Related papers: Tropical Poincar\'e bundle, Fourier-Mukai transfor…
Braid Floer homology is an invariant of proper relative braid classes. Closed integral curves of 1-periodic Hamiltonian vector fields on the 2-disc may be regarded as braids. If the Braid Floer homology of associated proper relative braid…
We define a theory of real $(p,q)$-forms and currents on Berkovich spaces which is parallel to the theory of differential forms on complex spaces. It is based on Lagerberg's theory of superforms in tropical geometry and on the consideration…
We describe the Chow homology and cohomology of toric variety bundles, with no restrictions on the singularities of the fibre. We present the ordinary and equivariant homologies as modules over the cohomology of the base, identify the…
We fix the supports A=(A_1,...,A_k) of a list of tropical polynomials and define the tropical resultant TR(A) to be the set of choices of coefficients such that the tropical polynomials have a common solution. We prove that TR(A) is the…
The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure…
We prove a deviation inequality for noncommutative martingales by extending Oliveira's argument for random matrices. By integration we obtain a Burkholder type inequality with satisfactory constant. Using continuous time, we establish…
Tropicalization is a procedure that assigns polyhedral complexes to algebraic subvarieties of a torus. If one fixes a weighted polyhedral complex, one may study the set of all subvarieties of a toric variety that have that complex as their…
We show that the construction of Bayer, Bertram, Macri and Toda gives rise to a Bridgeland stability condition on a principally polarized abelian threefold with Picard rank one by establishing their conjectural generalized…
In this article we describe the equivariant and ordinary topological $K$-ring of a toric bundle with fiber a $T$-{\it cellular} toric variety. This generalizes the results in \cite{su} on $K$-theory of smooth projective toric bundles. We…
We establish a partial extension of the Poincar\'e duality theorem of Jell-Rau-Shaw to tropical hypersurfaces arising from non-primitive triangulations. We introduce a notion of level of primitivity for triangulations of lattice polytopes…
This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…
We study infinitesimal Torelli problems and infinitesimal variations of Hodge structure for families of curves arising in singular and extrinsically constrained geometric settings. Motivated by the Green--Voisin philosophy, we develop an…
Elements of the tropical vertex group, introduced by Kontsevich and Soibelman, are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are…
The notion of Fourier transform is among the more important tools in analysis, which has been generalized in abstract harmonic analysis to the level of abelian locally compact groups. The aim of this paper is to further generalize the…
We show that the Poincar\'e lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincar\'e lemma for transversal crystals of level…
Given any toric subvariety $Y$ of a smooth toric variety $X$ of codimension $k$, we construct a length $k$ resolution of $\mathcal O_Y$ by line bundles on $X$. Furthermore, these line bundles can all be chosen to be direct summands of the…
We associate to an analytic subvariety of a torus a tropical variety. In the first part, we generalize the results from tropical algebraic geometry to this non-archimedean analytic situation. The periodic case is applied to a totally…
We characterise those real analytic mappings between any pair of tori which carry absolutely convergent Fourier series to uniformly convergent Fourier series via composition. We do this with respect to rectangular summation. We also…
Observing the special structure of the system and using the Poincar{\'{e}}-Sobolev inequality, we establish Liouville type theorems for the 3D steady tropical climate model under certain conditions on $u$, $v$, $\nabla \theta$. Our results…
This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In two previous papers, we develop the basic theory of formal manifolds,…