Geometry from Information Geometry

We use the method of maximum entropy to model physical space as a curved statistical manifold. It is then natural to use information geometry to explain the geometry of space. We find that the resultant information metric does not describe the full geometry of space but only its conformal geometry -- the geometry up to local changes of scale. Remarkably, this is precisely what is needed to model "physical" space in general relativity.Comment: Presented at MaxEnt 2015, the 35th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (July 19-24, 2015, Potsdam NY, USA

Single-Player and Two-Player Buttons & Scissors Games

We study the computational complexity of the Buttons \& Scissors game and obtain sharp thresholds with respect to several parameters. Specifically we show that the game is NP-complete for $C = 2$ colors but polytime solvable for $C = 1$. Similarly the game is NP-complete if every color is used by at most $F = 4$ buttons but polytime solvable for $F \leq 3$. We also consider restrictions on the board size, cut directions, and cut sizes. Finally, we introduce several natural two-player versions of the game and show that they are PSPACE-complete.Comment: 21 pages, 15 figures. Presented at JCDCG2 2015, Kyoto University, Kyoto, Japan, September 14 - 16, 201

Geometry over composition algebras : projective geometry

The purpose of this article is to introduce projective geometry over composition algebras : the equivalent of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in correspondance with Jordan algebras and that the points of a projective space correspond to rank one matrices in the Jordan algebra. A second part thus studies properties of rank one matrices. Finally, subvarieties of projective spaces are discussed.Comment: 24 page

Dagger Geometry As Banach Algebraic Geometry

In this article, we apply the approach of relative algebraic geometry towards analytic geometry to the category of bornological and Ind-Banach spaces (non-Archimedean or not). We are able to recast the theory of Grosse-Kl\"onne dagger affinoid domains with their weak G-topology in this new language. We prove an abstract recognition principle for the generators of their standard topology (the morphisms appearing in the covers). We end with a sketch of an emerging theory of dagger affinoid spaces over the integers, or any Banach ring, where we can see the Archimedean and non-Archimedean worlds coming together

The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions

Abstract. We show that the Morita equivalences Cliff(4) H, Cliff(7) Cliff(−1), and Clif

Geometry in the Transition from Primary to Post-Primary

This article is intended as a kind of precursor to the document Geometry for Post-primary School Mathematics, part of the Mathematics Syllabus for Junior Certicate issued by the Irish National Council for Curriculum and Assessment in the context of Project Maths. Our purpose is to place that document in the context of an overview of plane geometry, touching on several important pedagogical and historical aspects, in the hope that this will prove useful for teachers.Comment: 19 page