The most concise summary of my research interests is that I enjoy using topological and knot theoretic notions to try and make sense of some weird algebra stuff.
My current research interest is in attempting to bridge the gap between the two perspectives of the link between Drinfeld associators and Kashiwara-Vergne solutions.
The first perspective on this is the one that comes from the quantum algebra roots of both objects. There is a relatively well understood algebraic link between the two of them, but there are still open conjectures about the structure of their symmetry groups which are of particular interest.The second perspective uses bijections between certain universal finite type invariants ('knot' invariants) and these two objects to give a more topological angle.
The algebraic perspective hasn't fully been able to be translated over to the topological side, which is the area that I'm currently working on. Its hoped in making these translations, there'll be some progress made on the conjectures that were orginally made on the algebraic side of the problem.
Check out the sections below to have a look at my public-facing research activities (papers, talks, etc.).
Hogan, T. (2021, preliminary). The link between Drinfel’d associators and Kashiwara-Vergne solutions. MATRIX-MFO Tandem Workshop: Invariants and Structures in Low-Dimensional Topology (hybrid meeting), Oberwolfach Reports 42 (2021) pp 28-30.
First paper coming soon, stay tuned!
'Symmetries of trivalent tangles: approaching the link between Drinfeld associators and Kashiwara-Vergne solutions' (December 2020), presented at the Topology Special Session of the AustMS Meeting 2020, Online (20 minutes).
'A basic introduction to planar algebras' (April 2021), presented at the (GT)^2 PhD Student Symposium, Online (20 minutes).
'The link between Drinfel’d associators and Kashiwara-Vergne solutions' (September 2021), presented at MATRIX-MFO Tandem Workshop: Invariants and Structures in Low-Dimensional Topology, Online (10 minutes).
'Symmetries of 3D and 4D expansions' (December 2021), presented at the Topology Special Session of the AustMS Meeting 2021, Online (20 minutes).