Our interest lies in random projections of high dimensional objects, e.g., random vectors and sets. Such objects arise in a variety of contexts. In the statistics and machine learning literature, projections onto random lower-dimensional subspaces are employed for dimension reduction, clustering, regression, and topic discovery in the setting of high-dimensional data. We focus on projections of random vectors selected according to some probability measure on the n-dimensional lp-ball. More precisely, we study probabilistic limit theorems such as a central limit theorem, or a moderate and a large deviation principle for the volume of random projections of general lp-balls.

Phenomena that emerge from an interaction between random influences and geometric properties are ubiquitous and extremely diverse. They appear in physics (e.g., condensation or crystallisation in interacting random particle models for equilibrium situations), materials science (e.g., electrical conducting properties in metals with impurities), in telecommunication (e.g., connectivity in spatial multi-hop ad-hoc communication networks), and elsewhere. The origins and the mechanisms that lead to the phenomena are often deeply hidden. Bringing them to the surface often requires serious research activities, many of which have to be theoretical by the nature of the problem.

The aim of the CRC is to bring together, on the one hand, mathematicians who have been socialized in symplectic geometry and, on the other, scientists working in areas that have proved important for the cross-fertilization of ideas with symplectic geometry, notably dynamics and algebra. In addition, the CRC intends to explore connections with fields where, so far, the potential of the symplectic viewpoint has not been fully realized or, conversely, which can contribute new methodology to the study of symplectic questions (e.g. optimization, computer science). The CRC bundles symplectic expertise that will allow us to make substantive progress on some of the driving conjectures in the field, such as the Weinstein conjecture on the existence of periodic Reeb orbits, or the Viterbo conjecture on a volume bound for the symplectic capacity of compact convex domains in R2n. The latter can be formulated as a problem in systolic geometry and is related to the Mahler conjecture in convex geometry.

The Research Training Group (RTG) High-dimensional Phenomena in Probability - Fluctuations and Discontinuity offers excellent national and international graduates in the mathematical sciences the opportunity to conduct internationally visible doctoral research in probability theory. The goal of the RTG is to bring together the joint expertise on aspects of high dimension in probability. In the study of random structures in high dimensions, one frequently observes universality in limit theorems (fluctuations) as well as phase transitions (discontinuities). These aspects form the common focus of a large number of currently active research projects in stochastic processes. 

The scientific network cumulants, concentration and superconcentration focuses on the common interests of nine young scientists at the research locations Berlin, Bochum, Münster and Osnabrück. The three topics are fundamental in several research areas in current probability theory. The main focus of the first phase of the scientific network is on the one hand to explore new applications of the classical method of cumulants to prove concentration inequalities or moderate deviations. On the other hand the technical tools whose descriptions are scattered throughout the literature should be brought together in an survey article. We take this opportunity to review the optimality of the given assumptions and adapt them to new applications.