Abstracts for Conference on "Tropical aspects in Geometry and Topology"

In a joint work with V.Kharlamov, we gave an estimate for the numbers of lines on real algebraic n-dimensional hypersurfaces of degree (2n-1) using evaluation of a certain signed count of these lines. The signs involved can be viewed as a generalized version of the Welschinger indices.  Our approach allows also a generalization to obtain a similar estimate for the numbers of projective subspaces on real algebraic  hypersurfaces (of certain dimensions and degrees).

Surprisingly, in a quite a number of real enumerative problems the number of real solutions
happens to satisfy high lower bounds. For the moment, such a phenomenon is rather deep
studied in the case of interpolation of real points on a real rational surface by real rational curves. In this talk,
based on a, joint with Rares Rasdeaconu, work in progress, I intend to show that a similar phenomenon should
hold in the case of counting rational curves on K3 surfaces. As in other similar situations, our lower bounds

The study of singularities and singular varieties has been an important issue in tropical geometry from the very beginning, for example, the Mikhalkin's correspondence theorem deals with nodal plane tropical curves. In the talk we address the geometry of higher-dimensional singular tropical hypersurfaces and show that even in the case of one singularity one encounters
some unexpectedly interesting geometric phenomena involving combinatorics of lattice polytopes and metric geometry. Joint work with Hannah and Thomas Markwig.
 

A sufficiently generic bivariate Laurent polynomial with given Newton polygon Delta defines an algebraic curve C, many of whose numerical invariants are encoded in the combinatorics of Delta. These include the genus (classical), the gonality, the Clifford index and the Clifford dimension (an enhancement of recent results by Kawaguchi), the scrollar invariants and, for sufficiently nice instances of Delta, certain secondary scrollar invariants that were introduced by Schreyer (new observation).