arXiv subjects

Josephine Yu

Publications and source records attributed to Josephine Yu.

At least 19 recordsLinked to original sources

arXiv2026

Positivity and log concavity of the Links--Gould polynomial of knots

Motivated by the recent work of Harper--Kohli--Song--Tahar, we formulate a positivity, hole-free, and log-concavity conjecture for the Links--Gould polynomial of alternating links and verify it for all 51.3 million alternating knots with at most 19 crossings. All but 544 of those knots satisfy a stronger type-B log-concavity condition characterized by the slopes of edges in the subdivision of the monomial support induced by the log coefficients.

math.GT↗

arXiv2025

Ehrhart Theory over Abelian Group Rings

We introduce a unified framework for Ehrhart theory in which lattice point enumerators take coefficients in an Abelian group ring, encoding substantially richer algebraic data than classical counts. We prove that fundamental results of Ehrhart theory extend to this setting through a generalized Brion theorem, including rational generating functions, reciprocity phenomena, connections to volume, and vertex-cone decompositions. We further show how to derive $q$-enumerative and weighted theories from this setting, recasting several major refinements of Ehrhart theory as consequences of a single algebraic mechanism. We also show how our framework combines with equivariant Ehrhart theory.

math.CO↗

arXiv2025

Computing parametric weighted Ehrhart polynomials of smooth polytopes

We show that when integral polytopes are deformed while keeping the same facet normal vectors, the coefficients of weighted Ehrhart and $h^*$-polynomials are piecewise polynomial functions in the ``right hand sides'' of the linear inequalities defining the polytopes. We give an algorithm and an implementation in SageMath for computing these polynomials for smooth polytopes, such as type $A$ alcoved polytopes, using a weighted Euler-Maclaurin type formula by Khovanski\v{i} and Pukhlikov. We discuss some natural questions concerning signs of the coefficients of the weighted $h^*$-polynomials.

math.CO↗

arXiv2024

Tropicalizing Principal Minors of Positive Definite Matrices

We study the tropicalization of the image of the cone of positive definite matrices under the principal minors map. It is a polyhedral subset of the set of $M$-concave functions on the discrete $n$-dimensional cube. We show it coincides with the intersection of the affine tropical flag variety with the submodular cone. In particular, any cell in the regular subdivision of the cube induced by a point in this tropicalization can be subdivided into base polytopes of realizable matroids. We use this tropicalization as a guide to discover new algebraic inequalities among the principal minors of positive semidefinite matrices of a fixed size. We also extend our results to positive semidefinite matrices via taking closures in the tropical semifield $\mathbb{R}\cup\{-\infty\}$.

math.CO↗

arXiv2024

Real and Positive Tropicalizations of Symmetric Determinantal Varieties

We study real and positive tropicalizations of the varieties of low rank symmetric matrices over real or complex Puiseux series. We show that real tropicalization coincides with complex tropicalization for rank two and corank one cases. We also show that the two notions of positive tropicalization introduced by Speyer and Williams coincide for symmetric rank two matrices, but they differ for symmetric corank one matrices.

math.AG↗

arXiv2024

The Tropical Variety of Symmetric Rank 2 Matrices

We study the tropicalization of the variety of symmetric rank two matrices. Analogously to the result of Markwig and Yu for general tropical rank two matrices, we show that it has a simplicial complex structure as the space of symmetric bicolored trees and that this simplicial complex is shellable. We also discuss some matroid structures arising from this space and present generating functions for the number of symmetric bicolored trees.

math.CO↗

arXiv2023

Weighted Ehrhart Theory: Extending Stanley's nonnegativity theorem

We generalize R. P. Stanley's celebrated theorem that the $h^\ast$-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the $h^\ast$-polynomial as a real-valued function for a larger family of weights. We then target the case when the weight function is the square of a single (arbitrary) linear form. We show stronger results for two-dimensional convex lattice polygons and give concrete examples showing tightness of the hypotheses. As an application, we construct a counterexample to a conjecture by Berg, Jochemko, and Silverstein on Ehrhart tensor polynomials.

math.CO↗

arXiv2022

Quantum Electrodynamical Metamaterials

Recent experiments have revealed ultrastrong coupling between light and matter as a promising avenue for modifying material properties, such as electrical transport, chemical reaction rates, and even superconductivity. Here, we explore (ultra)strong coupling as a means for manipulating the optical response of metamaterials based on ensembles of constituent units individually in the ultrastrong coupling regime. We develop a framework based on linear response for quantum electrodynamical systems to study how light-matter coupling affects the optical response. We begin by applying this framework to find the optical response of a two-level emitter coupled to a single cavity mode, which could be seen as a "meta-atom" of a metamaterial built from repeated units of this system. We find optical behaviors ranging from that of a simple two-level system (Lorentz-oscillator) to effectively transparent, as the coupling goes from the weak to deep strong coupling regimes. We explore a one-dimensional chain of these meta-atoms, demonstrating the tunability of its optical behavior. Our scheme may ultimately provide a framework for designing new metamaterials with low-loss, highly-confined modes, as well as tunable (single-photon) nonlinearities.

physics.optics↗

arXiv2022

Random singlet-like phase of disordered Hubbard chains

Local moment formation is ubiquitous in disordered semiconductors such as Si:P, where it is observed both in the metallic and the insulating regimes. Here, we focus on local moment behavior in disordered insulators, which arises from short-ranged, repulsive electron-electron interactions. Using density matrix renormalization group and strong-disorder renormalization group methods, we study paradigmatic models of interacting insulators: one dimensional Hubbard chains with quenched randomness. In chains with either random fermion hoppings or random chemical potentials, both at and away from half-filling, we find exponential decay of charge and fermion 2-point correlations but power-law decay of spin correlations that are indicative of the random singlet phase. The numerical results can be understood qualitatively by appealing to the large-interaction limit of the Hubbard chain, in which a remarkably simple picture emerges.

cond-mat.str-el↗

arXiv2022

Moments, Sums of Squares, and Tropicalization

We use tropicalization to study the duals to cones of nonnegative polynomials and sums of squares on a semialgebraic set $S$. The truncated cones of moments of measures supported on the set $S$ is dual to nonnegative polynomials on $S$, while "pseudo-moments" are dual to sums of squares approximations to nonnegative polynomials. We provide explicit combinatorial descriptions of tropicalizations of the moment and pseudo-moment cones, and demonstrate their usefulness in distinguishing between nonnegative polynomials and sums of squares. We give examples that show new limitations of sums of squares approximations of nonnegative polynomials. When the semialgebraic set is defined by binomial inequalites, its moment and pseudo-moment cones are closed under Hadamard product. In this case, their tropicalizations are polyhedral cones that encode all binomial inequalities on the moment and pseudo-moment cones.

math.AG↗

arXiv2021

Toric and tropical Bertini theorems in positive characteristic

We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field \kk(t_1,\dots,t_d) of rational functions over an algebraically closed field \kk of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu to arbitrary characteristic, which removes the characteristic dependence from the d-connectivity result for tropical varieties from that paper.

math.AG↗

arXiv2020

On the hypergraph connectivity of skeleta of polytopes

We show that for every $d$-dimensional polytope, the hypergraph whose nodes are $k$-faces and whose hyperedges are $(k+1)$-faces of the polytope is strongly $(d-k)$-vertex connected, for each $0 \leq k \leq d- 1$.

math.CO↗

arXiv2019

Higher Connectivity of Tropicalizations

We show that the tropicalization of an irreducible d-dimensional variety over a field of characteristic 0 is (d-l)-connected through codimension one, where l is the dimension of the lineality space of the tropicalization. From this we obtain a higher connectivity result for skeleta of rational polytopes. We also prove a tropical analogue of the Bertini Theorem: the intersection of the tropicalization of an irreducible variety with a generic hyperplane is again the tropicalization of an irreducible variety.

math.AG↗

arXiv2019

Positively Hyperbolic Varieties, Tropicalization, and Positroids

A variety of codimension $c$ in complex affine space is called positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension $c$. Positively hyperbolic hypersurfaces are defined by stable polynomials. We give a new characterization of positively hyperbolic varieties using sign variations, and show that they are equivalently defined by being hyperbolic with respect to the positive part of the Grassmannian, in the sense of Shamovich and Vinnikov. We prove that positively hyperbolic projective varieties have tropicalizations that are locally subfans of the type $A$ hyperplane arrangement defined by $x_i = x_j$, in which the maximal cones satisfy a non-crossing condition. This gives new proofs of some results of Choe--Oxley--Sokal--Wagner and Br\"and\'en on Newton polytopes and tropicalizations of stable polynomials. We settle the question of which tropical varieties can be obtained as tropicalizations of positively hyperbolic varieties in the case of tropical toric varieties, constant-coefficient tropical curves, and Bergman fans. Along the way, we also give a new characterization of positroids in terms of a non-crossing condition on their Bergman fans.

math.CO↗

arXiv2018

Real tropicalization and analytification of semialgebraic sets

Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of semialgebraic subsets of $K^n$ under this map from a general point of view. For a semialgebraic set $S \subset K^n$ we define a space $S_r^{\text{an}}$ called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of $S$. We prove a real analogue of the tropical fundamental theorem and show that the tropicalization of any semialgebraic set is described by tropicalization of finitely many inequalities which are valid on the semialgebraic set. We also study the topological properties of real analytification and tropicalization. If $X$ is an algebraic variety, we show that $X_r^{\text{an}}$ can be canonically embedded into the real spectrum $X_r$ of $X$, and we study its relation with the Berkovich analytification of $X$.

math.AG↗

arXiv2018

Inverse Design of Discrete Mechanical Metamaterials

Mechanical and phononic metamaterials exhibiting negative elastic moduli, gapped vibrational spectra, or topologically protected modes enable precise control of structural and acoustic functionalities. While much progress has been made in their experimental and theoretical characterization, the inverse design of mechanical metamaterials with arbitrarily programmable spectral properties and mode localization remains an unsolved problem. Here, we present a flexible computational inverse-design framework that allows the efficient tuning of one or more gaps at nearly arbitrary positions in the spectrum of discrete phononic metamaterial structures. The underlying algorithm optimizes the linear response of elastic networks directly, is applicable to ordered and disordered structures, scales efficiently in 2D and 3D, and can be combined with a wide range of numerical optimization schemes. We illustrate the broad practical potential of this approach by designing mechanical bandgap switches that open and close pre-programmed spectral gaps in response to an externally applied stimulus such as shear or compression. We further show that the designed structures can host topologically protected edge modes, and validate the numerical predictions through explicit 3D finite element simulations of continuum elastica with experimentally relevant material parameters. Generally, this network-based inverse design paradigm offers a direct pathway towards manufacturing phononic metamaterials, DNA origami structures and topolectric circuits that can realize a wide range of static and dynamic target functionalities.

cond-mat.mtrl-sci↗

arXiv2017

Toric h-vectors and Chow Betti Numbers of Dual Hypersimplices

The toric h-numbers of a dual hypersimplex and the Chow Betti numbers of the normal fan of a hypersimplex are the ranks of intersection cohomology and Chow cohomology respectively of the torus orbit closure of a generic point in the Grassmannian. We give explicit formulas for these numbers. We also show that similar formulas hold for the coordinator numbers of type A^* lattices.

math.CO↗

arXiv2017

Beyond Polyhedral Homotopies

We present a new algorithmic framework which utilizes tropical geometry and homotopy continuation for solving systems of polynomial equations where some of the polynomials are generic elements in linear subspaces of the polynomial ring. This approach generalizes the polyhedral homotopies by Huber and Sturmfels.

math.AG↗