Vorträge des SFB 1442

A conjecture by Church-Farb-Putman predicts that the rational cohomology of SL_n(Z) vanishes in high degrees. In this series of two talks, we will explain why one might believe that this conjecture is true and describe recent progress that has been made for SL_n(Z) and related groups. The first talk focuses on SL_n(Z). The second talk puts the results in a broader context of arithmetic groups.

In the late 19th century Cantor discovered the existence of uncountable numbers, and then went on to define a whole hierarchy of infinite cardinal numbers, in other words there are "different levels of infinity". It is natural to ask if finitary and countably infinite combinatorial objects have uncountable analogues. It turns out that the answer is yes. We will focus on one such key combinatorial property, the tree property. A classical result from graph theory (König's infinity lemma) shows the existence of this property for countable trees. More precisely, the lemma says that every infinite, finitely branching tree has an infinite branch. We will discuss what happens in the case of uncountable trees.

A conjecture by Church-Farb-Putman predicts that the rational cohomology of SL_n(Z) vanishes in high degrees. In this series of two talks, we will explain why one might believe that this conjecture is true and describe recent progress that has been made for SL_n(Z) and related groups. The first talk focuses on SL_n(Z). The second talk puts the results in a broader context of arithmetic groups.

In joint work with Vytautas Pa?k?nas, we show that the universal framed deformation ring of an arbitrary mod p representation of the absolute Galois group of a p-adic local field valued in a possibly disconnected reductive group G is flat, local complete intersection and of the expected dimension. In particular, any such mod p representation has a lift to characteristic 0. The work extends results of Böckle, Iyengar and Pa?k?nas in the case G=GL_n. We give an overview of the proof of this main result.

We present a graph-theoretic model for dynamical systems given by a homeomorphism f on the Cantor set X. This construction gives a bijective correspondence between such dynamical systems (X,f) and a subclass of two-colored Bratteli separated graphs. We use this construction in order to write any dynamical system of our interest as an inverse limit of a sequence of (what we call) generalized finite shifts. This enables us to compute the associated Steinberg algebra (resp. C*-algebra) of the dynamical systems as colimits of the graph algebras (resp. graph C*-algebras) associated with the different levels of the corresponding separated graph. In subsequent work we plan to apply this theory to relate the type semigroup of the dynamical system with the graph monoid of the corresponding separated graph, and with the non-stable K-theory of the Steinberg algebra. This is joint work with Pere Ara (Universitat Autònoma de Barcelona).

Abstract: What has homological mirror symmetry ever done for you? I will give my personal answer to that question and discuss joint work in progress with Liam Watson and Artem Kotelskiy concerning the behaviour of Khovanov homology under Conway mutation.