# Five Gaps in Mathematics

@article{Bahri2015FiveGI, title={Five Gaps in Mathematics}, author={Abbas Bahri}, journal={Advanced Nonlinear Studies}, year={2015}, volume={15}, pages={289 - 319} }

Abstract In this article we present a careful survey and critique of 5 landmark results and methods that seem to have played a pivotal role in the development of Mathematics since 1984. This article conveys the slow development of our understanding over the years and could not have been possible without the numerous communications with some authors of these articles as well as conversations with other friends and collaborators. The key contribution and the kindness of these colleagues is hereby… Expand

#### Figures from this paper

#### One Citation

A counterexample to the second inequality of Corollary (19.10) in the monograph "Ricci Flow and the Poincare Conjecture" by J.Morgan and G.Tian

- Mathematics
- 2015

We provide here a counter-example to the second inequality of Corollary (19.10) in the Clay Institute Monograph by J.Morgan and G.Tian entitled "Ricci Flow and the Poincare Conjecture". We had… Expand

#### References

SHOWING 1-10 OF 41 REFERENCES

Instantons and Four-Manifolds

- Mathematics
- 1984

This volume has been designed to explore the confluence of techniques and ideas from mathematical physics and the topological study of the differentiable structure of compact four-dimensional… Expand

Width and finite extinction time of Ricci flow

- Mathematics
- 2007

This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a… Expand

DIFFERENTIAL TOPOLOGY

- 2002

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

Monopoles and Three-Manifolds

- Mathematics
- 2008

Preface 1. Outlines 2. The Seiberg-Witten equations and compactness 3. Hilbert manifolds and perturbations 4. Moduli spaces and transversality 5. Compactness and gluing 6. Floer homology 7.… Expand

Pseudo-orbits of Contact Forms

- Mathematics
- 1988

This is a brief summary of a paper to appear, where I developed some tools in order to study the Weinstein conjecture [1]. This conjecture states that any contact vector-field on a compact contact… Expand

Topological Remarks–Critical Points at Infinity and String Theory

- Mathematics
- 2009

Abstract We take the example of the standard geodesics problem on S2 to show how the point to circle Morse relations in the periodic orbits variational problem for contact vector-fields lead directly… Expand

A Course in Minimal Surfaces

- Mathematics
- 2011

Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential… Expand

Multiple Integrals in the Calculus of Variations

- Mathematics
- 1966

Semi-classical results.- The spaces Hmp and Hmp0.- Existence theorems.- Differentiability of weak solutions.- Regularity theorems for the solutions of general elliptic systems and boundary value… Expand

Another look at Sobolev spaces

- Mathematics
- 2001

The standard seminorm in the space $W^{s,p}$, with $s$<$1$, does not converge, when $s$ approaches $1$, to the corresponding $W^{1,p}$ seminorm. We prove that continuity is restored provided we… Expand

Riemannian Geometry

- Nature
- 1927

THE recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann proposed the generalisation, to spaces of any order, of Gauss's… Expand