28th August 2024

Speaker: Bailey Whitbread (University of Queensland)

Title: Geometry of character varieties

Abstract: A basic object linking the worlds of algebra and geometry is an algebraic group. These are spaces cut out by polynomial equations which are also groups. They are used to create character varieties, which play a central role in numerous areas of mathematics. I'll explain how we study character varieties using number theory, representation theory and Julia.

22nd May 2024

Speaker: Matthias Fresacher (Western Sydney University)

Title: What are diagram monoids?

Abstract: As the name suggests, diagram monoids are a family of monoids that have nice easy to draw pictorial representations as graphs. In fact, their multiplication is much easier to understand in their graphical form than the complex formal definition. Playing with diagrams is a fun and exciting way to do maths without numbers or even equations. It is all about drawing pretty pictures and following through paths in the maze that is a diagram. So whilst formally, diagram monoids only make their first appearance in postgraduate maths, they can easily be played with by kids and hence are very accessible. Are you ready for some maths by pictures?

3rd April 2024

Speaker: Kai Machida (University of Melbourne)

Title: Monoid schemes, Lambda schemes and resolutions of singularities

Abstract: F1-geometry is a hypothetical geometry over a base deeper than SpecZ. The origins of this theory can be attributed to Jacques Tits, who in the 1950’s postulated the existence of such a geometry that would explain analogues between projective geometries and finite sets. Since the 1990’s there have been many approaches to defining appropriate categories for F1-geometry with a consistent theme being the appearance of monoid schemes. We will discuss details of Borger’s theory of Lambda-schemes and how using pointed monoid schemes we can apply a resolution theorem of Bierstone and Milman for toric varieties to give Lambda-equivariant resolutions.

2nd November 2022

Speaker: Jackson Ryder (University of New South Wales)

Title: What is noncommutative algebraic geometry?

Abstract: Classical algebraic geometry is concerned with the study of varieties, which are sets of solutions to polynomials. The more modern approach is via schemes, which are (locally) prime spectra of commutative rings. One may then wonder if such a geometric object can be constructed from a noncommutative ring. The answer in most cases is no, as noncommutative rings generally have very few two-sided ideals. However, a theorem of Serre shows that all of the geometric data of a projective scheme can be encoded in a category constructed from purely algebraic data, free from any commutativity restrictions! This is our jumping off point for noncommutative algebraic geometry, where we work with the type of category that arises from Serre's theorem. In this talk we give an introduction to the commutative algebro-geometric notions required to understand Serre's theorem, and then use this theorem to motivate our notion of a `noncommutative projective scheme'. Finally, we show how this setting allows us new insight into areas such as the representation theory of finite-dimensional algebras.

12th October 2022

Speaker: Christopher Parker (University of Technology Sydney)

Title: What is intersection theory?

Abstract: Intersection theory studies the intersection of two schemes meeting in a third. It is perhaps one of the oldest areas of algebraic geometry, dating back to Bézout in the 1700s, with the well-known Bézout's theorem. We will explore intersection theory in its modern formulation - via an algebro-geometric analogue of a cohomology ring, called a Chow ring. We will highlight the use of intersection theory in enumerative geometry by applying this theory to moduli spaces, and use it to solve some classical problems in the field.

21st September 2022

Speaker: Alex Elzenaar (Max Planck Institut für Mathematik)

Title: What is a Kleinian group?

Abstract: They were first isolated in the very earliest days of modern geometry by Poincaré, played an integral role in Thurston's geometrisation programme in the 20th century, and continue to be studied in various forms due to their role as a bridge linking topology, geometry, algebra, and analysis. They appear as covering groups for Riemann surfaces and 3-manifolds, as simple examples of dynamical systems, as symmetry groups of tilings in hyperbolic geometry, and also in the study of automorphic forms in number theory. We will take a walk through a zoo of many wildly different examples of Kleinian groups in order to advertise this very rich field to people who may only know a little complex analysis and linear algebra; we hope that the talk will also be fun for more advanced students in geometry and algebra.

31st August 2022

Speaker: Stefano Giannini (University of Queensland)

Title: What is Hilbert's 14th Problem?

Abstract: Hilbert’s 14th Problem asks: Given the action of a group on a vector space V, is the ring of invariant functions finitely generated? For a certain class of groups, a proof of Hilbert shows this is always true. In the mid-1960s, Nagata provided an example of a unipotent group where the subspace of invariant functions is not finitely generated. One may still ask: For which pairs (G,V) is the invariant ring finitely generated? In this talk, I will describe Hilbert's 14th problem, present some classical results in invariant theory, and discuss how my work fits into this story.

2nd June 2022

Speaker: Tyson Klingner (University of Adelaide)

Title: Langlands Duality in the Hitchin System

Abstract: In 1987 Hitchin introduced Higgs bundles as a tool to solve self-dual Yang Mills equations on Compact Riemann surfaces. Since then, Higgs bundle have appeared in a wide array in area, even in the Langlands Program when Ngo used them to prove the fundamental lemma. In this talk we will demonstrate Langlands duality interacting with Higgs bundles.

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.

The study of Whitehead torsion in topology later motivated the definition of a different functor (now called K_1) which was found to fit an exact sequence that related the K_0 and K_1 functor, and it was this idea that lead mathematicians (particularly Daniel Quillen) to extend these K functors to all n > 0. We will look at the details of this construction as well as some important applications of these groups.

Speaker 2: Vandit Trivedi (Australian National University)

Title: Obtaining a biautomatic structure on a group via a group action

Abstract: Automatic groups are a special class of groups for which the word problem is decidable. It is not known whether every automatic group is biautomatic. Answering this question is further complicated by the fact that an automatic group may have multiple automatic structures. In some groups, a biautomatic structure is obvious. In other cases, there are methods to find a biautomatic structure on a group. In this talk, we look at biautomatic structures on some groups as well as one of the methods to determine a biautomatic structure, through a group acting on a systolic simplicial complex.

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. 

Surprisingly, these discrete groups were recently discovered by Vaughn Jones while working on the very continuous structures of conformal nets and subfactors. In the first part of this talk, I will explain how Jones’ work provided a new method for constructing unitary representations of Thompson’s groups and provided easy new proofs of certain known analytical properties of Thompson’s groups. In particular, I will talk about a specific family of Jones’ representations called Pythagorean representations. In the second part of the talk, I will describe a new continuous class of Pythagorean representations constructed with my Honours supervisor, Arnaud Brothier. We proved these representations are almost always irreducible and pairwise non-isomorphic. Further, we proved that they are not the induction of a finite representation of any subgroup of F.

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.

2nd November 2022

Speaker: Jackson Ryder (University of New South Wales)

Title: What is noncommutative algebraic geometry?

Abstract: Classical algebraic geometry is concerned with the study of varieties, which are sets of solutions to polynomials. The more modern approach is via schemes, which are (locally) prime spectra of commutative rings. One may then wonder if such a geometric object can be constructed from a noncommutative ring. The answer in most cases is no, as noncommutative rings generally have very few two-sided ideals. However, a theorem of Serre shows that all of the geometric data of a projective scheme can be encoded in a category constructed from purely algebraic data, free from any commutativity restrictions! This is our jumping off point for noncommutative algebraic geometry, where we work with the type of category that arises from Serre's theorem. In this talk we give an introduction to the commutative algebro-geometric notions required to understand Serre's theorem, and then use this theorem to motivate our notion of a `noncommutative projective scheme'. Finally, we show how this setting allows us new insight into areas such as the representation theory of finite-dimensional algebras.