Mathematical Institute, Oxford University,
July 9-11, 2011.
In Lie theory, many of the natural algebraic varieties one studies have a symplectic or Poisson
structure – the Springer resolution, the Kostant-Kirillov form on coadjoint orbits, are classical
examples, while recent work on quiver varieties has yielded a host of new examples. In particular,
one finds many examples of symplectic resolutions – roughly this is a singular variety which has a
Poisson structure usually with finitely many symplectic leaves together with a smooth symplectic
resolution.
In algebraic geometry, the study of Gromov-Witten invariants is now a rich but still perplexing
topic, guided by certain organising principles whose motivation largely comes from physics and
in particular the predictions of mirror symmetry. Recently, the work of a number of authors suggest
that in many cases where the varieties are symplectic resolutions then these structures should
be closely related to phenomenon arising in representation theory of certain noncommutative algebras
such as Hecke algebras and symplectic reflection algebras.
Evidence for such a connection is only recently emerging: we mention two examples: firstly,
pioneering work of Braverman-Maulik-Okounkov shows how the equivariant quantum cohomology
of the cotangent bundle of the flag variety (the Springer resolution) can be identifed
with the affine Cherednik-Matsuo connection. This allows them to relate predictions of mirror
symmetry to known representation-theoretic facts. Secondly, work of Pandharipande, Okounkov,
Bezrukavnikov and Etingof has revealed relations between the (small) equivariant quantum cohomology
of the Hilbert scheme of points in the plane and the representation theory of the rational
Cherednik algebras associated to the symmetric groups.
The scientific goal of the conference will be to bring together researchers in the UK, both
students and experts, in the different subjects – quantum cohomology, symplectic reflection algebras,
and the geometry of symplectic varieties – along with international experts to explore the
potential for interactions between their fields suggested by these recent developments.