arXiv subjects

Yelena Mandelshtam

Publications and source records attributed to Yelena Mandelshtam.

12 recordsLinked to original sources

arXiv2026

Nonnegative conorms, regular matroids, and the tropical Schottky problem

We prove that a positive-definite quadratic form has nonnegative conorms if and only if it is matroidal, resolving a conjecture of Dutour Sikirić and Kummer. We show that the positive conorm support determines a regular matroid, and the positive conorms are the coefficients of a unimodular integral decomposition of the form. As a result, we show that a positive-definite form lies in the tropical Schottky locus if and only if its conorms are nonnegative and its positive conorm support matroid is cographic. This answers a question of Chua, Kummer, and Sturmfels and provides algorithms for the tropical Schottky decision and recovery problems in every genus.

math.AG↗

arXiv2025

Tropical KP Theory on Banana Curves

The Kadomtsev-Petviashvili (KP) equation is the cornerstone of integrable systems, whose solutions reflect deep connections in algebraic geometry. Banana curves are reducible rational curves obtained as a degeneration of hyperelliptic curves. In this work, we relate the family of KP multi-solitons arising from banana curves together with non-special divisors of fixed degree to the combinatorics of the tropical theta divisor of the curve. We describe the Voronoi and Delaunay polytopes and show that the latter are combinatorially equivalent to uniform matroid polytopes. As a consequence, the combinatorics of the tropical theta divisor canonically encodes the matroid and Grassmannian structures underlying the associated KP multi-soliton solutions. We define the Hirota variety of a banana graph, which parametrizes all tau functions arising from such a graph. Starting from the matroid arising from Delaunay polytopes and the periods in the tropical limit, we construct an explicit parametrization of this variety which realizes the tau function as a multi-soliton. Our framework specializes naturally to real and positive settings.

math.AG↗

arXiv2025

Combinatorics of $m=1$ Grasstopes

A Grasstope is the image of the totally nonnegative Grassmannian $\text{Gr}_{\geq 0}(k,n)$ under a linear map $\text{Gr}(k,n)\dashrightarrow \text{Gr}(k,k+m)$. This is a generalization of the amplituhedron, a geometric object of great importance to calculating scattering amplitudes in physics. The amplituhedron is a Grasstope arising from a totally positive linear map. While amplituhedra are relatively well-studied, much less is known about general Grasstopes. We study Grasstopes in the $m=1$ case and show that they can be characterized as unions of cells of a hyperplane arrangement satisfying a certain sign variation condition, extending work of Karp and Williams. Inspired by this characterization, we also suggest a notion of a Grasstope arising from an arbitrary oriented matroid.

math.CO↗

arXiv2025

The positive orthogonal Grassmannian

The Plücker positive region $\mathrm{OGr}_+(k,2k)$ of the orthogonal Grassmannian emerged as the positive geometry behind the ABJM scattering amplitudes. In this paper we initiate the study of the positive orthogonal Grassmannian $\mathrm{OGr}_+(k,n)$ for general values of $k,n$. We determine the boundary structure of the quadric $\mathrm{OGr}_+(1,n)$ in $\mathbb{P}^{n-1}_{+}$ and show that it is a positive geometry. We show that $\mathrm{OGr}_+(k,2k+1)$ is isomorphic to $\mathrm{OGr}_+(k+1, 2k+2)$ and connect its combinatorial structure to matchings on $[2k+2]$. Finally, we show that in the case $n>2k+1$, the \emph{positroid cells} of $\mathrm{Gr}_+(k,n)$ do not induce a CW cell decomposition of $\mathrm{OGr}_+(k,n)$.

math.CO↗

arXiv2023

Hirota Varieties and Rational Nodal Curves

The Hirota variety parameterizes solutions to the KP equation arising from a degenerate Riemann theta function. In this work, we study in detail the Hirota variety arising from a rational nodal curve. Of particular interest is the irreducible subvariety defined as the image of a parameterization map, we call this the main component. Proving that this is an irreducible component of the Hirota variety corresponds to solving a weak Schottky problem for rational nodal curves. We solve this problem up to genus nine using computational tools.

math.AG↗

arXiv2023

Crossing the transcendental divide: from translation surfaces to algebraic curves

We study constructing an algebraic curve from a Riemann surface given via a translation surface, which is a collection of finitely many polygons in the plane with sides identified by translation. We use the theory of discrete Riemann surfaces to give an algorithm for approximating the Jacobian variety of a translation surface whose polygon can be decomposed into squares. We first implement the algorithm in the case of $L$ shaped polygons where the algebraic curve is already known. The algorithm is also implemented in any genus for specific examples of Jenkins-Strebel representatives, a dense family of translation surfaces that, until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves. Using Riemann theta functions, we give numerical experiments and resulting conjectures up to genus 5.

math.AG↗

arXiv2021

Non-Rigid Rank-One Infinite Measures on the Circle

For a class of irrational numbers, depending on their Diophantine properties, we construct explicit rank-one transformations that are totally ergodic and not weakly mixing. We classify when the measure is finite or infinite. In the finite case they are isomorphic to irrational rotations. We also obtain rank-one nonrigid infinite invariant measures for irrational rotations, and, for each Krieger type, nonsingular measures on irrational rotations. In the third version, in the infinite case we use the constructions to provide examples of non-weakly mixing infinite measure-preserving ergodic transformations which do not have any nontrivial probability preserving factors with discrete spectrum, thereby answering a questions of Aaronson and Nakada and of Glasner and Weiss.

math.DS↗

arXiv2021

Pencils of Quadrics: Old and New

Two-dimensional linear spaces of symmetric matrices are classified by Segre symbols. After reviewing known facts from linear algebra and projective geometry, we address new questions motivated by algebraic statistics and optimization. We compute the reciprocal curve and the maximum likelihood degrees, and we study strata of pencils in the Grassmannian.

math.AG↗

arXiv2021

KP Solitons from Tropical Limits

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 Hirota variety which parametrizes all tau functions arising from such a sum. We compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and we present an algorithm that finds a soliton solution from a rational nodal curve.

math.AG↗

arXiv2016

On Graphs Representable by Pattern-Avoiding Words

In this paper we study graphs defined by pattern-avoiding words. Word-representable graphs have been studied extensively following their introduction in 2000 and are the subject of a book published by Kitaev in 2015. Recently there has been interest in studying graphs represented by pattern-avoiding words. In particular, in 2016, Gao, Kitaev, and Zhang investigated 132-representable graphs, that is, word-representable graphs that can be represented by a word which avoids the pattern 132. They proved that all 132- representable graphs are circle graphs and provided examples and properties of 132-representable graphs. They posed several questions, some of which we answer in this paper. One of our main results is that not all circle graphs are 132-representable, thus proving that 132-representable graphs are a proper subset of circle graphs, a question that was left open in the paper by Gao et al. We show that 123-representable graphs are also a proper subset of circle graphs, and are different from 132-representable graphs. We also study graphs represented by pattern-avoiding 2-uniform words, that is, words in which every letter appears exactly twice.

math.CO↗

arXiv2015

Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex

The structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate an even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minors of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam and Postnikov. Here we investigate the entire set of arrangements and its relations with this triangulation. First, we show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation, and study its relations with the arrangement of t-th largest minors. Finally, we show that arrangements of largest minors induce a structure of partially ordered sets on the entire collection of minors. We use the Lam and Postnikov circuit triangulation of the hypersimplex to describe a 2-dimensional grid structure of this poset.

math.CO↗

arXiv2014

When Learners Surpass their Sources: Mathematical Modeling of Learning from an Inconsistent Source

We present a new algorithm to model and investigate the learning process of a learner mastering a set of grammatical rules from an inconsistent source. The compelling interest of human language acquisition is that the learning succeeds in virtually every case, despite the fact that the input data are formally inadequate to explain the success of learning. Our model explains how a learner can successfully learn from or even surpass its imperfect source without possessing any additional biases or constraints about the types of patterns that exist in the language. We use the data collected by Singleton and Newport (2004) on the performance of a 7-year boy Simon, who mastered the American Sign Language (ASL) by learning it from his parents, both of whom were imperfect speakers of ASL. We show that the algorithm possesses a frequency-boosting property, whereby the frequency of the most common form of the source is increased by the learner. We also explain several key features of Simon's ASL.

cs.CL↗