Symplectic, real, and tropical aspects of enumerative geometry – ENUMGEOM

Classical enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, for instance the number of curves in a variety that intersect given subvarieties in a prescribed manner. This field was revolutionized through ideas of theoretical physics which have led to deep interactions between diverse fields of mathematics such as symplectic topology, tropical geometry, category theory, integrable systems, algebraic geometry, representation theory.

The modern developments began with the introduction of the Gromov-Kontsevich moduli spaces of stable pseudo-holomorphic maps into a symplectic manifold and Witten's insight that the intersection numbers on these moduli spaces can be interpreted as partition functions of quantum field theories. This idea led to significant amount of deep conjectures in mathematics, among which are Witten's conjecture and the mirror symmetry conjecture.

A generalization of these ideas lies within the realm of real enumerative geometry. Even classically the progress in this field as compared to complex enumerative geometry has been slow since solutions in real geometry can disappear and only a lower bound of their count is foreseeable. Such lower bound was first introduced by Welschinger for rational curves in lower dimensions. A full theory in arbitrary genus based on moduli spaces of symmetric curves was only recently developed by Georgieva and Zinger. The intersection theories on these moduli spaces can be interpreted as partition functions of certain extended quantum field theories and with this perspective all conjectures in the complex setting have their real counterpart and are open for investigation.

The main ingredient in the proofs of mirror symmetry rely on our ability to calculate the corresponding invariants and this is one of the main difficulties in the general case. A proposal made by Kontsevich to use tropical curves introduces a combinatorial flavor to the question. The prediction of applications of tropical geometry in enumerative geometry was confirmed by Mikhalkin who established an appropriate correspondence theorem and found a
combinatorial algorithm for computation of Gromov-Witten type invariants for toric surfaces.
The tropical approach has important applications in real enumerative geometry. In particular, Mikhalkin's correspondence theorem allows one to calculate or estimate Welschinger invariants (real analogs of genus zero Gromov-Witten invariants) in some situations. Mikhalkin's seminal work paved the way to many important applications of tropical geometry in (real) enumerative geometry, and since then
a number of generalizations of Mikhalkin's correspondence theorem were proved.

An important recent development in complex, real, and tropical enumerative geometries is related to the Göttsche-Shende conjecture. Block and Göttsche proposed to attribute certain polynomial weights to planar tropical curves arising in tropical calculations of Gromov-Witten type invariants. This suggestion was motivated by the study of refined Severi degrees that was initiated by Göttsche and Shende. The Göttsche-Shende conjecture conjecture highlights a special class of smooth real algebraic varieties: those for which the Euler characteristic of the real part coincides with the signature of the complexification and it is of a particular interest to determine which moduli spaces appearing in enumerative problems belong to this class.

The aim of this project on one side is to further develop the above mentioned topics and on another to foster interactions between specialists in these different domains, to aide the exchange of ideas and help their transmission from one point of view to another.