Related papers: Geometric Monodromy around the Tropical Limit
We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the…
We consider the following question: given $A \in SL(2,R)$, which potentials $q$ for the second order Sturm-Liouville problem have $A$ as its Floquet multiplier? More precisely, define the monodromy map $\mu$ taking a potential $q \in…
We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data,…
Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. The purpose of this paper is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the…
In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the…
In this paper we study certain families of motives, which arise as direct summands of the cohomology of the Dwork family. We computationally find examples of interesting families with the following three properties. Firstly, their geometric…
The purpose of this survey is to summarize known results about tropical hypersurfaces and the Cayley Trick from polyhedral geometry. This allows for a systematic study of arrangements of tropical hypersurfaces and, in particular,…
In this paper, we present a numerical method for rigorously finding the monodromy of linear differential equations. Beginning at a base point where certain particular solutions are explicitly given by series expansions, we first compute the…
We give a survey on some aspects of the topological investigation of isolated singularities of complex hypersurfaces by means of Picard-Lefschetz theory. We focus on the concept of distinguished bases of vanishing cycles and the concept of…
New hyperfields, that is fields in which addition is multivalued, are introduced and studied. In a separate paper these hyperfields are shown to provide a base for the tropical geometry. The main hyperfields considered here are classical…
We compute the monodromy groups of real Enriques surfaces of hyperbolic type. The principal tools are the deformation classification of such surfaces and a modified version of Donaldson's trick, relating real Enriques surfaces and real…
Curve singularities are classical objects of study in algebraic geometry. The key player in their combinatorial structure is the {\it value semigroup}, or its compactification, the {\it value semiring}. One natural problem is to explicitly…
Using our previous work we give a tropical formula for disk potentials for Lagrangian tori in almost toric four-manifolds, that is, fibrations by Lagrangian tori with only toric and focus-focus singularities, generalizing results of…
We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our…
We determine necessary and sufficient conditions for real algebraic curves near the non-singular tropical limit to be hyperbolic with respect to a point, thus generalising Speyer's classification of stable curves near the tropical limit. In…
Let $G$ be a connected reductive algebraic group. We develop a Gr\"obner theory for multiplicity-free $G$-algebras, as well as a tropical geometry for subschemes in a spherical homogeneous space $G/H$. We define the notion of a spherical…
Given a tropical fan curve $\Sigma$ and a family of algebraic curves $X \rightarrow \mathbb{A}^k$ we define the realization locus $\mathop{Real}_\Sigma \subseteq \mathbb{A}^k$ as the set of fibers $X_a$ whose tropicalization is $\Sigma$. We…
This paper studies the space of monodromy data of second order $q$-difference equations through the framework of WKB analysis. We compute the connection matrices associated to the Stokes phenomenon of WKB wavefunctions and develop a general…
The purpose of this thesis is to introduce two new kinds of hyperplane arrangements, inspired by the graphic arrangements and $q$-deformations of graphic arrangements. In this thesis, the author extends the definition of $q$-deformation to…
We generalize the notion of tight geodesics in the curve complex to tight trees. We then use tight trees to construct model geometries for certain surface bundles over graphs. This extends some aspects of the combinatorial model for doubly…