Related papers: The extended tropical vertex group

Bound state excitations of the spin 1/2-XYZ model are considered inside the Bethe Ansatz framework by exploiting the equivalent Non-Linear Integral Equations. Of course, these bound states go to the sine-Gordon breathers in the suitable…

Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will…

In the previous article we introduced the new concept of mixed representations of quivers and described the generators of their algebras of invariants. In this article we describe the defining relations of these algebras. Some applications…

In the 1990s, Kita--Yoshida and Cho--Matsumoto introduced intersection forms on the twisted (co)homologies of hyperplane arrangement complements. We give a closed combinatorial formula for these intersection pairings. We show that these…

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\mathbb{R}^{n}$ with linear boundary conditions on (a relatively open part of) a…

We introduce the polygonalisation complex of a surface, a cube complex whose vertices correspond to polygonalisations. This is a geometric model for the mapping class group and it is motivated by works of Harer, Mosher and Penner. Using…

We give a topological interpretation of the core group invariant of a surface embedded in S^4. We show that the group is isomorphic to the free product of the fundamental group of the double branch cover of S^4 with the surface as a…

We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of…

This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and…

We study monodromies of plane curve singularities and pseudo-periodic homeomorphisms of oriented surfaces with boundary, following an original idea of the first author: t\^ete-\`a-t\^ete graphs and twists. We completely characterize mapping…

We present applications of tropical geometry to some integrable piecewise-linear maps, based on the lecture given by one of the authors (R. I.) at the workshop "Tropical Geometry and Integrable Systems" (University of Glasgow, July 2011),…

We define the class of multivariate group entropies as a novel set of information - theoretical measures, which extends significantly the family of group entropies. We propose new examples related to the "super-exponential" universality…

Recent breakthroughs in the study of scattering amplitudes have uncovered profound and unexpected connections with combinatorial geometry. These connections range from classical structures -- such as polytopes, matroids, and Grassmannians…

Quantum graphs can be extended to scattering systems when they are connected by leads to infinity. It is shown that for certain extensions, the scattering matrices of isospectral graphs are conjugate to each other and their poles…

We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of…

The Doran-Harder-Thompson "gluing/splitting" conjecture unifies mirror symmetry conjectures for Calabi-Yau and Fano varieties, relating fibration structures on Calabi-Yau varieties to the existence of certain types of degenerations on their…

We study the spectral properties of the Laplace operator associated to a hyperbolic surface in the presence of a unitary representation of the fundamental group. Following the approach by Guillop\'e and Zworski, we establish a factorization…

We use techniques from Gromov-Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane…

In this article we describe relations of the topology of closed 1-forms to the group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CW- complexes and show that many…

We develop the Lie theory of Lie-admissible algebras whose product is enriched with higher operations modeled on directed graphs with a view to apply it to the deformation theories controlled by this kind of Lie algebras. We produce…