From the point of view of mathematics, the aim of the workshop was to give a contribution to the clarification of the relations between the theory of moduli spaces, representation theory and integrable systems. We believe that a more fruitful interaction between the different communities of people working in these fields will strengthen them individually and collectively. These different mathematical topics have important applications in theoretical physics, notably gauge theory and conformal field theory. Thus, the goal of the workshop on the physical side was to study the implications of the mathematical results on those areas of theoretical physics.

 

A survey about moduli spaces of sheaves was given by Justin Sawon (Mukai moduli spaces). A detailed description of moduli spaces of one-dimensional pure sheaves was provided by Mario Maican (Homology of the moduli spaces of plane sheaves of multiplicity 4 and 5). The relations between moduli spaces and integrable systems were addressed by Justin Sawon (Holomorphic Lagrangian fibrations) and Peter Dalakov (Donagi-Markman cubic for the G-generalised Hitchin system). On the other hand, Giulia Saccà (Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties) described a characterization of the singularities of some moduli spaces of sheaves on K3 surfaces in terms of quiver representations. 

 

Hilbert schemes of points are example of some moduli spaces of rank one sheaves. Weiping Li (Cohomological crepant resolution conjecture for the Hilbert scheme of points on surfaces) described a proof of the so-called Cohomological crepant resolution conjecture for these moduli spaces and Eugene Gorsky (Refined knot invariants and Hilbert schemes) showed the relations between them and the refined knot invariants.

 

Moduli spaces of sheaves are strictly related to the enumerative invariants for gauge theories in 4-dimensions and 6-dimensions. The recent work of Nekrasov and Okounkov about a mathematical derivation of the so-called geometric engineering was explained by Duiliu-Emanuel Diaconescu (Gauge theory and enumerative geometry). A related talk was given by Alberto Cazzaniga (Higher rank phenomena in Donaldson-Thomas theory). Computations of instanton partition functions by using moduli spaces of sheaves were described by Richard J. Szabo (Supersymmetric gauge theories on ALE spaces) and  Duiliu-Emanuel Diaconescu (Character varieties and Donaldson-Thomas invariants). A related talk for gauge theories on toric Sasaki-Einstein manifolds was given by Jian Qiu (Super Yang-Mills on toric Sasaki-Einstein manifolds and the factorization of partition functions).

 

The Alday-Gaiotto-Tachikawa conjecture was one of the hot topics of the workshop. A survey about the relations between representation theory, moduli spaces of monopoles and instantons was given by Alexander Braverman (Intersection cohomology of instanton and monopole moduli spaces and representation theory I and II). A new result related to moduli spaces of monopoles was described by Michael Finkelberg (Twisted Whittaker sheaves on zastava, after Gaitsgory, Gaiotto and Witten); on the other hand, Erik Carlsson (AGT and the Segal-Sugawara construction) described a construction of a vertex operator associated with moduli spaces of SU(2) instantons. A more physical perspective on AGT and its applications was given by Alessandro Tanzini (Quantum cohomology and quantum hydrodynamics from supersymmetric gauge theories), Mikhail Bershtein (Bilinear equations on Painlevé-functions from CFT) and Rubik Poghossian (AGT in use: 2d Renormalization group flows in next to leading order). Related talks, with an emphasis on the conformal field theory point of view, were given by Alexander Belavin (Correlation functions in minimal Liouville gravity from Douglas string equation) and Nicolai Reshetikhin (On semiclassical asymptotics of 6j-symbols).

 

It is the opinion of the organizers that the workshop fulfilled its scopes in a fully satisfactory way. The talks were interesting, and the structure of the workshop  left space for personal discussion. Hopefully the  workshop has allowed many of the participants to get in touch with the most recent advances in the field.

 

A special thank is due to the Lorentz Center and its personnel, for hosting and financing this event,  and for their splendid work.