A modular functor is defined as a system of mapping class group representations on vector spaces (the so-called conformal blocks) that is compatible with the gluing of surfaces. The notion plays an important role in the representation theory of quantum groups and conformal field theory. In my talk, I will give an introduction to the theory of modular functors and recall some classical constructions. Afterwards, I will explain the approach to modular functors via cyclic and modular operads and their bicategorical algebras. This will allow us to extend the known constructions of modular functors and to classify modular functors by certain cyclic algebras over the little disk operad for which an obstruction formulated in terms of factorization homology vanishes. (The talk is based to a different extent on different joint works with Adrien Brochier, Lukas Müller and Christoph Schweigert.)

Around 30 years ago, Lin defined an analog of the Casson invariant for knots. This invariant counts representations of the knot group into SU(2) which satisfy tr(ρ(m)) = c for some fixed c. As a function of c, the Casson-Lin invariant turns out to be given by the Levine-Tristram signature function.

If K is a small knot in S³, I'll describe a version of the Casson-Lin invariant which counts representations of the knot group into SL₂(R) with tr(ρ(m)) = c for c in [-2,2]. The sum of the SU(2) and SL₂(R) invariants is a constant h(K), independent of c. I'll discuss the proof of this fact and give some applications to the existence of real parabolic representations and left-orderings. This is joint work with Nathan Dunfield.

Ideal triangulations were introduced by Thurston as a tool for studying hyperbolic three-manifolds.  Taut ideal triangulations were introduced by Lackenby as a tool for studying "optimal" representatives of second homology classes.

After these applications in geometry and topology, it is time for dynamics. Veering triangulations (taut ideal triangulations with certain decorations) were introduced by Agol to study the mapping tori of pseudo-Anosov homeomorphisms.  Gueritaud gave an alternative construction, and then Agol and Gueritaud generalised it to find veering triangulations of three-manifolds admitting pseudo-Anosov flows (without perfect fits).

We prove the converse of their result: that is, from any veering triangulation we produce a canonical dynamic pair of branched surfaces (in the sense of Mosher).  These give flows on appropriate Dehn fillings of the original manifold.  Furthermore, our construction and that of Agol--Gueritaud are inverses.  This then gives a "perfect" combinatorialisation of pseudo-Anosov flow (without perfect fits).

This is joint work with Henry Segerman.

In this talk, I will first give a brief introduction to the Landau-Ginzburg -- Conformal Field Theory (LG-CFT) correspondence, a prediction from physics. This prediction links aspects of Landau-Ginzburg models, described by matrix factorisations for a polynomial known as the potential, with Conformal Field Theories, described by for example vertex operator algebras. While both sides of the correspondence have good mathematical descriptions, it is an open problem to give a mathematical formulation of the correspondence. 

After this introduction, I will discuss the only known realisation of this correspondence, for the potential $x^d$. For even $d$ this is a recent result, and I will give a sketch of the proof which uses the tools of module tensor categories

 I will not assume prior knowledge of matrix factorisations, CFTs, or module tensor categories. This talk is based on joint work with Ana Ros Camacho.

I will present classification results for 4-manifolds with boundary and infinite cyclic fundamental group, obtained in joint work with Anthony Conway and with Conway and Lisa Piccirillo.  Time permitting, I will describe applications to knotted surfaces in simply connected 4-manifolds, and to investigating the difference between the relations of homotopy equivalence and stable homeomorphism. These will also draw on work with Patrick Orson and with Conway,  Diarmuid Crowley, and Joerg Sixt.

Bestvina--Feighn proved that Aut(F_n) is a rational duality group, i.e. there is a Q[Aut(F_n)]-module, called the rational dualizing module, and a form of Poincare duality relating the rational cohomology of Aut(F_n) to its homology with coefficients in this module. Bestvina--Feighn's proof does not give an explicit combinatorial description of the rational dualizing module of Aut(F_n). But, inspired by Borel--Serre's description of the rational dualizing module of arithmetic groups, Hatcher--Vogtmann constructed an analogous module for Aut(F_n) and asked if it is the rational dualizing module. In work with Miller, Nariman, and Putman, we show that Hatcher--Vogtmann's module is not the rational dualizing module.

I will talk about work in progress with Ana Caraiani aimed at proving Ihara’s lemma for quaternionic Shimura varieties, generalising the strategy of Manning-Shotton for Shimura curves. As an arithmetic motivation, in the first part of the talk I will recall an approach to the Birch and Swinnerton-Dyer conjecture based on congruences between modular forms, relying crucially on Ihara’s lemma.

The Zilber-Pink conjecture predicts that there should be only finitely
many algebraic numbers t such that the three elliptic curves with
j-invariants t, -t, 2t are all isogenous to each other.  Using previous
work of Habegger and Pila, it suffices to prove a height bound for such
t.  I will outline the proof of this height bound by viewing periods of
the elliptic curves as values of G-functions.  An innovation in this
work is that both complex and p-adic periods are required.  This is
joint work with Christopher Daw.