Past Algebra Seminar

19 October 2010
17:00
Desi Kochloukova
Abstract
We discuss homological finiteness Bredon types FPm with respect to the class of finite subgroups and seperately with respect to the class of virtually cyclic subgroups. We will concentrate to the case of solubles groups and if the time allows to the case of generalized R. Thompson groups of type F. The results announced are joint work with Brita Nucinkis (Southampton) and Conchita Martinez Perez (Zaragoza) and will appear in papers in Bulletin of LMS and Israel Journal of Mathematics.
12 October 2010
17:00
Kevin McGerty
Abstract
Recently Frenkel and Hernandez introduced a kind of "Langlands duality" for characters of semisimple Lie algebras. We will discuss a representation-theoretic interpretation of their duality using quantum analogues of exceptional isogenies. Time permitting we will also discuss a branching rule and relations to Littelmann paths.
15 June 2010
17:00
Detlev Hoffmann
Abstract
An important problem in algebra is the study of algebraic objects defined over fields and how they behave under field extensions, for example the Brauer group of a field, Galois cohomology groups over fields, Milnor K-theory of a field, or the Witt ring of bilinear forms over a field. Of particular interest is the determination of the kernel of the restriction map when passing to a field extension. We will give an overview over some known results concerning the kernel of the restriction map from the Witt ring of a field to the Witt ring of an extension field. Over fields of characteristic not two, general results are rather sparse. In characteristic two, we have a much more complete picture. In this talk, I will explain the full solution to this problem for extensions that are given by function fields of hypersurfaces over fields of characteristic two. An important tool is the study of the behaviour of differential forms over fields of positive characteristic under field extensions. The result for Witt rings in characteristic two then follows by applying earlier results by Kato, Aravire-Baeza, and Laghribi. This is joint work with Andrew Dolphin.
1 June 2010
17:00
Peter Jorgensen
Abstract
\ \ The cluster category of Dynkin type $A_\infty$ is a ubiquitous object with interesting properties, some of which will be explained in this talk. \\ \ \ Let us denote the category by $\mathcal{D}$. Then $\mathcal{D}$ is a 2-Calabi-Yau triangulated category which can be defined in a standard way as an orbit category, but it is also the compact derived category $D^c(C^{∗}(S^2;k))$ of the singular cochain algebra $C^*(S^2;k)$ of the 2-sphere $S^{2}$. There is also a “universal” definition: $\mathcal{D}$ is the algebraic triangulated category generated by a 2-spherical object. It was proved by Keller, Yang, and Zhou that there is a unique such category. \\ \ \ Just like cluster categories of finite quivers, $\mathcal{D}$ has many cluster tilting subcategories, with the crucial difference that in $\mathcal{D}$, the cluster tilting subcategories have infinitely many indecomposable objects, so do not correspond to cluster tilting objects. \\ \ \ The talk will show how the cluster tilting subcategories have a rich combinatorial structure: They can be parametrised by “triangulations of the $\infty$-gon”. These are certain maximal collections of non-crossing arcs between non-neighbouring integers. \\ \ \ This will be used to show how to obtain a subcategory of $\mathcal{D}$ which has all the properties of a cluster tilting subcategory, except that it is not functorially finite. There will also be remarks on how $\mathcal{D}$ generalises the situation from Dynkin type $A_n$ , and how triangulations of the $\infty$-gon are new and interesting combinatorial objects.

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