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Visions in Mathematics

Part of the Modern Birkhäuser Classics Book Series

Editors: Alon, N.; Bourgain, J.; Connes, A.; Gromov, M.; Milman, V.

DOI: 10.1007/978-3-0346-0425-3

ISBN: 978-3-0346-0424-6

Publisher: Birkhäuser Basel - Springer Journals

Apr 13, 2011

1 - 10 of 13 Chapters

LitStream

Chapter 1Algebraic and Probabilistic Methods in Discrete Mathematics

Alon, Noga

Pages: 455 –470

[Combinatorics is an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. This survey contains a discussion of two of the main general techniques that played a crucial role in the development of modern combinatorics: algebraic methods...
LitStream

Chapter 2Challenges in Analysis

Coifman, R.

Pages: 471 –480

[Mathematical analysis, and in particular Harmonic Analysis, has traditionally been tied to physical modeling — providing the language to describe the infinitesimal laws of nature through calculus and partial differential expressions as well as descriptions of field effects through integral...
LitStream

Chapter 3Noncommutative Geometry Year 2000

Connes, Alain

Pages: 481 –559

[Our geometric concepts evolved first through the discovery of Non-Euclidean geometry. The discovery of quantum mechanics in the form of the noncommuting coordinates on the phase space of atomic systems entails an equally drastic evolution. We describe a basic construction which extends the...
LitStream

Chapter 4Introduction to Symplectic Field Theory

Eliashberg, Y.; Glvental, A.; Hofer, H.

Pages: 560 –673

[We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field theory, and at the same time serves as a rich source of new...
LitStream

Chapter 5Holomorphic Curves and Real Three-Dimensional Dynamics

Hofer, H.

Pages: 674 –704

[In this paper we describe new tools for studying smooth dynamical systems on three-manifolds. We also mention some interesting open problems.]
LitStream

Chapter 6Perspectives on the Analytic Theory of L-Functions

Iwaniec, H.; Sarnak, P.

Pages: 705 –741

[To the general mathematician L-functions might appear to be an esoteric and special topic in number theory. We hope that the discussion below will convince the reader otherwise. Time and again it has turned out that the crux of a problem lies in the theory of these functions. At some level it is...
LitStream

Chapter 7Combinatorics with a Geometric Flavor

Kalai, Gil

Pages: 742 –791

[In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg’s theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen’s upper bound theorem...
LitStream

Chapter 8Topics in Asymptotic Geometric Analysis

Milman, V.

Pages: 792 –815

[The term “Geometric Analysis” is a recent one but it has quickly become fashionable and is used too often and for very different mathematics. So, we added the adjective “asymptotic” to be more specific, and we will first explain the subject of this talk.]
LitStream

Chapter 9Quantum Information Theory: Results and Open Problems

Shor, Peter

Pages: 816 –838

[The discipline of information theory was founded by Claude Shannon in a truly remarkable paper [Sh] which laid down the foundations of the subject. We begin with a quote from this paper which is an excellent summary of the main concern of information theory: The fundamental problem of...
LitStream

Chapter 10Universality, Phase Transitions and Statistical Mechanics

Spencer, Thomas

Pages: 839 –858

[This survey will describe some mathematical results and conjectures related to phase transitions and statistical mechanics. In particular, we will focus on the principle of universality which asserts that singularities associated with second order phase transitions are universal. This means that...

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