Quantum Traces and Degenerations

Abstract: Skein algebras of surfaces describe a multiplication for curves on surfaces, which remembers the poisson structure on the ring of regular functions of the character variety of the surface. Quantum trace maps, introduced by Bonahon and Wong, show how skein algebras of punctured surfaces can be embedded into well-behaved algebras called quantum tori. Our discussion will focus on a joint generalization of skein algebras, which captures the hyperbolic geometry seen in Roger-Yang skein algebras and the quantum group comodule structure seen in stated skein algebras. This generalization is a key tool in building a quantum trace map for degenerations (coming from filtrations) of skein algebras of closed surfaces. As time permits we will discuss some applications. A strong effort will be made to introduce these topics at the expense of some technical details. This work is joint with Thang Le and Hiroaki Karuo.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | video )

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Noncommutative geometry in NYC

Series comments: Noncommutative Geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. Our seminar welcomes talks in Number Theory, Geometric Topology and Representation Theory linked to the context of Operator Algebras. All talks are kept at the entry-level accessible to the graduate students and non-experts in the field. To join us click meet.google.com/zjd-ehrs-wtx (5 min in advance) and igor DOT v DOT nikolaev AT gmail DOT com to subscribe/unsubscribe for the mailing list, to propose a talk or to suggest a speaker. Pending speaker's consent, we record and publish all talks at the hyperlink "video" on speaker's profile at the "Past talks" section. The slides can be posted by providing the organizers with a link in the format "myschool.edu/~myfolder/myslides.pdf". The duration of talks is 1 hour plus or minus 10 minutes.

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