Pure Math Honours presentations

Abstract:

Honours presentations, in order, of Aditya Ganguly, Dilshan Wijesena, Ethan Brown, Chesta Wu, and Michael Horton. 

Each talk is 20 minutes long followed by 5 minutes of questions and by 5 minutes break.

We will start at 2:35.

Speaker: Aditya Ganguly

Title: Existence of 2-factors in random uniform regular hypergraphs

Abstract:  In the theory of random graphs, it is a common objective to prove that almost all graphs of a certain type contain a specified subgraph. To this end, the second moment method is an elementary tool from probability theory which can sometimes show that, with 'high probability', the discrete random variable counting the number of occurrences of a given subgraph in a random graph is strictly positive. When this technique fails, however, the Small Subgraph Conditioning Method (SSCM), introduced by Robinson and Wormald in 1992, has been used to prove the existence of subgraphs such as perfect matchings, Hamilton cycles and spanning trees in almost all regular graphs.

Our paper extends a result of Robalewska (1996), that almost all regular graphs contain a 2-factor (2-regular subgraph), to the world of uniform regular hypergraphs by using the SSCM. We also explain the consequences of this result for a version of qualitative equivalence between random hypergraph models, known as 'contiguity'.

Speaker: Dilshan Wijesena

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 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. With my supervisor, Arnaud Brothier, we constructed a new continuous class of Pythagorean representations. These representations were proven to be 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.

Speaker: Ethan Brown

Title: Permutation Automorphisms of the Cuntz Algebras

Abstract: We will discuss automorphisms of the Cuntz algebras - specifically, those which arise from permutations acting upon products of generators. We will follow work by Conti and Szymański in finding such automorphisms, by turning the problem into an equivalent combinatorial problem, and see how this process can be implemented computationally. Finally, we will briefly discuss further directions, and how this method could be used to efficiently determine substructures of the automorphism groups.

Speaker: Chesta Wu

Title: TBA

Abstract: TBA

Speaker: Michael Horton

Title: Relative position of subspaces in a Hilbert space

Abstract: In this talk, we will look at the ways in which you can classify systems of subspaces in a Hilbert space by their relative position to each other. My talk briefly covers the work by Halmos regarding 2 subspaces and how their relative position can be classified up to unitary equivalence. I then go over some of the more recent work by Enemoto and Watatani, who develop a classification of n subspaces by relaxing the equivalence relation to isomorphism classes, looking more specifically at the cases for 3 and 4 subspaces.