Research.
Broadly speaking, my research interests thus far have laid in using specific kinds of invariants in low-dimensional topology and knot theory to approach questions about Lie theory and quantum algebra. Of particular interest to my work are Drinfel’d associators, which correspond with invariants of braids enriched with some combinatorial data, and Kashiwara-Vergne solutions, which correspond with invariants of specific 4D knotted objects.
My early work focused on producing an equivalent topological construction for the unique injective map between the automorphism groups of Drinfel'd associators and Kashiwara-Vergne solutions. More recently, my work has been directed more towards the topological side of Kashiwara-Vernge theory. In particular, I have been thinking about how to unify multiple seemingly unrelated topological models for Kashiwara-Vergne theory, as well as developing a 4D analogue of the classical 'wheeling theorem' using invariants of 4D knotted objects.
My full research statement is available upon request.
Publications and pre-prints.
Z. Dancso, T. Hogan, and M. Robertson. (2022). A knot-theoretic approach to comparing the Grothendieck-Teichmüller and Kashiwara-Vergne groups. arXiv:2211.11370
T. Hogan. 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 2354-2356.
Talks.
A knot-theoretic interpretation of the Goldman-Turaev Lie bialgebra, presented at the University of Melbourne Topology Seminar, September 2023. In-person (50 minutes).
On a lift of the Goldman-Turaev Lie bialgebra in genus zero, presented at the University of Sydney Mathematical Postgradute Society seminar, May 2023. In-person (50 minutes).
A knot-theoretic approach to comparing the Grothendieck-Teichmuller and Kashiwara-Vergne groups, presented at the Topology Special Session of the AustMS Meeting 2022, December 2022. In-person (20 minutes).
A knot-theoretic approach to comparing the Grothendieck-Teichmuller and Kashiwara-Vergne groups, presented at the University of Melbourne Topology Seminar, September 2022. In-person (50 minutes).
Symmetries of 3D and 4D expansions, presented at the Topology Special Session of the AustMS Meeting 2021, December 2021. Online (20 minutes).
The link between Drinfel’d associators and Kashiwara-Vergne solutions, presented at MATRIX-MFO Tandem Workshop: Invariants and Structures in Low-Dimensional Topology, September 2021. Online (10 minutes).
A basic introduction to planar algebras, presented at the (GT)ˆ2 PhD Student Symposium, April 2021. Online (20 minutes).
Symmetries of trivalent tangles: approaching the link between Drinfeld associators and Kashiwara-Vergne solutions, presented at the Topology Special Session of the AustMS Meeting 2020, December 2020. Online (20 minutes).