In this paper, we will introduce the reader to the field of topology given a background of Calculus and Analysis. To familiarize the reader with topological concepts, we will present a proof of Brouwer’s Fixed Point Theorem. The end result of this paper will be a proof of the Poincaré-Hopf Theorem, an important theorem equating the index of a vector field on a manifold, and the Euler characteristic, an invariant of the manifold itself. We will conclude this paper with some useful applications… Expand

2017 IEEE International Conference on Robotics and Automation (ICRA)

2017

TLDR

This paper gives each point on the surface a uv-coordinates naturally represented by a complex number, except for a small number of zero points (singularities), and shows that natural, efficient robot paths can be obtained by using such coordinate systems.Expand

This paper is to propose solutions to selected exercises in Differential Topology by Guillemin and Pollack, [1], and to comment on certain proofs in the book. Although the sections covered in this… Expand

1. Introduction 2. The alternating algebra 3. De Rham cohomology 4. Chain complexes and their cohomology 5. The Mayer-Vietoris sequence 6. Homotopy 7. Applications of De Rham cohomology 8. Smooth… Expand

This paper presents a meta-thesis on the basis of a model derived from the model developed in [Bouchut-Boyaval, M3AS (23) 2013] that states that the mode of action of the Higgs boson is determined by the modulus of the E-modulus.Expand

MANIFOLDS Introduction Review of topological concepts Smooth manifolds Smooth maps Tangent vectors and the tangent bundle Tangent vectors as derivations The derivative of a smooth map Orientation… Expand