Geometry and Topology Commons^™

Discovering Geometric And Topological Properties Of Ellipsoids By Curvatures, Lina Wu, Shihshu Walter Wei, Jia Liu, Ye Li

Publications and Research

Aims/ Objectives: We are interested in discovering the geometric, topological and physical properties of ellipsoids by analyzing curvature properties on ellipsoids. We begin with studying ellipsoids as a starting point. Our aim is to find a way to study geometric, topological and physical properties from the analytic curvature properties for convex hyper-surfaces in the general setting.

Study Design: Multiple-discipline study between Differential Geometry, Topology and Mathematical Physics.

Place and Duration of Study: Department of Mathematics (Borough of Manhattan Community College-The City University of New York), Department of Mathematics (University of Oklahoma), Department of Mathematics and Statistics (University of West Florida), …

Non-Planar On-Shell Diagrams, Sebastían Franco, Daniele Galloni, Brenda Penante, Congkao Wen

Publications and Research

We initiate a systematic study of non-planar on-shell diagrams in N = 4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the d log form of the on-shell form explicit. We make significant progress towards a general classification of arbitrary on-shell diagrams by means of two classes of combinatorial objects: generalized matching and matroid polytopes. We propose a boundary measurement that connects general on-shell diagrams to the Grassmannian. Our proposal exhibits two important and non-trivial properties: positivity in the planar case and it matches the combinatorial description of the diagrams in …

Elementary College Geometry, Henry Africk

Open Educational Resources

This text is intended for a brief introductory course in plane geometry. It covers the topics from elementary geometry that are most likely to be required for more advanced mathematics courses. The only prerequisite is a semester of algebra.

The emphasis is on applying basic geometric principles to the numerical solution of problems. For this purpose the number of theorems and definitions is kept small. Proofs are short and intuitive, mostly in the style of those found in a typical trigonometry or precalculus text. There is little attempt to teach theorem-proving or formal methods of reasoning. However the topics are …

On Groups Of Homological Dimension One, Jonathan Cornick

Publications and Research

It has been conjectured that the groups of homological dimension one are precisely the nontrivial locally free groups. Some algebraic, geometric and analytic properties of any potential counter example to the conjecture are discussed.

On C^1 Robust Singular Transitive Sets For Three-Dimensional Flows, Carlos Arnoldo Morales, Maria José Pacífico, Enrique Ramiro Pujals

Publications and Research

Abstract:

The main goal of this paper is to study robust invariant transitive sets containing singularities for C¹ flows on three-dimensional compact boundaryless manifolds:they are partially hyperbolic with volume expanding central direction. Moreover, they are either attractors or repellers. Robust here means that this property cannot be destroyed by small C¹-perturbations of the flow.

Résumé:

Le but de ce travail est d'étudier des ensembles invariants robustes ayant des singularités pour des flots C¹ sur des variétés tridimensionelles : ce sont des ensembles hyperboliques singuliers. << Robuste >> veut dire ici que cette propriété ne peut être détruite par des …

Finite-Difference Approach To The Hodge Theory Of Harmonic Forms, Jozef Dodziuk

Publications and Research

No abstract provided.