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SFB 878:  Groups, Geometry and Actions

Subject Area Mathematics
Term from 2010 to 2019
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 129719356
 
Final Report Year 2019

Final Report Abstract

The CRC 878 ”Groups, Geometry and Actions” applied a uniform approach for studying a wide range of mathematical topics. We will now describe some of the most notable results obtained in 2018/2–2019/1. The research areas were A. Algebra and logic, B. Geometry and topology, C. Noncommutative geometry and probability theory. A. In the modular representation theory of p-adic groups a differential graded Hecke algebra was constructed whose derived category is equivalent to the derived category of smooth group representations. Moduli spaces of global G-shtukas were constructed for arbitrary flat affine group schemes G of finite type. The analogue of the Langlands-Rapoport conjecture for global G-shtukas was proved. The Rapoport-Zink conjecture concerning the image of the period morphism for p-divisible groups was established. The universal family of semistable p-adic Galois representations was constructed. A p-adic analogue of the Uhlenbeck-Yau theory of semistable vector bundles was established. For the first time natural dynamical systems attached to arithmetic schemes were constructed whose period orbits correspond to the closed points. This had been an open problem even in the simplest case. In the representation theory of algebras, tilting modules were classified by combinatorical data and polytopes. The Hodge theory of vanishing cycles was used for a partial proof of the quantum cluster positivity conjecture. Natural examples of ample stable theories were constructed. The first examples of sharply 2- and 3-transitive groups not containing a nontrivial abelian normal subgroup were found. The existence of such groups had been an open question for decades. B. In differential geometry the phenomena were clarified that occur when passing from the smooth to the singular space. Using the Ricci flow the classification of non-collapsed manifolds with almost non-negative curvature operator was achieved. It was also shown that immortal homogeneous Ricci flow solutions subconverge to a non-gradient Ricci soliton. The Farrell-Jones conjecture was proved for many classes of groups. E.g. CAT (0)-groups, cocompact lattices in Lie groups, GLn (Z), relative hyperbolic groups, mapping class groups of surfaces. Using index theory and cobordism theory it was shown that the global structure of the space R+ (M) of Riemannian metrics of positive scalar curvature on high-dimensional manifolds has a very rich topology. In particular, R+ (Sd) has the structure of an infinite loop space. A classical question about the Pontrjagin classes of topological bundles could be answered: The higher Pontrjagin classes of nonlinear Rn-bundles do not vanish, in contrast to the linear situation. C. New classes of super expanders were found. These graphs are very useful and notoriously difficult to construct. The Rosenberg conjecture was solved: A discrete group is amenable if and only if its reduced group-C*-algebra is quasidiagonal. This result was the final ingredient for the classification of simple, separable, unital, UCT C*-algebras with finite nuclear dimension. The K-theory of the semigroup C*-algebras of ax + b-semigroups for rings of integers of a number field was calculated. The two-point function of the φ4 -QFT model on a noncommutative geometry was calculated explicitely. Previously, this had been achieved only in very rare cases. Limits of random recursive structures were determined using stochastic fixpoint equations. This led to a complete description of all solutions of the equations and to an analysis of leader-election procedures and their connection to coalescence processes. In the theory of random matrices with correlated entries a phase transition was discovered. In the case of weak correlation Wigner’s semicircle law still holds but strong correlation of the entries on the diagonal leads to free convolutions of this law with some other distributions known from random matrix theory.

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