Related papers: A note on the pentagram map and tropical geometry
In the present paper, we construct an invariant of braids in the real projective plane which corresponds to an ``action'' of braids on certain graphs in $\R{}P^{2}$ with labels. This paper is a sequel of papers \cite{M},\cite{KM}. It…
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on…
In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study…
This article discusses a combinatorial extension of tropical intersection theory to spaces given by glueing quotients of partially open convex polyhedral cones by finitely many automorphisms. This extension is done in terms of linear…
We organize fundamental properties of quasi-Hamiltonian spaces on which a finite group acts, and we apply them to the theory of moduli spaces of flat connections on an oriented compact surface with boundary.
We construct a converging geometric iterated function system on the moduli space of ordered triangles, for which the involved functions have geometric meanings and contain a non-contraction map under the natural metric.
In this note, we investigate some topological properties of probabilistic modular spaces.
We introduce the modular class of a Poisson map. We look at several examples and we use the modular classes of Poisson maps to study the behavior of the modular class of a Poisson manifold under different kinds of reduction. We also discuss…
This contribution belongs to a combinatorial approach to hyperbolic geometry and it is aimed at possible applications to computer simulations. It is based on the splitting method which was introduced by the author and which is reminded in…
We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves…
We describe a framework to construct tropical moduli spaces of rational stable maps to a smooth tropical hypersurface or curve. These moduli spaces will be tropical cycles of the expected dimension, corresponding to virtual fundamental…
An overview is given of the construction of a differential polynomial ring of functions on the moduli space of Calabi-Yau threefolds. These rings coincide with the rings of quasi modular forms for geometries with duality groups for which…
We construct the moduli space for equivalence classes of n-pointed tropical curves of genus g, together with its compactification given by weighted tropical curves, and establish some of its basic topological properties. We compare it to…
We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The…
Given a lattice polygon, we study the moduli space of all tropical plane curves with that Newton polygon. We determine a formula for the dimension of this space in terms of combinatorial properties of that polygon. We prove that if this…
The deep diagonal map $T_k$ acts on planar polygons by connecting the $k$-th diagonals and intersecting them successively. The map $T_2$ is the pentagram map, and $T_k$ is a generalization. We study the action of $T_k$ on two subsets of the…
It often happens in mathematics that one and the same equation is known under different names in different areas of mathematics. The famous pentagon identity appears in low-dimensional topology in different ways. In this paper, we use the…
We propose to study the tropical geometry specifically arising from convergent Hahn series in multiple indeterminates. One application is a new view on stable intersections of tropical hypersurfaces. Another one is perturbations of rank one…
Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in…
In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my…