Minh-Tam Trinh - The Topology and Arithmetic of Unipotent Braid Varieties - IPAM at UCLA
Recorded 27 October 2026. Minh-Tam Trinh of Howard University presents "The Topology and Arithmetic of Unipotent Braid Varieties" at IPAM's Contact Geometry, Cluster Algebras and Skein Theory Workshop.Abstract: Using Turaev’s observation that the HOMFLYPT skein of the annulus is the ring of q-symmetric functions, we can expand the annular closure of any braid in any basis for the latter. For positive braid closures, I gave a geometric interpretation of the modified Hall–Littlewood expansion in terms of braid varieties with unipotent monodromy. I will explain how this yields an equivalence between a 1975 point-counting identity of Kawanaka and a 2009 theorem of Kálmán about HOMFLYPT, which turns out to lift to an equivalence of respective categorifications. In more recent work, Nathan Williams and I interpret Kawanaka's most general identity in terms of partial Springer resolutions. I will explain how this allows us to decompose Springer-theoretic braid varieties into pieces with the point counts, though not the geometry, of Deodhar cells. If time permits, I will show how these results predict surprising homotopy equivalences between varieties over the complex numbers.
Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-ii-contact-geometry-cluster-algebras-and-skein-theory/ Receive SMS online on sms24.me
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