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Talks take place in lecture hall SRZ 216/217 in Seminar building, University of Münster, Orléansring 12, 48149 Münster (which is close to the Math building).
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Cornelia Drutu (University of Oxford and MPI Bonn):
Coarse non-positive curvature and rank rigidity
The notion of curvature made
its way from differential geometry to synthetic
geometry and, more recently, to combinatorics,
algebra and topology. After having modelled the notion of hyperbolic
groups and spaces on negatively curved geometry,
M. Gromov pointed out the need for a similar
notion for non-positive curvature, that should
ideally cover Hilbert geometries and
Banach spaces. Various possible definitions
have been formulated since, with usefulness depending on the viewpoint and the problem addressed.
Generally speaking, non-positive curvature displays its most striking and informative manifestations when confronted with problems of rigidity. The brand of rigidity considered in this talk comes from the Rank Rigidity Theorem of W. Ballmann. I shall explain how, under some mild curvature condition, a generalized form of this result is satisfied. The theorem applies to discrete groups of projective automorphisms of convex cones, Helly groups (answering a question of Genevois), Coxeter groups, Garside groups, hierarchically hyperbolic groups. This is joint work with Davide Spriano and Stefanie Zbinden.
Christian Kremer (MPI Bonn):
The Nielsen Realisation Problem for high-dimensional aspherical manifolds
The classical Nielsen Realisation Problem asks whether a finite subgroup of the mapping class group of a surface
can be realised by an actual group action on the surface. It was answered affirmatively by Kerckhoff.
In my talk, I will present a generalisation of this problem to high-dimensional aspherical manifolds,
and discuss results for group actions of cyclic groups of prime order, based on the newly developed
technique of equivariant and isovariant Poincaré duality.
Markus Land (University of Mainz):
The homotopy limit problem for schemes
The homotopy limit problem asks under which circumstances a canonical map from Grothendieck-Witt theory to C2-homotopy fixed points of algebraic K-theory is an equivalence (possibly after 2-adic completion). This question has been essentially answered for schemes defined over Z[1/2]. I will report on joint work with Akhil Mathew where we give an affirmative answer to this question for schemes defined over Z, subject to usual finiteness conditions and regularity conditions at points of residue characteristic 2. Along the way, we reprove Milnor’s conjecture for fields of characteristic 2 which was originally solved by Kato.
Phil Pützstück (University of Münster):
Weight Completion and Anderson Duality
Weight structures on stable categories abstract cellular decompositions: they describe objects by filtrations whose graded pieces lie in an additive weight heart. While bounded weight structures are controlled by their hearts, many natural examples in homotopy theory are unbounded. I will explain a notion of weight completeness, analogous to completeness for t-structures, which gives reconstruction results from the weight heart even in the unbounded case. Examples include chain complexes and modules over ring spectra. I will then turn to a more intricate example related to Anderson duality. Besides the standard weight structure on spectra, whose heart consists of arbitrary sums of the sphere, there is an Anderson weight structure whose heart consists of arbitrary sums of the Anderson dual of the sphere. These two weight structures have equivalent weight completions, described by modules over the spectral integral Steenrod algebra. I will conclude with a glimpse of related results on extending Anderson duality to all spectra using condensed mathematics, and on using weight completions to understand the resulting category of Anderson-reflexive condensed spectra. This is based on joint work with Thomas Nikolaus.
Steffen Sagave (University of Nijmegen):
Logarithmic topological cyclic homology
Log ring spectra are a homotopical generalization of the logarithmic rings appearing in logarithmic algebraic geometry. In this talk, I will present a version of log ring spectra that covers many interesting examples. I will also outline how to construct topological Hochschild homology and topological cyclic homology for log ring spectra. Moreover, I will show how localization sequences for these theories help to identify good candidates for fraction fields of topological K-theory spectra. This is report on joint work in progress with John Rognes and Christian Schlichtkrull.
Leonard Tokic (University of Bochum):
A family completion theorem in tempered cohomology
The classical Atiyah-Segal completion theorem identifies the K-theory of BG with a certain algebraic completion of the representation ring of G. The family version of this, proved by Haeberly and Jackowski, identifies the G-equivariant K-theory cohomology of TxEF, where T is a G-space and EF is a universal G-space with isotropy in F, with a completion of the G-equivariant K-theory cohomology of T. In this talk I want to highlight a categorical perspective on these results, and present a generalization to the equivariant tempered cohomology theories of Lurie. Time permitting, I also want to talk about how such a result could look like for compact Lie group equivariant TMF, based on joint work with Jack Davies and Sil Linskens.
Xiyan Zhong (MPI Bonn):
Rigidity of the period map up to finite covers
The period map is an important holomorphic map from the moduli space Mg of genus g curves to the moduli space Ag of g-dimensional principally polarized abelian varieties, sending a curve to its Jacobian. Benson Farb proved that the period map is the unique nonconstant holomorphic map from Mg to Ah for h at most g. In recent work, we study holomorphic maps from a certain finite cover Rg of Mg to Ah for h at most g, and prove that the unique nonconstant holomorphic map from Rg to Ah is the lift of the period map to Rg. The proof proceeds by first classifying linear representations of certain finite-index subgroups of the mapping class group.
Claudius Zibrowius (University of Bochum):
Fukaya Categories of Surfaces and Categorified Link Invariants: An Introduction
I will describe a classification result from higher representation theory and explain its role in recent developments in Heegaard Floer and Khovanov theories. The emphasis will be on the conceptual picture rather than technical details, and no previous background in either theory will be assumed.
