ANALYSIS AND CALCULATIONS

2016-05-05 2016-06-10 2016-07-15 9/30/2024 10:34:59 AM Advanced Calculation and Analysis 1 2016 Original research article 1 2 2016 .1:1.2016 http://dx.doi.org/10.21065/ English Stavros Anastassiou Existence theorem, Geometrical study and Topological properties. The importance of analysis in mathematics cannot be overestimated. Many areas of modern mathematics may be seen as parts of analysis while, at the same time, many areas of mathematics heavily rely on analysis to built on and further comment on their problems. Indeed many mathematicians, who consider themselves to be analysts, work on differential equations and prove numerous and powerful results regarding their behavior (enumerating these results, one could begin by mentioning the existence theorem of solutions in the O.D.E's case, and various results concerning solutions of boundary value problems in the P.D.E.'s case). Such was the success of analysis in geometry, that the term \global analysis" was coined to describe the work of those mathematicians who use analytical techniques to study geometrical and topological properties \in the large". The proof of the Geometrization and Poincare conjectures is probably the most famous result of using analytical tools (properties of the Ricci flow) to achieve a geometrical result (classification of manifolds).