Georgia Institute of Technology College of Sciences

It is of interest that researchers study competing risks in which subjects may fail from any one of k causes. Comparing any two competing risks with covariate effects is very important in medical studies. In this talk, we develop omnibus tests for comparing cause-specific hazard rates and cumulative incidence functions at specified covariate levels. The omnibus tests are derived under the additive risk model by a weighted difference of estimates of cumulative cause-specific hazard rates. Simultaneous confidence bands for the difference of two conditional cumulative incidence functions are also constructed. A simulation procedure is used to sample from the null distribution of the test process in which the graphical and numerical techniques are used to detect the significant difference in the risks. In addition, we conduct a simulation study, and the simulation result shows that the proposed procedure has a good finite sample performance. A melanoma data set in clinical trial is used for the purpose of illustration.

Linear boundary value problems occur ubiquitously in many areas of science and engineering, and the cost of computing approximate solutions to such equations is often what determines which problems can, and which cannot, be modelled computationally. Due to advances in the last few decades (multigrid, FFT, fast multipole methods, etc), we today have at our disposal numerical methods for most linear boundary value problems that are "fast" in the sense that their computational cost grows almost linearly with problem size. Most existing "fast" schemes are based on iterative techniques in which a sequence of incrementally more accurate solutions is constructed. In contrast, we propose the use of recently developed methods that are capable of directly inverting large systems of linear equations in almost linear time. Such "fast direct methods" have several advantages over existing iterative methods: (1) Dramatic speed-ups in applications involving the repeated solution of similar problems (e.g. optimal design, molecular dynamics). (2) The ability to solve inherently ill-conditioned problems (such as scattering problems) without the use of custom designed preconditioners. (3) The ability to construct spectral decompositions of differential and integral operators. (4) Improved robustness and stability. In the talk, we will also describe how randomized sampling can be used to rapidly and accurately construct low rank approximations to matrices. The cost of constructing a rank k approximation to an m x n matrix A for which an O(m+n) matrix-vector multiplication scheme is available is O((m+n)*k). This cost is the same as that of the well-established Lanczos scheme, but the randomized scheme is significantly more robust. For a general matrix A, the cost of the randomized scheme is O(m*n*log(k)), which should be compared to the O(m*n*k) cost of existing deterministic methods.

We will discuss a new method of asymptotic analysis of matrix-valued Riemann-Hilbert problems that involves dispensing with analyticity in favor of measured deviation therefrom. This method allows the large-degree analysis of orthogonal polynomials on the real line with respect to varying nonanalytic weights with external fields having two Lipschitz-continuous derivatives, as long as the corresponding equilibrium measure has typical support properties. Universality of local eigenvalue statistics of unitary-invariant ensembles in random matrix theory follows under the same conditions. This is joint work with Ken McLaughlin.

The eigenvalues of the Laplacian are the squares of the frequencies of the normal modes of vibration, according to the wave equation. For this reason, Bers and Kac referred to the problem of determining the shape of a domain from the eigenvalue spectrum of the Laplacian as the question of whether one can "hear" the shape. It turns out that in general the answer is "no." Sometimes, however, one can, for instance in extremal cases where a domain, or a manifold, is round. There are many "isoperimetric" theorems that allow us to conclude that a domain, curve, or a manifold, is round, when enough information about the spectrum of the Laplacian or a similar operator is known. I'll describe a few of these theorems and show how to prove them by linking geometry with functional analysis.