Related papers: High-dimensional convex sets arising in algebraic …
The simplex method in Linear Programming motivates several problems of asymptotic convex geometry. We discuss some conjectures and known results in two related directions -- computing the size of projections of high dimensional polytopes…
This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of…
We present a study of cubic surfaces from the novel perspective of positive geometry. Our positive geometries have dimension two (the surface minus its 27 lines), dimension three (its complement in 3-space), and dimension four (the moduli…
We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…
We consider asymptotic dimension of coarse spaces. We analyse coarse structures induced by metrisable compactifications. We calculate asymptotic dimension of coarse cell complexes. We calculate the asymptotic dimension of certain negatively…
We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.
In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was…
We introduce the notion of negative topological dimension and the notion of weight for the asymptotic topological dimension. Quantizing of spaces of negative dimension is applied to linguistic statistics.
Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article…
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on…
This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set $A$ containing $\Omega(|A|^{3/2})$ three-term arithmetic progressions.
Efficient representations of convex sets are of crucial importance for many algorithms that work with them. It is well-known that sometimes, a complicated convex set can be expressed as the projection of a much simpler set in higher…
The nonsingular real plane algebraic curves of given degree $d$ are considered either up to isotopy or up to deformation. The asymptotic behavior of the number $I_d$ of isotopy classes and the number $D_d$ of deformation classes are…
We study asymptotics of various Euclidean geometric phenomena as the dimension tend to infinity.
This paper develops asymptotic methods to count faces of random high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have surprising implications - in statistics, probability, information theory, and signal processing…
We review the role that infinite-dimensional symmetries arising at the boundary of asymptotically flat spacetimes play in the context of the celestial holography program. Once recast into the language of conformal field theory, asymptotic…
This chapter is a tutorial on techniques and results in free convex algebraic geometry and free real algebraic geometry (RAG). The term free refers to the central role played by algebras of noncommutative polynomials R<x> in free (freely…
We consider conformal gravity boundary conditions and outline the highest dimensional non-trivial asymptotic symmetry algebras of conformal gravity. The highest among them is five dimensional and leads to a global geon solution.
The asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly one when an outer…
An asymptotic theory is developed for computing volumes of regions in the parameter space of a directed Gaussian graphical model that are obtained by bounding partial correlations. We study these volumes using the method of real log…