John Calabrese (University of Oxford) - Curve counting invariants and flops


This talk will be a simplified and floppy version of a talk by Bridgeland. For a smooth and projective CY3 I'll explain (very roughly) how one can extract identities between Donaldson-Thomas and Pandharipande-Thomas invariants out of relations between various subcategories of sheaves. This is done using the machinery of motivic Hall algebras of Joyce and Song. Time permitting I'll explain an approach to comparing Donaldson-Thomas invariants of two CY3's related by a flop.



Stephen Casey (University of Cambridge) - Projective, optical and path geometries


In this talk, I will introduce a problem of Roger Liouville; to determine when a given family of curves on some open set in R^n can arise as the set of unparametrised geodesics of some metric. I'll show how to derive some necessary conditions for metrisability and apply these results to some examples in three dimensions. I'll also look at an application of this work to relativity and discuss the notion of optical equivalence of two Riemannian metrics in the context of spacetimes which are static in more than one way. In particular, I will highlight the link between optical equivalence and projective equivalence.



Julian Gibbons (Imperial College London) - Unknotting number one


I will be talking about unknotting number of knots, and some of the methods used to compute it. For a simple invariant, it is surprisingly difficult to calculate, and (surprisingly!) relates to the theory of 3- and 4-manifolds. I will discuss this connection, in generality, but giving some examples along the way.



Roberto Rubio (University of Oxford) - From the Cayley transform to Higgs bundles


A guided and geometrical tour of the Cayley transform, from its origins in the XIX century to the statement of a recent Cayley correspondence for Higgs bundles. No previous experience required.



Robert Tang (University of Warwick) - The curve complex and covers


The curve complex associated to a surface is a simplicial complex which captures intersection information about simple closed curves on the surface in question. It has had applications to various topics such as mapping class groups, Teichmuller space and the study of hyperbolic 3-manifolds. After introducing some basic notions and ideas, I will talk about the coarse geometry of maps between curve complexes induced by covers between their respective surfaces.