arXiv subjects

Stephanie Alexander

Publications and source records attributed to Stephanie Alexander.

5 recordsLinked to original sources

arXiv2019

Alexandrov geometry: foundations

Alexandrov spaces are defined via axioms similar to those given by Euclid. The Alexandrov axioms replace certain equalities with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above and curvature bounded below. The definitions of the two classes of spaces are similar, but their properties and known applications are quite different. Our approach is novel in its attention to the interrelatedness of the two fields, and its emphasis on the way each illuminates the other. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the basic material as well as selected topics inspired by considering the two contexts simultaneously. We only consider the intrinsic theory, leaving applications aside. This book includes material up to the definition of dimension. Another volume still in preparation will cover further topics.

math.DG↗

arXiv2017

An invitation to Alexandrov geometry: CAT(0) spaces

Our goal is to show the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. The topics include 1. Reshetnyak's gluing theorem, 2. Estimates on the number of collisions in billiards, 3. Reshetnyak's majorization theorem, 4. Hadamard--Cartan globalization theorem, 5. Polyhedral spaces, 6. Construction of exotic aspherical manifolds, 7. The geometry of two-convex sets in Euclidean space, 8. Barycenters and dimension theory.

math.DG↗

arXiv2015

Augur: a decentralized oracle and prediction market platform

Augur is a trustless, decentralized oracle and platform for prediction markets. The outcomes of Augur's prediction markets are chosen by users that hold Augur's native Reputation token, who stake their tokens on the actual observed outcome and, in return, receive settlement fees from the markets. Augur's incentive structure is designed to ensure that honest, accurate reporting of outcomes is always the most profitable option for Reputation token holders. Token holders can post progressively-larger Reputation bonds to dispute proposed market outcomes. If the size of these bonds reaches a certain threshold, Reputation splits into multiple versions, one for each possible outcome of the disputed market; token holders must then exchange their Reputation tokens for one of these versions. Versions of Reputation which do not correspond to the real-world outcome will become worthless, as no one will participate in prediction markets unless they are confident that the markets will resolve correctly. Therefore, token holders will select the only version of Reputation which they know will continue to have value: the version that corresponds to reality.

cs.CR↗

arXiv2013

An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds

It is proved that a convex hypersurface in a Riemannian manifold of sectional curvature at least k is an Alexandrov's space of curvature at least k. This theorem provides an optimal lower curvature bound for an older theorem of Buyalo.

math.DG↗

arXiv2010

Alexandrov meets Kirszbraun

We give a simplified proof of the generalized Kirszbraun theorem for Alexandrov spaces, which is due to Lang and Schroeder. We also discuss related questions, both solved and open.

math.DG↗