Mathematics Colloquia and Seminars

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The usual Temperley-Lieb 2-category (embedded strings in a disk modulo some local relations) leads to many things, including (a) a sequence of integers with rich combinatorial properties (Catalan numbers), (b) a family of orthogonal polynomials (Chebyshev polynomials), and (c) a family of TQFTs with applications to low-dimensional topology and quantum computing. I'll introduce a family of n-categories C(n,k) built out of codimension k submanifolds of an n-ball. C(2,1) is the Tempeley-Lieb category, and most of the talk will be about C(3,1) and its quotients. We will see that (a), (b) and (c) above have analogues for C(3,1). Namely, we will obtain (a) a collection of higher dimensional Catalan numbers, indexed by unrooted trees; (b) a family of orthogonal polynomials, with the variables indexed by labeled rooted trees and the polynomials indexed by a different sort of labeled rooted tree; and (c) a (3+ε)-dimensional TQFT, whose Hilbert space is built out of surfaces in a 3-manifold modulo some local relations. When computing this Hilbert space, one is lead to structures reminiscent of weighted branched surfaces and normal surfaces.