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Cohomology in Algebraic Geometry and Representation Theory

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.


Organisers


Abstract

The theme of the FRIAS Research Focus is cohomology, a set of ideas for generating and computing invariants in a variety of geometric settings. Today, the idea of cohomology is central to all geometric disciplines of pure mathematics, as well as arithmetic and representation theory. It certainly appears prominently in each of the applicants long-term research plans.

The use of a common method allows to transfer ideas and results from one discipline to another, though prima facie, these might appear unrelated. The Research Focus aims to draw on these synergies. Learning from each other, and exploiting the similarities and differences between the participating researcher’s specialty fields, we expect to see progress in each concrete research project.


News and Events

Recent results and open problems about Oeljeklaus-Toma manifolds, December 8, 2017

Talk given by Victor Vuletescu, Professor at the University of Bucharest

FRIAS Lecture Hall, 10:15 am - 11:45 am 

More information here.

 

Summer School "Degeneration of Calabi-Yau varieties and arithmetic", October 9-13, 2017

Hodge-theoretic and arithmetic aspects of families of Calabi-Yau varieties (mainly in dimensions 2 and 3), periods, degenerations at the boundary, compactifications of moduli spaces, and relations to physics.

Lecture series by
  • Charles Doran (University of Alberta)
  • Matt Kerr (University of Washington)
  • Radu Laza (Stony Brook University)
  • Johannes Walcher (University of Heidelberg)
  • Don Zagier (MPIM Bonn)

 

https://www.gk1821.uni-freiburg.de/cy2017

Organisers