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We introduce a growth process which samples sections of uniform infinite causal triangulations by elementary moves in which a single triangle is added. A relation to a random walk on the integer half line is shown. This relation is used to…
This paper deals with a generalized length-preserving flow for convex curves in the plane. It is shown that the flow exists globally and deforms convex curves into circles as time tends to infinity.
This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include…
Our knowledge and ignorance concerning the geometry of quantum states are discussed.
We study an abstract setting for cutting planes for integer programming called the infinite group problem. In this abstraction, cutting planes are computed via cut generating function that act on the simplex tableau. In this function space,…
We present a brief review on information processing, computing and inference via quantum fluctuation, and clarify the relationship between the probabilistic information processing and theory of quantum spin glasses through the analysis of…
Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…
In this expository note we discuss a class of graded algebras named Cox rings, which are naturally associated to algebraic varieties generalizing the homogeneous coordinate rings of projective spaces. Whenever the Cox ring is finitely…
We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical…
We explicitly compute limit shapes for several grand canonical Gibbs ensembles of partitions of integers. These ensembles appear in models of aggregation and are also related to invariant measures of zero range and coagulation-fragmentation…
The aim is to give a geometric characterization of the finite generation of the Cox ring of anticanonical rational surfaces. This characterization is encoded in the finite generation of the effective monoid. Furthermore, we prove that in…
This paper gives a complete characterization of infinitely divisible semimartingales, i.e., semimartingales whose finite dimensional distributions are infinitely divisible. An explicit and essentially unique decomposition of such…
This paper presents a mathematical analysis of an elliptic partial differential equation (PDE) designed to compute the geometric thickness of a given shape. The PDE-based formulation provides a direct and systematic approach to evaluate…
We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.
We consider one-dimensional hierarchical coalescence processes (in short HCPs) where two or three neighboring domains can merge. An HCP consists of an infinite sequence of stochastic coalescence processes: each process occurs in a different…
We consider a natural notion of equivalence for bounded linear operators on $H^p,$ for $p\neq 2.$ We determine which isometries of finite codimension are equivalent. For these isometries , we classify those which have the Crownover…
In this paper we introduce the concept of infinite pointwise dense lineability (spaceability), and provide a criterion to obtain density from mere lineability. As an application, we study the linear and topological structures within the set…
The following are notes on the geometry of the bidisk. In particular, we examine the properties of equidistant surfaces in the bidisk.
The rise of quantum information science has opened up a new venue for applications of the geometric phase (GP), as well as triggered new insights into its physical, mathematical, and conceptual nature. Here, we review this development by…
In this small paper we bring together various open problems on geometric multidimensional continued fractions.