Past speakers will be listed here along with their talk slides and a recording of their talk.
2nd November at 2pm AEDT
12th October 2022
21st September 2022
31st August 2022
2nd June 2022
12th May 2022
Speaker 1: Joshua Graham (University of New South Wales)
Talk title: A (brief) Introduction to Algebraic K-theory and why it matters
Abstract: Algebraic K-theory arguably had its first appearance, albeit rudimentary, in the work of Grothendieck whilst he was studying a reformulation of the Riemann-Roch theorem. He did this by placing a group structure on the category of locally free coherent sheaves on a given scheme/variety (now called K_0). It was later found that the techniques behind his ideas had implications in other areas, particularly in vector bundles over compact Hausdorff topological spaces and also classifying finitely generated projective modules.
21st April 2022
Speaker: Dilshan Wijesena (University of New South Wales)
Talk title: A New Continuous Class of Irreducible Representations of R. Thompson’s Groups using Jones’ Machinery
Abstract: Richard Thompson’s groups F, T and V are one of the most fascinating discrete groups for their several unusual properties and their analytical properties have been challenging experts for many decades. Most notably, it was conjectured by Ross Geoghegan in 1979 that F is not amenable and thus another rare counterexample to the von Neumann problem. However, surprisingly despite many attempts, the question about amenability remains unanswered along with even more elementary questions such as Cowling-Haagerup weak amenability.
7th April 2022
Speaker: Saul Freedman (University of St. Andrews)
Talk title: The non-commuting, non-generating graph and intersection graph of a group
Abstract: Given a binary relation on the elements (or subgroups) of a group, it is natural to study the properties of the graph encoding this relation. A well-known example is the generating graph, whose vertices are the non-identity elements of the group, and whose edges are its generating pairs. Famous results here include the fact that the generating graph of a non-abelian finite simple group is connected with diameter 2 (Breuer, Guralnick and Kantor, 2008), and more generally, if the generating graph of a finite group has no isolated vertices, then its diameter is at most 2 (Burness, Guralnick and Harper, 2021).
Consider now the non-commuting, non-generating graph of a group, obtained by taking the complement of the generating graph, removing edges between elements that commute, and finally removing all vertices corresponding to central elements. We will explore the connectedness and diameter of this graph for various families of (finite and infinite) groups. We will also discuss the diameter of a related graph: the intersection graph of a finite simple group. Here, the vertices are the proper nontrivial subgroups, with edges corresponding to pairs of subgroups that intersect nontrivially.
24th March 2022
Speaker: Sebastian Bischof (Justus-Liebig Universität Gießen)
Talk title: Introduction to (twin) buildings
Abstract: Buildings have been introduced by Tits in order to study semi-simple algebraic groups from a geometrical point of view. One of the most important results in the theory of buildings is the classification of thick irreducible spherical buildings of rank at least 3. In particular, any such building comes from an RGD-system. The decisive tool in this classification is the Extension theorem for spherical buildings, i.e. a local isometry extends to the whole building.
Twin buildings were introduced by Ronan and Tits in the late 1980s. Their definition was motivated by the theory of Kac-Moody groups over fields. Each such group acts naturally on a pair of buildings and the action preserves an opposition relation between the chambers of the two buildings. This opposition relation shares many important properties with the opposition relation on the chambers of a spherical building. Thus, twin buildings appear to be natural generalizations of spherical buildings with infinite Weyl group. Since the notion of RGD-systems exists not only in the spherical case, one can ask whether any twin building (satisfying some further conditions) comes from an RGD-system. In 1992 Tits proves several results that are inspired by his strategy in the spherical case and he discusses several obstacles for obtaining a similar Extension theorem for twin buildings. In this talk I will speak about the history and developments of the Extension theorem for twin buildings.
10th March 2022
Speaker: Giulian Wiggins (University of Sydney)
Talk title: Stratifications of Module Categories
Abstract: We define a stratification of an abelian category as a formal way to express an abelian category as a "gluing" of smaller categories. This definition generalises the definitions of (equivariant) perverse sheaves as well as highest weight categories. In this talk we give necessary and sufficient conditions for an abelian category with a stratification to be equivalent to a category of modules over some algebra, and we discuss some features of the representation theory of such an algebra.