Enumerative geometry and knot invariants

A New Point of View in the Theory of Knot and Link Invariants

Recent progress in string theory has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to construct the new polynomials and we conjecture their general structure. This leads to new conjectures on the algebraic structure of the quantum-group polynomi...

Enumerative invariants of stongly semipositive real symplectic six-manifolds

Following the approach of Gromov and Witten [4, 22], we define invariants under deformation of stongly semipositive real symplectic six-manifolds. These invariants provide lower bounds in real enumerative geometry, namely for the number of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of points.

A short overview of the ”Topological recursion”

This is the long version of the ICM2014 proceedings. It consists in a short overview of the ”topological recursion”, a relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems. We recall how computing large size asymptotics in random matrices, has allowed to discover some fascinating and ubiquitous geometric invariants. Specializations of this method...

Degenerate Contributions, Enumerative Geometry, and Gromov-Witten Invariants

Enumerative invariants of stongly semipositive real symplectic manifolds

Following the approach of Gromov and Witten [3, 20], we define invariants under deformation of stongly semipositive real symplectic manifolds provided essentially that their real locus is Pin. These invariants provide lower bounds in real enumerative geometry, namely for the number of real rational J-holomorphic curves which realize a given homology class and pass through a given real configura...