Sunday, 18 January, 2015 to Friday, 23 January, 2015, Engelberg, Switzerland

The program will consist of two minicourses, given by Lothar Göttsche (On refined enumerative invariants), and by Lenny Ng (Conormal bundles, knot invariants, and topological strings) as well as research talks by other participants.

Scientific Committee of the conference series: Anton Alekseev (Geneva); Anna Beliakova (Zürich); Paul Biran (ETH); Jérémy Blanc (Basel); Tobias Ekholm (Uppsala); Ilia Itenberg (Paris); Conan Leung (Hong Kong); Grigory Mikhalkin (Geneva); Oleg VIro (Stony Brook).

The conference will take place in Treff Hotel Sonnwendhof, Gerschniweg 1, 6390 Engelberg.

Organizers: Ilia Itenberg (Paris), Grigory Mikhalkin (Geneva), Oleg Viro (Stony Brook).

Participants: Anton Alekseev (UNIGE) Jan 19-23; Anna Beliakova (UNIZH); Christan Blanchet (Paris, FR); Tobias Ekholm (Uppsala, SE) Jan 18-22; Vladimir Fock (Strasbourg, FR); Sergey Galkin (Moscow, RU); Lothar Göttsche (Trieste, IT); Ilia Itenberg (Paris, FR); Felix Janda (ETHZ) Jan 19-23; Andrés Jaramillo (Paris, FR); Johannes Josi (UNIGE); Nikita Kalinin (UNIGE); Natalia Kolokolnikova (UNIGE); Sergei Lanzat (UNIGE); Conan Leung (Hong Kong, HK); Grigory Mikhalkin (UNIGE); Lenny Ng (Durham, US); Alina Pavlikova (St. Petersburg, RU); Maria Podkopaeva (SwissMAP); Arthur Renaudineau (Paris, FR); Christoph Schiessl (ETHZ); Mikhail Shkolnikov (UNIGE); Andras Szenes (UNIGE); Oleg Viro (Stony Brook, US); Jean-Yves Welschinger (Lyon, FR) Jan 18-21.

Schedule:

Abstracts

Lenny NG Conormal bundles, knot invariants, and topological strings
In this minicourse, we will explore a method for studying topological knots through the symplectic/contact geometry of their conormal bundles. This leads to a knot invariant called knot contact homology, which is quite strong as an invariant and can be combinatorially described. Knot contact homology is still fairly mysterious after more than a dozen years of study, but we will discuss two recently discovered relations, one to representations of the knot group, and another (conjectured) to colored HOMFLY knot polynomials. To see this last relation, we will describe a surprising connection (due to joint work with Mina Aganagic, Tobias Ekholm, and Cumrun Vafa) between knot contact homology and string theory, involving mirror symmetry and topological strings on the resolved conifold.

Anton ALEKSEEV On Geometry and Topology of Moment Maps
In this talk, we first review the classical moment map theory including symplectic reduction, convexity properties and Duistermaat-Heckman localization. We then pass to more exotic moment map theories with values in solvable and compact Lie groups.

Christian BLANCHET Non semi-simple TQFTs
In common work with François Costantino, Nathan Geer and Bertrand Patureau, we have obtained new TQFTs based on an “unrolled” variant of quantum sl(2). We will present the relevant representation category, sketch the construction of the TQFT vector spaces, and describe the new Mapping Class Groups representations.

Tobias EKHOLM Cotangent bundles, knot contact homology, and physics
Knot contact homology is based on transporting phenomena in smooth topology (knots in a 3-manifold) to symplectic geometry (Lagrangian conromals in the cotangent bundle). This is a rather general scheme that can be applied also in other situations. We survey some recent results in that direction about cotangent bundles of high-dimensional homotopy spheres and about knot contact homology in other dimensions and codimensions. As will be clear, the 3-dimensional case has many special features. In particular we explain that it is related to topological string theory in a 3-dimensional Calabi-Yau manifold as well as to Chern-Simons gauge theory.

Jean-Yves WELSCHINGER Betti numbers of random nodal sets of elliptic pseudo-differential operators
I will explain how to bound from above the expected Betti numbers of the vanishing loci of random linear combinations of eigenvalues of any self adjoint positive elliptic pseudo-differential operator on some smooth closed manifold. This is a joint work with Damien Gayet.