Past Junior Topology and Group Theory Seminar

4 May 2011
16:00
Moritz Rodenhausen
Abstract
A factorability structure on a group G is a specification of normal forms of group elements as words over a fixed generating set. There is a chain complex computing the (co)homology of G. In contrast to the well-known bar resolution, there are much less generators in each dimension of the chain complex. Although it is often difficult to understand the differential, there are examples where the differential is particularly simple, allowing computations by hand. This leads to the cohomology ring of hv-groups, which I define at the end of the talk in terms of so called "horizontal" and "vertical" generators.
  • Junior Topology and Group Theory Seminar
16 February 2011
16:00
Lars Scheele
Abstract
The construction of the asymptotic cone of a metric space which allows one to capture the "large scale geometry" of that space has been introduced by Gromov and refined by van den Dries and Wilkie in the 1980's. Since then asymptotic cones have mainly been used as important invariants for finitely generated groups, regarded as metric spaces using the word metric. However since the construction of the cone requires non-principal ultrafilters, in many cases the cone itself is very hard to compute and seemingly basic questions about this construction have been open quite some time and only relatively recently been answered. In this talk I want to review the definition of the cone as well as considering iterated cones of metric spaces. I will show that every proper metric space can arise as asymptotic cone of some other proper space and I will answer a question of Drutu and Sapir regarding slow ultrafilters.
  • Junior Topology and Group Theory Seminar
9 February 2011
16:00
Alessandro Sisto
Abstract
After a quick-and-dirty introduction to nonstandard analysis, we will define the asymptotic cones of a metric space and we will play around with nonstandard tools to show some results about them. For example, we will hopefully prove that any separable asymptotic cone is proper and we will classify real trees appearing as asymptotic cones of groups.
  • Junior Topology and Group Theory Seminar

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