Related papers: Semigroups, Cartier divisors and convex bodies

The aim of this note is to give two new conceptual proofs of Ionescu-Weitzenb\"ock's inequality. The first one, which is a vector proof, provides us a geometric interpretation of the difference between the two sides of this inequality and…

In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying…

A variant of Gromov's H{\"o}lder-equivalence problem, motivated by a pinching problem in Riemannian geometry, is discussed. A partial result is given. The main tool is a general coarea inequality satisfied by packing energies of maps.

The affine quermassintegrals associated to a convex body in $\mathbb{R}^n$ are affine-invariant analogues of the classical intrinsic volumes from the Brunn-Minkowski theory, and thus constitute a central pillar of affine convex geometry.…

We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton--Okounkov body of a certain semigroup, and if the filtration is induced by a divisorial valuation, then the associated graded…

The purpose of this paper is to investigate the close relation between Okounkov bodies and Zariski decompositions of pseudoeffective divisors on smooth projective surfaces. Firstly, we completely determine the limiting Okounkov bodies on…

We define the quantum $p$-divergences and introduce Beckner's inequalities for primitive quantum Markov semigroups on a finite-dimensional matrix algebra satisfying the detailed balance condition. Such inequalities quantify the convergence…

We shall discuss a higher-rank Khovanskii-Teissier inequality, generalizing a theorem of Li. In the course of the proof, we develop new Hodge-Riemann bilinear relations in certain mixed settings, which in themselves slightly extend the…

We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such…

In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum $P_1+\dots+P_d$ of $d$-dimensional lattice polytopes is bounded from above by a…

In this note we give an answer to a question about mixed volumes asked by Gromov in his paper "Convex Sets and Kahler Manifolds". For reader's convenience we remind definitions and some of the properties of mixed volumes and mixed…

We construct symmetric convex bodies that are not intersection bodies, but all of their central hyperplane sections are intersection bodies. This result extends the studies by Weil in the case of zonoids and by Neyman in the case of…

We express the Segre class of a monomial scheme -- or, more generally, a scheme monomially supported on a set of divisors cutting out complete intersections -- in terms of an integral computed over an associated body in euclidean space. The…

For $(n-2)$ free divisor classes on a smooth projective variety of dimension $n$, the product of these free divisor classes induces a Lefschetz type operator acting on the N\'{e}ron-Severi space or the cohomology group of $(1,1)$ classes.…

We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective)…

Let $D$ be a big integral divisor on a smooth projective surface $X$. In this paper, we study Noether-type inequalities for $D$. The key ingredient is the introduction of a numerical invariant $\mathfrak{e}(D)$, which depends only on the…

Recently, there are many developments on the second main theorem for holomorphic curves into algebraic varieties intersecting divisors in general position or subgeneral position. In this paper, we refine the concept of subgeneral position…

This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual…

We develop computer assisted arguments for proving the existence of transverse homoclinic connecting orbits, and apply these arguments for a number of non-perturbative parameter and energy values in the spatial equilateral circular…

We show that the "topological BF-type" term introduced by Slavnov in order to cure the infrared divergences of gauge theories in noncommutative space can be characterized as the consequence of a new symmetry. This symmetry is a…