10:00 to 10:50 
Tudor Dimofte (University of Edinburgh, UK) 
3d gauge theories: vortices and vertex algebras (Lecture 1) 3d N=4 gauge theories admit two topological twists, often called A and B, that are expected to lead mathematically to fully extended 3d topological quantum field theories (TQFTs). I will review some aspects of these putative TQFTs, some of their known and expected connections to representation theory, and (especially) connections to moduli spaces of vortices and their cohomology. I will then present some recent work on accessing these 3d TQFTs via boundary vertex algebras  much as was done for ChernSimons TQFT using the WZW model in the '80s and '90s. In particular, I will discuss using boundary vertex algebras to define braided tensor categories of line operators and to prove their equivalence under 3d mirror symmetry. (These developments in joint work with Andrew Ballin, Thomas Creutzig and Wenjun Niu.)



11:30 to 12:20 
Jeongseok Oh (Imperial College, UK) 
Quasimaps, their wallcrossings and mirror symmetry (Lecture 1) Inspired by the role of loop spaces in the proof of the mirror theorem by his advisor (Alexander Givental), Bumsig Kim and his collaborators developed the theory of spaces of quasimaps. Interestingly, the change of their generating functions of invariants (defined by integration over these spaces) according to their stability conditions, known as a wallcrossing formula, has exactly the same form as mirror symmetry for certain potential functions predicted by physics. So it gives a geometric interpretation of mirror symmetry. A huge computational advantage is that the moduli spaces of quasimaps have less boundary components than those of stable maps, so that they become easier to work with. Together with the wallcrossing formula, we can compute genus g GromovWitten invariants for quintics, up to a few low degree invariants which remain unknown.
In this tribute talk to Bumsig Kim, I shall review his achievements in mirror symmetry through quasimaps.



14:30 to 15:20 
Andrea Ferrari (Durham University, UK) 
Vortices, other saddles, and wallcrossing The enumerative geometry of vortex moduli spaces plays a key role in the study of supersymmetric gauge theories in three dimensions. For instance, in the presence of eight supercharges, field configurations contributing to the path integral can often be localised to moduli spaces of solutions of generalised vortex equations. In the presence of less supercharges, however, the path integral can receive additional contributions, which interplay in a remarkable way with those originating from vortex moduli spaces. In this talk, I will introduce some of these phenomena in simple, abelian examples, and comment on the expected relation to and potential implications for topics in mathematics such as wallcrossing.



16:30 to 17:20 
Mathew Bullimore (Durham University, UK) 
Vortices and generalised symmetry There is much recent interest in generalised or categorical symmetries that go beyond the paradigm of groups and into the realm of higher groups and fusion categories. I will discuss aspects of such symmetries in the context of threedimensional abelian GLSMs and moduli spaces of vortices. I will then speculate on generalised notions of equivariance in quasimap Ktheory of toric stacks.



17:30 to 18:20 
Nick Manton (University of Cambridge, UK) 
Statistical mechanics of vortices Classical or quantized statistical mechanics of criticallycoupled Abelian Higgs vortices can be modelled by free dynamics on the Nvortex moduli space, with N large. Vortex interactions are captured by the nontrivial moduli space geometry. To avoid boundary effects and satisfy Bradlow’s constraint, the vortices are defined on a compact surface of large area A, with A/N > 4π. The classical partition function depends only on the moduli space volume, and the first quantum correction at high temperature T depends on the integrated scalar curvature. Using these known geometrical quantities, we deduce the highT equation of state of the vortex gas. When A/N is only slightly larger than 4π, the moduli space simplifies to complex projective space with its Fubini–Study geometry. Here the quantum partition function and equation of state can be calculated for any temperature. (NSM thanks S. Nasir, J. Baptista, J.M. Speight and S. Wang for their collaboration and contributions.)


