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.