I will give an introduction to factorization homology using configuration spaces, and discuss the nonabelian Poincaré duality theorem of Segal, Salvatore, Lurie, and Ayala–Francis​, which relates factorization homology to mapping spaces. Time permitting, I will also talk about the Ayala–Francis axiomatic approach to factorization homology, which positions factorization homology as a "homology theory for manifolds."

Spectral networks are a combinatorial tool consisting of labelled lines on a Riemann surface. They have a surprising amount of applications and are intimately linked to non-Abelianization of flat connections, Fock–Goncharov cluster coordinates, exact WKB theory, etc. After reviewing this story for the SL(2) and SL(3) cases, I will describe this is in detail for the group $G_2$. Time permitting, I will give as an application a concrete parametrization of the nonabelian Hodge correspondence for the Hitchin component of the split real form of $G_2$. This is joint work with Andy Neitzke.

Modular functors are collections of projective representations of mapping class groups of surfaces, compatible with cutting and gluing operations. They can be thought of as categorified, anomalous 2d topological field theories (TFT) where the "anomaly" is responsible for the projectiveness of the representations.

A well-known folklore theorem states that ordinary 2d TFT are classified by (commutative) Frobenius algebras. In a similar way, any modular functor yields a "categorified Frobenius algebra", of which ribbon categories form a large class of examples. In this talk, we'll explain a necessary and sufficient condition for such a structure to extend to a modular functor, formulated in terms of certain generalized skein modules attached to handlebodies. A key observation is that this is, indeed, a condition, not extra structure, so that such an extension is essentially unique whenever it exists.

This construction should be thought of as a far reaching generalization of the construction by Masbaum and Roberts of a modular functor from Kauffman skein modules. As a special case it also recovers, in a purely topological way, the construction of a modular functor from a (not necessarily semisimple) modular category by Lyubachenko, and the uniqueness result is new even in those cases. This is based on joint work with Lukas Woike.

Starting with a counting problem for elements of the symmetric group, we introduce the so-called shifted symmetric functions. These functions, which also occur naturally in enumerative geometry, have the remarkable property that the corresponding generating series are quasimodular forms. We discuss another family of functions on partitions with the same property. In particular, using certain Hamiltonian operators associated to cohomological field theories, we explain how this seemingly different family of functions turns out to be closely related to the shifted symmetric functions.

The coloured Jones and Alexander polynomials are quantum invariants that come from representation theory. There are important open problems in quantum topology regarding their geometric information. Our goal is to describe these invariants from a topological viewpoint, as intersections between submanifolds in configuration spaces. We show that the Nth coloured Jones and Alexander polynomials of a knot can be read off from Lagrangian intersections in a fixed configuration space. At the asymptotic level, we geometrically construct a universal ADO invariant for links as a limit of invariants given by intersections in configuration spaces. The parallel question of providing an invariant unifying the coloured Jones invariants is the subject of the universal Habiro invariant for knots. The universal ADO invariant that we construct recovers all of the coloured Alexander invariants (in particular, the Alexander polynomial in the first term).