2018, Schlein, Benjamin, Sohinger, Vedran

In this mini-workshop we brought together leading experts working on the application of Gibbs measures to the study of nonlinear PDEs. This framework is a powerful tool in the probabilistic study of solutions to nonlinear dispersive PDEs, in many ways alternative or complementary to deterministic methods. Among the special topics discussed were the construction of the measures, applications to dynamics, as well as the microscopic derivation of Gibbs measures from many-body quantum mechanics.

2019, Kostenko, Aleksey, Pankrashkin, Konstantin

The main focus of the workshop is on the analysis of boundary value problems for differential and difference operators in some non-classical geometric settings, such as fractal graphs, sub-Riemannian manifolds or non-elliptic transmission problems. Taking into account their importance in modern mathematical analysis, we aim at developing suitable tools in the operator theory to deal with the new problem settings.

2020, Hohage, Thorsten, Kaltenbacher, Barbara

Inverse problems in partial differential equations (PDEs) consist in reconstructing some part of a PDE such as a coefficient, a boundary condition, an initial condition, the shape of a domain, or a singularity from partial knowledge of solutions to the PDE. This has numerous applications in nondestructive testing, medical imaging, seismology, and optical imaging. Whereas classically mostly boundary or far field data of solutions to deterministic PDEs were considered, more recently also statistical properties of solutions to random PDEs have been studied. The study of numerical reconstruction methods of inverse problems in PDEs is at the interface of numerical analysis, PDE theory, functional analysis, statistics, optimization, and differential geometry. This workshop has mainly addressed five related topics of current interest: model reduction, control-based techniques in inverse problems, imaging with correlation data of waves, fractional diffusion, and model-based approaches using machine learning.

2014, Guta, Madalin, Nussbaum, Michael

The classical metric theory of statistical models (experiments) has recently been extended towards an asymptotic equivalence paradigm, allowing to classify and relate problems which are essentially infinite dimensional and ill-posed. Modern statistical concepts like these are also being integrated into the emerging field of quantum statistics, which is developing on the background of technological breakthroughs in quantum engineering. The workshop brought together leading experts in these areas, with the goal of establishing a common language, and fostering collaborations between mathematical statisticians, theoretical physicists and experimentalists.

2006, Biskup, Marek, van der Hofstad, Remco, Sidoravicius, Vladas

The workshop focused on the broad area of spatial random processes and their connection to statistical mechanics. The subjects of interest included random walk in random environment, interacting random walks, polymer models, random fields and spin systems, dynamical problems, metastability as well as problems involving two-dimensional conformal geometry. The workshop brought together many leading researchers in these fields who reported to each other on their recent achievements and exchanged ideas for new problems and potential solutions.

2006, Hulek, Klaus, Peternell, Thomas

The main aim of this workshop was to discuss recent developments in several complex variables and complex geometry. The topics included: classification of higher dimensional varieties, mirror symmetry, hyperbolicity, Kähler geometry and classical geometric questions.

2018, Hotz, Thomas, Huckemann, Stephan, Miller, Ezra

Statistics for data with geometric structure is an active and diverse topic of research. Applications include manifold spaces in directional data or symmetric positive definite matrices and some shape representations. But in some cases, more involved metric spaces like stratified spaces play a crucial role in different ways. On the one hand, phylogenetic trees are represented as points in a stratified data space, whereas branching trees, for example of veins, are data objects, whose stratified structure is of essential importance. For the latter case, one important tool is persistent homology, which is currently a very active area of research. As data sets become not only larger but also more complex, the need for theoretical and methodological progress in dealing with data on non-Euclidean spaces or data objects with nontrivial geometric structure is growing. A number of fundamental results have been achieved recently and the development of new methods for refined, more informative data representation is ongoing. Two complimentary approaches are pursued: on the one hand developing sophisticated new parameters to describe the data, like persistent homology, and on the other hand achieving simpler representations in terms of given parameters, like dimension reduction. Some foundational works in stochastic process theory on manifolds open the doors to this field and stochastic analysis on manifolds, thus enabling a well-founded treatment of non-Euclidean dynamic data. The results presented in the workshop by leading experts in the field are great accomplishments of collaboration between mathematicians from statistics, geometry and topology and the open problems which were discussed show the need for an expansion of this interdisciplinary effort, which could also tie in more closely with computer science.

2006, Smereka, Peter, Voigt, Axel

Anisotropic motion laws play a key role in many applications ranging from materials science, biophysics to image processing. All these highly diversified disciplines have made it necessary to develop common mathematical foundations and framworks to deal with anisotropy in geometric motion. The workshop brings together leading experts from various fields to address well-posedness, accuracy, and computational efficiency of the mathematical models and algorithms.

2008, Werner, Wendelin, Zeitouni, Ofer

[no abstract available]

2012, Goldreich, Oded, Sudan, Madhu, Vadhan, Salil

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, and pseudorandomness. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes.