maths document

May 11th, 2015
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A VIEW OF MATHEMATICSAlain CONNESMathematics is the backbone of modern science and a remarkably efficient sourceof new concepts and tools to understand the reality in which we participate.It plays a basic role in the great new theories of physics of the XXth century suchas general relativity, and quantum mechanics.The nature and inner workings of this mental activity are often misunderstoodor simply ignored even among scientists of other disciplines. They usually onlymake use of rudimentary mathematical tools that were already known in the XIXthcentury and miss completely the strength and depth of the constant evolution ofour mathematical concepts and tools.I was asked to write a general introduction on Mathematics which I ended up doingfrom a rather personal point of view rather than producing the usual endless litanyX did this and Y did that. The evolution of the concept of space in mathematics serves as a unifying theme starting from some of its historical roots andgoing towards more recent developments in which I have been more or less directlyinvolved.Contents1. The Unity of Mathematics2. The concept of Space2.1. Projective geometry2.2. The Angel of Geometry and the Devil of Algebra2.3. Noneuclidean geometry2.4. Symmetries2.5. Line element and Riemannian geometry2.6. Noncommutative geometry2.7. Grothendiecks motives2.8. Topos theory3. Fundamental Tools3.1. Positivity3.2. Cohomology3.3. Calculus3.4. Trace and Index Fo

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