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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: 10.1007/978-3-0346-0422-2

ISBN: 978-3-0346-0421-5

Publisher: Birkhäuser Basel - Springer Journals

Mar 24, 2011

1 - 10 of 17 Chapters

LitStream

Chapter 1Gauß—Manin Determinant Connections and Periods for Irregular Connections

Bloch, Spencer; Esnault, Hélène

Pages: 1 –31

[Gauß-Manin determinant connections associated to irregular connections on a curve are studied. The determinant of the Fourier transform of an irregular connection is calculated. The determinant of cohomology of the standard rank 2 Kloosterman sheaf is computed modulo 2 torsion. Periods...
LitStream

Chapter 2Problems in Hamiltonian PDE’S

Bourgain, J.

Pages: 32 –56

[The purpose of this exposé is to describe a line of research and problems, which I believe, will not be by any means completed in the near future. As such, we certainly hope to encourage further investigations. The list of topics in this field is fairly extensive and only a few will be commented...
LitStream

Chapter 3On a Classical Limit of Quantum Theory and the Non-Linear Hartree Equation

Fröhlich, Jürg; Tsai, Tai-Peng; Yau, Horng-Tzer

Pages: 57 –78

[Historically, the description of Nature in theoretical physics began with an encoding of rather special, simple natural phenomena exhibiting high approximate symmetry in special, unstable mathematical structures. For example, the shape of the orbits of planets around the sun was conceived to be...
LitStream

Chapter 4Rough Structure and Classification

Gowers, W. T.

Pages: 79 –117

[When I was first asked to speak at the “Visions in Mathematics” conference, I had what I believe was a typical reaction. I wanted to try to emulate Hilbert a century ago, but since I knew that I could not possibly match his breadth of vision, I was forced to make some sort of compromise. In this...
LitStream

Chapter 5Spaces and Questions

Gromov, Misha

Pages: 118 –161

[Our Euclidean intuition, probably inherited from ancient primates, might have grown out of the first seeds of geometry in the motor control systems of early animals who were brought up to sea and then to land by the Cambrian explosion half a billion years ago. The primates' brain had been idling...
LitStream

Chapter 6Classification of Infinite-Dimensional Simple Groups of Supersymmetries and Quantum Field Theory

Kac, Victor G.

Pages: 162 –183

[This work was motivated by two seemingly unrelated problems: Lie’s problem of classification of “local continuous transformation groups of a finite-dimensional manifold”.The problem of classification of operator product expansions (OPE) of chiral fields in 2-dimensional conformal field theory.]
LitStream

Chapter 7Geometrization in Representation Theory

Kazhdan, D.

Pages: 184 –187

[The common theme of the articles “γ-function of Representations and Lifting” and “Geometric and Unipotent Crystals” is the geometrization of familiar objects. In the first article we interpret some special functions appearing in the theory of representations as materializations of geometric...
LitStream

Chapter 8Geometric and Unipotent Crystals

Berenstein, Arkady; Kazhdan, David

Pages: 188 –236

[Let G be a split semisimple algebraic group over ℚ, g be the Lie algebra of G and Uq(g) be the corresponding quantized enveloping algebra. Lusztig has introduced in [Lul] canonical bases for finite-dimensional Uq(g)-modules. About the same time E as hi war a introduced in [Kl] crystal bases as a...
LitStream

Chapter 9γ-Functions of Representations and Lifting

Braverman, A.; Kazhdan, D.

Pages: 237 –278

[Let F be a local non-Archimedean field and let ψ: F → ℂ× be a non-trivial additive character of F. To this data one associates a meromorphic function γψ: π → γψ(π) on the set of irreducible representations of the group GL(n, F) in the following way. Consider the invariant distribution Φψ:=...
LitStream

Chapter 10PDE as a Unified Subject

Klainerman, Sergiu

Pages: 279 –315

[Given that one of the goals of the conference is to address the issue of the unity of Mathematics, I feel emboldened to talk about a question which has kept bothering me all through my scientific career: Is there really a unified subject of Mathematics which one can call PDE? At first glance...

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