Related papers: Birational sequences and the tropical Grassmannian
We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the…
Let Gr be the affine Grassmannian for a connected complex reductive group G. Let C_G be the complex vector space spanned by (equivalence classes of) Mirkovic-Vilonen cycles in Gr. The Beilinson-Drinfeld Grassmannian can be used to define a…
Working over various graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over…
We produce new combinatorial methods for approaching the tropical maximal rank conjecture, including inductive procedures for deducing new cases of the conjecture on graphs of increasing genus from any given case. Using explicit…
Relying on the theory of agrarian invariants introduced in previous work, we solve a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are…
Let X be a smooth complex projective variety of dimension n equipped with a very ample Hermitian line bundle L. In the first part of the paper, we show that if there exists a toric degeneration of X satisfying some natural hypotheses (which…
In this paper, we continue our study of blade arrangements and the positroidal subdivisions which are induced by them on $\Delta_{k,n}$. A blade is a tropical hypersurface which is generated by a system of $n$ affine simple roots of type…
Hilbertian kernel methods and their positive semidefinite kernels have been extensively used in various fields of applied mathematics and machine learning, owing to their several equivalent characterizations. We here unveil an analogy with…
Grassmann tensors arise from classical problems of scene reconstruction in computer vision. Trifocal Grassmann tensors, related to three projections from a projective space of dimension k onto view-spaces of varying dimensions are studied…
We propose an iterated version of the Gilbert model, which results in a sequence of random mosaics of the plane. We prove that under appropriate scaling, this sequence of mosaics converges to that obtained by a classical Poisson line…
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the…
We show that, after the change of variables $q=e^{iu}$, refined floor diagrams for $\mathbb{P}^2$ and Hirzebruch surfaces compute generating series of higher genus relative Gromov-Witten invariants with insertion of a lambda class. The…
We study convolution algebras associated with Heckman-Opdam polynomials. For root systems of type BC we derive three continuous classes of positive convolution algebras (hypergroups) by interpolating the double coset convolution structures…
We prove Pieri formulas for the multiplication with special Schubert classes in the K-theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian…
This is a survey article on Morse theory based on lectures to graduate students and advanced undergraduates. After a brief review of standard material, mostly without proofs, the Morse theory of complex Grassmannian manifolds is worked out…
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 construct a combinatorial model that is described by the cube recurrence, a nonlinear recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in $\mathbb{Z}^3$. In the process, we prove…
Local properties of families of algebraic subsets $W_g$ in Birkhoff strata $\Sigma_{2g}$ of Gr$^{(2)}$ containing hyperelliptic curves of genus $g$ are studied. It is shown that the tangent spaces $T_g$ for $W_g$ are isomorphic to linear…
In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams. We give here detailed proofs in the tropical geometry…
Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Gr\"obner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$ that assembles the Gr\"obner…