Since their inception, the study of symplectic structures and the applications of symplectic techniques (as well as their odd-dimensional geometric counterparts) have benefited from a strong external motivation. Symplectic concepts have been developed to solve problems in other fields that have resisted more traditional approaches, or they have been used to provide alternative and often conceptually simpler or unifying arguments for known results. Prominent examples are the property P for knots, Cerf's theorem on diffeomorphisms of the 3-sphere, and Lyusternik-Fet's theorem on periodic geodesics.
The CRC aims to bring together mathematicians socialized in symplectic geometry on the one hand, and scientists working in areas that have proven important for cross-fertilization of ideas with symplectic geometry, in particular dynamics and algebra, on the other. In addition, the CRC aims to explore connections to areas where the potential of the symplectic view has not yet been fully exploited or, conversely, which can contribute new methods to the study of symplectic questions (e.g. optimization, computer science). The SFB brings together symplectic expertise that will allow us to make significant progress on some of the most important conjectures in this field, such as the Weinstein conjecture on the existence of periodic Reeb orbits or the Viterbo conjecture on a volume limit for the symplectic capacity of compact convex domains in R2n. The latter can be formulated as a problem of systolic geometry and is related to the Mahler conjecture in convex geometry. The focus on symplectic structures and techniques will provide a coherent structure for a group of mathematicians with a wide range of interests.

This is the priority program "Random Geometric Systems" (SPP 2265 "Random Geometric Systems") of the German Research Foundation (DFG).

Phenomena that arise from an interaction between random influences and geometric properties are ubiquitous and extremely diverse. They occur in physics (e.g. condensation or crystallization in models with interacting random particles for equilibrium situations), in materials science (e.g. electrically conductive properties in metals with impurities), in telecommunications (e.g. connectivity in spatial ad hoc multihop communication networks) and elsewhere. The origins and mechanisms leading to these phenomena are often deeply hidden. Bringing them to light often requires serious research activities, many of which must be theoretical due to the nature of the problem.

[This content is not available in "Englisch" yet]

The aim of this priority program is to co-design robust, computational, continuum biomechanical models by developing new methods that combine research in modeling, numerics and medical applications. The focus is on models of active biological systems in the human organism in order to develop methods that can later be integrated into a clinical environment and to define the interfaces between model and clinical application. However, the priority program does not aim to establish the transfer of models to the clinic via clinical trials. The program will focus on coupling strategies for "active" biological systems. The definition of "active" refers to systems that undergo a change of state due to physical, chemical and/or biological phenomena or stimuli. Examples include metabolic processes, growth and remodeling or electrical stimulation.
The SPP can strengthen the existing expertise in the German research landscape in a sustainable and internationally visible way, leading to a pioneering role in the field of continuum biomechanical modelling of active biological systems.

Laser beam welding has become increasingly important as a flexible and non-contact joining technique. The processing of alloys with a large melting range poses a challenge due to their tendency to solidify and crack. Despite the high industrial relevance, current approaches only deal with individual aspects of the problem. The FOR5134 research group has therefore been working on the development of a predictive, efficient and highly scalable multiscale and multiphysical simulation framework in seven subprojects since 2021 in order to develop a quantitative process understanding of the mechanisms of solidification cracking. This website provides an overview of the objectives, the structure and the researchers involved.

Over the past decades, scientific computing and mathematical modeling have successfully derived equations and developed numerical schemes to simulate numerous variants of processes in all conceivable application areas. The success is based on the ability to identify the mathematical core of the applications and to construct models and algorithms that are sometimes highly specialized to the specific process and scenario. One of the next challenges is therefore the coupling of different mathematical models and different calculation methods in order to be able to use the achievements for complex simulations with many different processes.

At the University of Cologne, we research the complex interplay between human cultural evolution and the Earth's diverse ecosystems. Our interdisciplinary team is pioneering research that transcends traditional boundaries. Join us as we unravel the intricate connections that shape our world and determine our future. Be part of a groundbreaking endeavor that is redefining our understanding of the interactions between humans and the Earth system. Learn more about our mission and our ongoing research in the HESCOR project.

The HESCOR project at the University of Cologne, which is part of the "Profilbildung 2022" initiative of the Ministry of Culture and Science of the State of North Rhine-Westphalia, aims to develop a new field of research into the coupling of humans and the Earth system. This interdisciplinary project focuses on the question of how the interactions between the human and earth systems have influenced human cultural evolution.