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Mathematics and Geometry

FRIAS research focus "Cohomology in Algebraic Geometry and Representation Theory"

Prof. Dr. Annette Huber-Klawitter (University of Freiburg); Prof. Dr. Stefan Kebekus (University of Freiburg); Prof. Dr. Wolfgang Soergel (University of Freiburg)

The Research Focus “Cohomology in Algebraic Geometry and Representation Theory” with Annette Huber-Klawitter (Theory of Numbers), Stefan Kebekus (Algebraic Geometry) and Wolfgang Soergel (Representation Theory) as principal investigators deals with a topic from Pure Mathematics. The linking element of their work in different sub-disciplines of mathematics is cohomology, a concept that originally served to explore geometrical spaces with the help of linear algebraic structures. A particular challenge in mathematics is to explain when two things (for example, two geometrical objects) are “different”. One possibility to show these differences is to simply count holes. While a circle has one hole, the geometrical form of the number 8 has two. The same applies to spheres and toruses (which has the form and surface of a bagel). Cohomology enables mathematicians to give a systematic definition of the illustrative concept of “holes” and provides methods for their analysis and calculation. In this way, it provides answers to questions like “What happens when we ‘glue’ two spaces together?”, “When do new holes emerge?”, or “How many holes does a complex space have?”. This is especially interesting when analysing high-dimensional spaces, which easily exceed human imagination.

The Research Focus aims to use cohomology as a common ground for “Algebraic Geometry”, “Representation Theory” and “Number Theory,” and to exchange ideas across these mathematic disciplines together with the guest researchers of the group. The Research Focus will invite a number of fellows and guest researchers and will also closely collaborate with the Mathematical Research Institute in Oberwolfach and the DFG Research Training Group 1821.



FRIAS-USIAS project group "Linking Finance and Insurance: Theory and Applications"

Prof. Dr. Jean Berard (Universität Strasbourg); Prof. Dr. Karl-Theodor Eisele (Universität Strasbourg); Prof. Dr. Thorsten Schmidt (Universität Freiburg); Prof. Dr. Ernst Eberlein (Universität Freiburg)

The goal of this research group is to tackle problems which lie in the intersection of finance and insurance. Under the current market situation this is of particular interest, as the present low interest rate environment is both a big challenge for insurance companies and a key driving factor of stock markets. This shows the high topicality of this endeavor on one side and the enormous potential for future developments on the other side. 

The main topics we aim at are hybrid derivatives which have equity and interest rates as underlying instruments. This type of derivatives appears naturally in equity-linked insurance products, variable annuities and other financial products from the area of pensions and life-insurance. Our first step is to develop fundamental results on assets of this type, in particular we are looking for valuation and risk-management methodologies. We will also cover the important question of model risk utilizing methods from robust finance and Bayesian finance. The second step is to apply these results by studying specific industry-relevant problems and developing tailor-made solutions.



FRIAS project group "Model Risk"

Prof. Dr. Patrick Dondl (Universität Freiburg); JProf. Dr. Philipp Harms (Universität Freiburg); Prof. Dr. Eva Lütkebohmert-Holtz (Universität Freiburg); Prof. Dr. Thorsten Schmidt (Universität Freiburg)

In the aftermath of the recent financial crises, model risk was identied as a main concern, triggering the rise of robust methods, robustness being linked intrinsically to nonlinearity. It is our goal to work on nonlinear methods in Finance and Probability, which are both applicable in practice and robust. Interesting connections to the physical sciences appear in models where evolution processes depend nonlinearly on randomness. Our proposal is a rst step towards a rich agenda with numerous applications. The topic has a high potential for future developments because an ecient treatment of model risk plays a crucial role in many other sciences such as Medicine, Physics, Biology, Informatics, where mathematical models are used. Our primary application in Finance is of highest social relevance, as illustrated by the enormous losses in the recent crises and their dramatic consequences for related economies. An important goal of this research group is to strengthen the connection to the applied sciences.