Overview

The aim of this project is the investigation of the asymptotic behaviour of distance measures of product densities, pair correlation functions and other second order characteristics and their nonparametric estimators in the setting of stationary (and isotropic) point processes in R^d. Starting from the assumption that one realization of a point field is observed in a convex window expanding in every direction, approved empirical second order quantities ( kernel-type product density estimators, Horvitz-Thompson estimator and others) and their distance to corresponding hypothetical second order quantities (measured, for instance, by the integrated squared error) are determined. Assuming that the hypothetical point process satisfies certain mixing conditions, (functional) central limit theorems are derived such that the asymptotic distribution depends only on the second order characteristic employed. These central limit theorems will result in ancillary goodness-of-fit tests. We plan to extend this approach to other stationary random measures such as fiber processes. The power of the suggested test procedures will be investigated by simulation studies, where particular interest lies in moderate window sizes. The test procedures will be implemented in the statistics software R.

This project within the area of PDE constrained optimization focuses on the development, analysis and implementation of optimization algorithms that combine efficient solution techniques from the numerics of PDEs, namely multilevel iterative solvers, and state-of-the-art optimization approaches, the so-called `all-at-once' optimization methods. It is well-known that multilevel techniques provide efficient PDE solvers of optimal algorithmic complexity. On the other hand, optimization methods within the all-at-once approach, such as sequential quadratic programming (SQP) methods and primal-dual Newton interior-point methods, have the appealing feature that in contrast to more traditional approaches, the numerical solution of the state equations is an integral part of the optimization routine. This is realized by incorporating the PDEs as constraints into the optimization routine. These strategies allow to save a considerable amount of computational work compared to methods that treat the PDE solution as an implicit function of the control/design variables. Moreover, the proper combination of multilevel techniques and optimization algorithms makes it possible to extract essential structural information from the originally infinite dimensional optimization problem. This can not be done with respect to a single grid. We aim to develop and analyze multilevel preconditioners for optimization subproblems arising in SQP and primal-dual Newton interior-point methods including strategies to control the level of inexactness allowable in optimization subproblems, when using iterative subproblem solvers. Moreover, we will investigate strategies to use multilevel methods for detection of negative curvature and in path following methods.

Organisation of an international scientific conference in Augsburg, centering around recent developments concerning the publication "Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes" by Bershadsky, Cecotti, Ooguri and Vafa. Specifically, the latter work investigates the geometry underlying topological quantum field theories and their deformations. It relates ideas from algebraic geometry and singularity theory (Frobenius manifolds and their generalisations, mixed twistor structures, primitive forms, harmonic bundles), with ideas from the theory of integrable systems and quantum field theory. The aim of the conference was in particular to address recent developments in the areas listed above, which on first sight seem to be far apart but all have common roots in the cited publication. These apparently disjoint strands were pulled together, and in particular the dialogue between the experts from the mathematics and the physics communities was triggered.