FIRST SWISSMAP GEOMETRY&TOPOLOGY CONFERENCE
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 1923;
Anna Beliakova (UNIZH);
Christan Blanchet (Paris, FR);
Tobias Ekholm (Uppsala, SE) Jan 1822;
Vladimir Fock (Strasbourg, FR);
Sergey Galkin (Moscow, RU);
Lothar Göttsche (Trieste, IT);
Ilia Itenberg (Paris, FR);
Felix Janda (ETHZ) Jan 1923;
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);
JeanYves Welschinger (Lyon, FR) Jan 1821.
Schedule:
Monday  Tuesday  Wednesday  Thursday  Friday  

09: 30 – 10: 30  Minicourse Lenny NG  Minicourse Lenny NG  Minicourse Lenny NG  Minicourse Lenny NG  Talk Andras SZENES 
11: 00 – 12: 00  Minicourse Lothar GÖTTSCHE  Minicourse Lothar GÖTTSCHE  Minicourse Lothar GÖTTSCHE  Minicourse Lothar GÖTTSCHE  Talk Vladimir FOCK 
16: 30 – 17: 30  Talk Conan LEUNG  Talk Sergey GALKIN  Talk Christian BLANCHET  Departure 

17: 45 – 18: 45  Talk Tobias EKHOLM  Talk JeanYves WELSCHINGER  Talk Anna BELIAKOVA  Talk Anton ALEKSEEV 
Minicourses 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.
Lothar GÖTTSCHE Refined curve counting and enumerative geometry
(1) We first introduce the Severi degrees, which count nodal curves in linear systems on surfaces.
Then, using relative Hilbert schemes of points, we introduce refined invariants. These are are polynomials in a variable y, which specialize
for sufficiently ample linear systems at y=1 to the Severi degrees and are given by nice generating functions.
(2) We review Welschinger invariants, which count real curves. We review plane tropical curves and the tropical definition of Severi degrees and Welschinger invariants.
Then using tropical geometry we introduce refined Severi degrees, polynomials in a variable y that interpolate between Severi degrees
and Welschinger invariants, which are closely related to the refined invariants from before.
(3) We introduce Floor diagrams, which under certain assumptions encode the relevant combinatorial information about plane tropical curves, and can be used
to compute refined Severi degrees. We relate these to Feynman diagrams for an action of a Heisenberg algebra, and use this to show that
the refined Severi degrees can be computed in terms of this Heisenberg algebra action.
(4) Time permitting we will introduce refined Broccoli invariants and discuss some of their properties.
The Welschinger invariants studied above were totally real, i.e. counting real curves through a number of real points.
The Broccoli invariants are one way to count real curves through a number of real points and a number of pairs of complex conjugate points, using tropical geometry.
We give a refined version of these invariants, again replacing numbers by polynomials, which specialize to the Broccoli invariants at y=1.
The description and properties are considerably nicer than for the nonrefined version.
Talks abstracts
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 DuistermaatHeckman localization. We then pass to more exotic moment map theories with values in solvable and compact Lie groups.
Anna BELIAKOVA Trace of the categorified quantum groups
In this talk I will give a gentle introduction to the categorified quantum groups and show that the trace (or 0th Hochschild homology) of the KhovanovLauda 2category is isomorphic to the current algebra. Then I'll discuss some applications of this fact to link homology theories.
(Coauthors: Zaur Guliyev, Kazuo Habiro, Aaron Lauda, and Ben Webster.)
Christian BLANCHET Non semisimple 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 3manifold) 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 highdimensional homotopy spheres and about knot contact homology in other dimensions and codimensions. As will be clear, the 3dimensional case has many special features. In particular we explain that it is related to topological string theory in a 3dimensional CalabiYau manifold as well as to ChernSimons gauge theory.
Vladimir FOCK Fay's trisecant formula and discrete integrable systems
Fay's trisecant identity is a quadratic relation satisfied by theta
functions on Jacobians of curves. We will present these relation in
different forms and show that they play a key role in solution of
discrete integrable system. An application to abelianization of local
systems on a Riemann surface will be also discussed.
Conan LEUNG Witten deformations and scattering in Amodel
In this talk, I will describe the differential graded algebra structure on differential forms under Witten deformations. An application on scattering in Mirror Symmetry will be explained. This research is supported by research grant from HK Government.
Andras SZENES Ktheoretic Thom polynomials
Thom polynomials are obstructions to avoiding singularities of maps between manifolds.
They are expressed as polynomials in the Chern classes of the relative tangent bundle.
I will present joint work with R. Rimanyi, where we have calculated the Ktheoretical versions of these invariants for certain simple contact singularity loci.
JeanYves WELSCHINGER Betti numbers of random nodal sets of elliptic pseudodifferential 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 pseudodifferential operator
on some smooth closed manifold. This is a joint work with Damien Gayet.