Related papers: The tropical Poincar\'e-Hopf theorem
We use the equivariant cohomology ring of the permutohedral variety to study matroids and their invariants. Investigating the pushforward of matroid Chern classes defined by A. Berget, C. Eur, H. Spink and D. Tseng to the product space…
In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliations in dimension two and their invariant curves. It appears a characterization of second type foliations and…
We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…
In this note, we prove that intrinsic characteristic classes of Lie algebroids - which in degree one recover the modular class - behave functorially with respect to arbitrary transverse maps, and in particular are weak Morita invariants. In…
We give a definition of tropical orbit spaces and their morphisms. We show that, under certain conditions, the weighted number of preimages of a point in the target of such a morphism does not depend on the choice of this point. We equip…
A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin-Kasteleyn random…
We present a higher-dimensional version of the Poincar\'e-Birkhoff theorem which applies to Poincar\'e time maps of Hamiltonian systems. The maps under consideration are neither required to be close to the identity nor to have a monotone…
We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the…
We describe the tropical curves in toric varieties and define the tropical Gromov-Witten invariants. We introduce amplitudes for the higher topological quantum mechanics (HTQM) on special trees and show that the amplitudes are equal to the…
We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…
In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of L consisting of all vectors v that…
The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of products of two simplices. Applications…
The Turing-Hopf type spatiotemporal patterns in a diffusive Holling-Tanner model with discrete time delay is considered. A global Turing bifurcation theorem for $\tau=0$ and a local Turing bifurcation theorem for $\tau>0$ are given by the…
We construct an obstruction theory for relative Hilbert schemes in the sense of Behrend-Fantechi and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface…
Given an undirected graph, the non-empty subgraph polytope is the convex hull of the characteristic vectors of pairs (F, S) where S is a non-empty subset of nodes and F is a subset of the edges with both endnodes in S. We obtain a strong…
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 apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.
Given two tropical polynomials $f, g$ on $\mathbb{R}^n$, we provide a characterization for the existence of a factorization $f= h \odot g$ and the construction of $h$. As a ramification of this result we obtain a parallel result for the…
The phase space of an integrable Hamiltonian system is foliated by invariant tori. For an arbitrary Hamiltonian H such a foliation may not exist, but we can artificially construct one through a parameterised family of surfaces, with the…
The main mathematical focus of this paper is a class of parametrised polynomial systems that we refer to as being tropically transverse. We show how their generic number of solutions can be expressed as the mixed volume of a modified…