundary partial differential equation (PDE) problem, a problem in which along with the option price, a boundary not known a priori (known as the optimal-exercise boundary) needs to be determined. Furthermore, empirical evidence suggests that some assumptions made under the Black-Scholes model do not hold. Thus, allowing for more-complex stochastic models to represent asset price movement is desirable, but complicates the option pricing problem even further.The Black-Scholes model assumes that volatilities of asset prices are constant, and that asset price processes are continuous. Empirical evidence, however, suggests otherwise. In the first talk, we therefore develop moving-boundary approaches that price the option in the presence of stochastic volatility and jumps. These moving-boundary approaches transform the free-boundary problem into a sequence of monotone fixed-boundary problems which are significantly easier to solve. In the second talk, we discuss two possible ways of overcoming the Œcurse of dimensionality inherent in numerical methods, with the aim of extending the moving-boundary approaches to price multidimensional American options.

The Representative Domain of a Homogeneous Bounded Domain
Abstract: The representative domain introduced by S. Bergman gives a nice realization for a homogeneous bounded domain, which is a generalization of the Harish-Chandra realization for a symmetric bounded domain. We show that the representative domain coincides with the image of the Cayley transform introduced by R. Penney and T. Nomura. As an application, we see that a homogeneous bounded domain is symmetric if and only if its representative domain is convex.This is a joint work with H. Ishi.

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High-order Boundary Integral Method for Surface Diffusions on Elastically Stressed Axisymmetric Rods

Volume Growth, Brownian Motion, and Stochastic Completeness of a Complete Riemannian Manifold
Abstract: A geodesically complete Riemannian manifold is called stochastically complete if its heat kernel (the minimal fundamental solution of the parabolic Laplace-Beltrami operator) is integrated to one. Since the heat kernel is the transition density function of Riemannian Brownian motion, a manifold is stochastically complete if and only if Brownian motion does not explode. To find a proper geometric condition for stochastic completeness is an old geometric problem. The first result in this direction was due to S. T. Yau, who proved that a Riemannian manifold is stochastically complete if its Ricci curvature is bounded from below by a constant. It has been know for quite some time that the property of stochastic completeness is intimately related to the volume growth of a Riemannian manifold. We study stochastic completeness by looking at the more refined question of upper escaping rates of Riemannian Brownian motion. We show how the Neumannheat kernel, time reversal of reflecting Brownian motion, and volumes of geodesic balls come together and give an elegant and often sharp upper bound of the escaping rate solely in terms of the volume growth function without any extra geometric restriction. This is a joint work with Guang Nan Qin of Institute of Applied Mathematics of the Chinese Academy of Sciences.

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Refreshments will be served in the Math Library Lounge at 4:00 p.m.

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Refreshments will be served in the Math Library Lounge at 4:00 p.m.

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Some of the Mathematical Legacy of Richard A. Hunt: The Years 1968-1973The talk will focus on three topics in Harmonic Analysis studied by Richard Hunt early in his mathematical career: boundary values of harmonic functions, weighted norm inequalities for the Hilbert transform, and pointwise convergence of Fourier series. It will be generally non-technical and include a sketch of the history of the topics as well as some of the outgrowth of his research on them. Refreshments will be served in the Math Library Lounge at 4:00 p.m.

Joint with the Graduate Student Organization (GSO)

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Patterns Patterns Everywhere
Abstract: Regular patterns appear all around us: from vast geological formations to the ripples in a vibrating coffee cup, from the gaits of trotting horses to tongues of flames, and even in visual hallucinations. The mathematical notion of symmetry is a key to understanding how and why these patterns form. In this lecture the speaker will show some of these fascinating patterns and explain how mathematical symmetry enters the picture.

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