Infinite tri-symmetric group, multiplication of double cosets, and checker topological field theories

We consider a product of three copies of infinite symmetric group and its representations spherical with respect to the diagonal subgroup. We show that such representations generate functors from a certain category of simplicial two-dimensional surfaces to the category of Hilbert spaces and bounded linear operators.Comment: 29 pages, 10 figure

Strength and Noetherianity for infinite Tensors

This thesis has four chapters. Chapter 1 introduces the concepts of Noetherianity up to the action of a group and polynomial functors. In Chapter 2, we prove that certain spaces associated to sequences of diagonal embeddings are Noetherian up to the action of the groups acting on them. Chapter 3 is devoted to investigating the strength of polynomials. In Chapter 4, we study closed subsets of polynomial functors. We prove that strict closed subsets always arise as unions of images from smaller polynomial functors

Goodwillie's Calculus of Functors and Higher Topos Theory

We develop an approach to Goodwillie's calculus of functors using the techniques of higher topos theory. Central to our method is the introduction of the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality which allows us to give a number of useful characterizations of the class of $n$-excisive maps. We use these results to show that the pushout product of a $P_n$-equivalence with a $P_m$-equivalence is a $P_{m+n+1}$-equivalence. Then, building on our previous work, we prove a Blakers-Massey type theorem for the Goodwillie tower. We show how to use the resulting techniques to rederive some foundational theorems in the subject, such as delooping of homogeneous functors.Comment: 40 pages, (a slightly modified version of) this paper is accepted for publication by the Journal of Topolog