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Research
Research: Welcome
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.).
Research: About
My work
Publications and pre-prints
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Z. Dancso, T. Hogan, and M. Robertson. (2022). A knot-theoretic approach to comparing the Grothendieck-Teichmüller and Kashiwara-Vergne groups. arXiv:2211.11370
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T, Hogan. (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.
Research: About
Talks
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'A knot-theoretic approach to comparing the Grothendieck-Teichmuller and Kashiwara-Vergne groups' (December 2022), presented at the Topology Special Session of the AustMS Meeting 2022, in-person (20 minutes).
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'A knot-theoretic approach to comparing the Grothendieck-Teichmuller and Kashiwara-Vergne groups' (September 2022), University of Melbourne Topology Seminar, in-person (50 minutes).
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'Symmetries of 3D and 4D expansions' (December 2021), presented at the Topology Special Session of the AustMS Meeting 2021, Online (20 minutes).
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'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).
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'A basic introduction to planar algebras' (April 2021), presented at the (GT)^2 PhD Student Symposium, Online (20 minutes).
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'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).
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