# 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.

University of Freiburg |
Algebraic K-theory is a theory encoding deep arithmetic information. At the advent of this theory in the 1950s it was not even expected that it would possess this remarkable feature, and there has been a success story over the following decades, ever again surprising with new connections to other fields of pure mathematics. A poorly understood aspect is that in positive characteristic p > 0, the structure of the etale fundamental group (at the same prime p) is known to be extremely complicated and mysterious. Correspondingly, one should expect that the structure of p-torsion etale constructible sheaves is very rich. One may investigate the latter using positive characteristic versions of the Riemann-Hilbert correspondence and categories of crystals. I want to understand how this arithmetic complexity interacts with the K-theory of these categories. |

Stony Brook University |
The Hitchin fibration is of fundamental importance in algebraic geometry, number theory and representation theory. I propose to determine the supports of Hitchin fibrations and to establish and explain certain remarkable hidden symmetries, P=W and TMS, on the cohomology of the moduli spaces of Higgs bundles. |

Institut Mathématique de Bourgogne |
In this research project, part of a bigger research program that I have made to compete in a Directeur de recherche position at the CNRS in France, I propose to develop a tool that hopefully will help to get new computations of the so-called motivic cohomology. More precisely, I propose to study a new spectral sequence which is inspired by the Leray spectral sequence, whose usefulness and versatility is universally acknowledge, in every branches of mathematics that uses cohomology. This go through the general concept of t-structures, introduced by Beilinson, Bernstein and Deligne and its application to motivic complexes whose first appearance is due to Voevodsky, and was developed further in several works by Morel, Ayoub and finally Bondarko and Déglise. After Voevodsky, this structure is called the homotopy t-structure. The first task that I proposed to undertake is to study special objects induced by that t-structure, of a grouplike nature, the objects of the heart. This is a first step in order to determine and compute a spectral sequence that arises naturally from a t-structure. The second part of the program, widely open, is to start computing with the spectral sequence in special geometric situations. |

Université Grenoble Alpes |
The Serre problem aims at identifying the fundamental groups of smooth complex algebraic varieties among finitely presented groups and admits several variants (the projective case, the Kähler case and their orbifold versions). Considerable progress on the Kähler case has been achieved in the last 30 years, but further progress will require new examples. The constraints governing the Serre problem or more generally the homotopy type of smooth complex algebraic varieties originate typically in Hodge theory: many natural homotopical invariants carry mixed Hodge structures. However this is not enough. Indeed the part of a Kähler groups that is killed by all finite dimensional representations seems to be invisible when wearing just Hodge theoretical glasses. |

Université de Rennes |
In recent years, the development of motivic stable homotopy theory and motivic integration has led to important applications and many new connections between different fields of mathematics such as algebraic and non-archimedean geometry, topology and singularities theory. These developments represent now a very active field of research that led to many new results but also to new problems and exciting new points of view on some older open problems. In this project, I intend to explore connections between non-archimedean geometry, stable homotopy theory, arc schemes and nearby cycles with potential applications to birational geometry and singularities theory in mind. |

Imperial College London |
The aim of this project is to develop a notion of Morse theory on Berkovich spaces. This problem is motivated by recent work of the PI with Mircea Mustaţă and Chenyang Xu on the relations between the minimal model program (MMP) and the non-archimedean approach to the SYZ conjecture in mirror symmetry by Kontsevich and Soibelman. In particular, we want to obtain an intrinsic geometric explanation for the fact that the non-archimedean SYZ fibration is a strong deformation retract. These results would open new perspectives on the interactions between Berkovich spaces, mirror symmetry and the MMP. |

IMAP Rio de Janeiro |
The project “Birational Geometry of foliations” aims at developing foliation theory through the study of their singularities and through investigations on the structure of foliations of numerical Kodaira dimension zero. |

ETH Zürich |