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We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has…

We explore the applications of Lorentzian polynomials to the fields of algebraic geometry, analytic geometry and convex geometry. In particular, we establish a series of intersection theoretic inequalities, which we call rKT property, with…

In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces.…

In this paper, we develop a tropical analog of the classical flag variety that we call the flag Dressian. We find relations, which we call "tropical incidence relations", for when one tropical linear space is contained in another, and show…

Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical…

We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how…

Tropical geometry has recently found several applications in the analysis of neural networks with piecewise linear activation functions. This paper presents a new look at the problem of tropical polynomial division and its application to…

We consider polynomials expressing the cohomology classes of subvarieties of products of projective spaces, and limits of positive real multiples of such polynomials. We study the relation between these covolume polynomials and Lorentzian…

Lorentzian polynomials are a fascinating class of real polynomials with many applications. Their definition is specific to the nonnegative orthant. Following recent work, we examine Lorentzian polynomials on proper convex cones. For a…

Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces…

We examine the incidence geometry of lines in the tropical plane. We prove tropical analogs of the Sylvester-Gallai and Motzkin-Rabin theorems in classical incidence geometry. This study leads naturally to a discussion of the realizability…

Lorentzian polynomials serve as a bridge between continuous and discrete convexity, connecting analysis and combinatorics. In this article, we study the topology of the space $\mathbb{P}\textrm{L}_J$ of Lorentzian polynomials on $J$ modulo…

We prove a transverse diameter theorem in the context of Lorentzian foliations, which can be interpreted as a Hawking--Penrose-type singularity theorem for timelike geodesics transverse to the foliation. In order to develop the necessary…

I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume…

Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with…

In this paper we consider the characteristic polynomial of not necessarily ranked posets. We do so by allowing the rank to be an arbitrary function from the poset to the nonnegative integers. We will prove two results showing that the…

In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of L consisting of all vectors v that…

In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in…

L\'opez de Medrano, Rinc\'on and Shaw defined the Chern classes on tropical manifolds as an extension of their theory of the Chern-Schwartz-MacPherson cycles on matroids. This makes it possible to define the Riemann-Roch number of tropical…

We construct a logarithmic version of the Hilbert scheme, and more generally the Quot scheme, of a simple normal crossings pair. The logarithmic Quot space admits a natural tropicalisation called the space of tropical supports, which is a…