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The shellability of the boundary complex of an unbounded polyhedron is investigated. To this end, it is necessary to pass to a suitable compactification, e.g., by one point. This observation can be exploited to prove that any tropical…
We give an explicit finite-dimensional model for the derived moduli stack of flat connections on $\mathbb{C}^k$ with logarithmic singularities along a weighted homogeneous Saito free divisor. We investigate in detail the case of plane…
We study the concentration of the norm of a random vector $Y$ uniformly sampled in the centered zero cell of two types of stationary and isotropic random mosaics in $\mathbb{R}^n$ for large dimensions $n$. For a stationary and isotropic…
We use the tropical geometry approach to compute absolute and relative Gromov-Witten invariants of complex surfaces which are $\CC P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be…
While the construction of symplectic integrators for Hamiltonian dynamics is well understood, an analogous general theory for Poisson integrators is still lacking. The main challenge lies in overcoming the singular and non-linear geometric…
For the directed landscape, the putative universal space-time scaling limit object in the (1+1) dimensional Kardar-Parisi-Zhang (KPZ) universality class, consider the geodesic tree -- the tree formed by the coalescing semi-infinite…
We investigate the tree gonality of a genus-$g$ metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this…
To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson…
Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties,…
Our start point is a 3D piecewise smooth vector field defined in two zones and presenting a shared fold curve for the two smooth vector fields considered. Moreover, these smooth vector fields are symmetric relative to the fold curve, giving…
We introduce and prove some basic results about a combinatorial model which produces embedded polygons in the plane. The model is closely related to outer billiards on kites, and also is related to corner percolation, to Hooper's Truchet…
We prove that Viro's patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic $M$-curves realize many isotopy types. The strategy we…
A method is presented to include irregular domain boundaries in a geometric multigrid solver. Dirichlet boundary conditions can be imposed on an irregular boundary defined by a level set function. Our implementation employs quadtree/octree…
Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among…
We describe families of plane-filling curves on any edge-to-edge tiling of the plane with regular polygons and finitely many classes of edges. It is shown how to partition the minimal number of edge classes from the group G of symmetries of…
A closed plane meander of order $n$ is a closed self-avoiding curve intersecting an infinite line $2n$ times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm,…
We study discrete complex analysis and potential theory on a large family of planar graphs, the so-called isoradial ones. Along with discrete analogues of several classical results, we prove uniform convergence of discrete harmonic…
We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem…
In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when…
The site percolation on the triangular lattice stands out as one of the few exactly solved statistical systems. By initially configuring critical percolation clusters of this model and randomly reassigning the color of each percolation…