*Francis Bonahan (University of Southern California)*. **Character varieties and Kauffman brackets. **Friday May 6th. Huxley 130, 1.30-2.30pm.

Many areas of mathematics involve character varieties of surfaces, consisting of all homomorphisms from the fundamental group of the surface to a Lie Group G. When G=SL(2,C), the corresponding character variety can be quantised by Kauffman brackets, a knot-theoretic concept. I will discuss the classification problem for Kauffman brackets, and in particular some interesting results and constructions that arise in this context.

*King’s College Geometry Day. *Friday May 13th 11am-6.30pm. Speakers: Colding, Mazzeo, Neves, Pacard and Topping.

For further details see: http://www.mth.kcl.ac.uk/%7Etinaglia/GeometryDay/

*Melanie Rupflin (Warwick). ***A****bout uniqueness and non-uniqueness for the harmonic map heat flow. **Friday May 20th. Huxley 130, 1.30-2.30pm.

We will discuss the issue of uniqueness for weak solutions of theharmonic map flow in the critical dimension as well as for certain supercritical settings. On the one hand, we present a sharp uniqueness criterion for the flow from any closed Riemannian surface into arbitrary compact targets. On the other hand we will explain that for symmetric, selfsimilar solutions in supercritical dimensions the issue of uniqueness is determined by the properties of the so called equator maps. We obtain in particular that for appropriate initial data the number of (genuinely different) evolutions under the harmonic map heat flow can be arbitrarily large, even infinite.

*Jon Woolf (Liverpool). ***Stratified spaces and the Tangle Hypothesis. **Friday May 27th. Huxley 130, 1.30-2.30pm.

I will sketch a proposed geometric definition of `n- category with duals’, based on ideas of Kevin Morrison and Scott Walker. One motivation for this definition is that in this context (i.e. interpreted appropriately as a statement about n-categories of this flavour) Baez-Dolan’s Tangle Hypothesis reduces to a consequence of the Pontrjagin-Thom construction. This is joint work with Conor Smyth.

*Sean Keel (University of Texas at Austin).* **Mirror Symmetry for affine Calabi-Yau Manifolds.** Friday June 3rd. Huxley 130, 1.30-2.30pm.

I’ll explain a conjecture (and theorem in dim 2), joint with Hacking and Gross, which gives the mirror to an affine CY of any dimension as the Spec as an explicit ring: it comes with a basis parameterized by a natural generalisation of Thurston’s boundary sphere to Teichmuller space, and a multiplication rule in terms of counts of rational curves. I’ll explain some surprising applications in dimension two — for example that a del Pezzo of degree 3 has a canonical cubic equation, and a theory of theta functions for the complement to a plane cubic curve, and then our expectation that the Fock-Goncharov basis of universally positive Laurant polynomials on a cluster algebra is a special case of our conjecture (and thus exist on any affine Calabi-Yau whose boundary contains a zero stratum).

*John Etnyre (Georgia Tech). ***Contact geometry and smooth embeddings. ** Friday June 10th. Clore Lecture Theatre, 1.30-2.30pm.

Given an embedding of one manifold in another, the co-normal construction produces a Legendrian submanifold of a contact manifold, thus allowing one to use contact geometry to study isotopy classes of embeddings. While this construction has been around for some time — and used to great effect by Arnold in studying “wave fronts” — it has had little use in studying knot theory, arguably one of the best known isotopy problems. Recent advances in Legendrian Contact Homology have renewed interest in this construction and in particular its application to knot theory. In this talk I will outline the definition and computation of invariants of knots coming from the co-normal construction and Legendrian Contact Homology. If time permits I will also discuss recent work constructing invariants of transverse knots in the standard contact structure on Euclidean 3-space. This talk is based on joint work with Tobias Ekholm, Lenny Ng and Mike Sullivan.

*Chris Hull (Imperial). ***Generalized Kahler Geometry, Monge-Ampere Equations and Flows****.**** **Friday June 17th, Clore Lecture Theatre, 1.30-2.30pm.

Supersymmetry and string theory have led to interesting new complex geometries that have a torsion arising from the curvature of a gerbe. One class of these was later rediscovered as generalized Kahler geometry, another is sometimes called Kahler with torsion. This talk will focus on these geometries and some of their mathematical structure that has been revealed by physics approaches, including generalizations of the Monge-Ampere equation and of Kahler Ricci flow.

*Vivek Shende (Princeton). *

**Hilbert schemes of plane curve singularities and knot invariants.**Friday June 24th, Clore Lecture Theatre, 1.30-2.30pm.

To a singular point on a complex plane curve in a surface, one associates a link by intersecting the curve with a small sphere around the singularity. It has long been understood that there is a close relationship between the topology of the link and the geometry of the singularity. We will discuss here a conjectural relationship between the Khovanov-Rozansky invariants of the link and the cohomology of Hilbert schemes of points on the singular curve.