A version of the mixed volume taking values in Z/2Z, Alexandr Esterov I will introduce a version of the mixed volume of lattice polytopes taking values in Z/2Z (very differently from the usual mixed volume modulo 2). It comes from arithmetic geometry, is constructed using tropical techniques, and appears in real algebraic geometry whenever the usual mixed volume appears over the complex numbers. For instance, it governs the parity of roots with the negative first coordinate for generic real sparse systems of equations (while the total number of roots is governed by the usual mixed volume) and defines the signs of the leading coefficients of sparse resultants.

Enumerative geometry and the quantum torus. Michael Polyak A number of classical enumerative problems can be easily solved by methods of tropical geometry. Tropical curves are planar metric graphs with certain requirements of balancing, rationality of slopes and integrality. We consider a generalization of tropical curves, removing requirements of rationality of slopes and integrality. One of the classical enumerative problems is that of counting of (either complex or real) rational curves through a collection of points in a toric variety. We explain this counting procedure as a construction of some simple top-dimensional cycles on certain moduli of (pseudo)tropical curves. Cycles on these moduli turn out to be closely related to maps of spheres and associative and Lie algebras. In particular, counting of both complex and real curves is related to the quantum torus algebra. More complicated counting invariants (the so-called Gromov-Witten descendants) are similarly related to the super-Lie structure on the quantum torus.

Viro's writhe and Ulrich sheaves. Mario Kummer Viro's writhe is an invariant of rigid isotopy for real algebraic curves in projective three-space. We show that it agrees with the topological degree of a natural map from a certain projective bundle over the second symmetric product of the curve to projective three-space. This map admits a relative Ulrich line bundle. The space of global sections of this line bundle carries a symmetric bilinear form in a natural way. The signature of this bilinear form again agrees with Viro's writhe. This is a joint work in progress with Daniele Agostini.

Tropical hypersurfaces with mild singularities in sandpile models. Nikita Kalinin A small perturbation of the maximal stable state in sandpile models on big subsets of the standard grid exhibits tropical curves. A proof of the planar case uses certain dynamics on tropical curves, while curves appearing in this dynamic are smooth or nodal. A direct generalisation of the problem for higher dimensions leads to tropical hypersurfaces which have only mild singularities. This means that the dual lattice polyhedron for each face (of any dimension) of such a hypersurface contains no lattice points except vertices. I explain the motivation from sandpiles and give an overview of the aforementioned dynamics.

800 conics on a smooth quartic surface. Alex Degtyarev I will discuss the recent progress concerning the problem on the maximal number of smooth rational curves of a fixed degree (most notably, conics) on polarized K3-surfaces. The most intriguing open question concerns conics on quartics. I will explain several constructions of the current champion, one with 800 conics, and try to substantiate the conjecture that 800 is indeed the maximum.

A tropical version of jacobian conjecture. Dima Grigoriev We prove that, for a tropical rational map if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.

Real loci of generic complex varieties. Oleg Viro The set of real points of a generic non-real complex algebraic variety is a real algebraic variety of special type. It is co-oriented in the real part of the ambient variety, realize there an integer cohomology class and has integer intersection and linking numbers with oriented submanifolds (even if the real part of the ambient space is not orientable). If the codimension of the original complex variety is one, the intersection is a base for a real pencil of hypersurfaces.