Advanced Courses Of Mathematical Analysis Vi - Proceedings Of The Sixth International School

Proceedings of the Sixth International School

Francisco Javier Martín-Reyes, Francisco Javier Martín-Reyes;María Lorente;Cristóbal González

• Verlag: WSPC
• Jahr: 2016
• ISBN:
  9789813147652
• Thema: Mathematics
• Thema: Mathematical Analysis

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This volume contains short courses and recent papers by several specialists in different fields of Mathematical Analysis. It offers a wide perspective of the current state of research, and new trends, in areas related to Geometric Analysis, Harmonic Analysis, Complex Analysis, Functional Analysis and History of Mathematics. The contributions are presented with a remarkable expository nature and this makes the discussed topics accessible to a more general audience.

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PART A

Courses

Convex inequalities, isoperimetry and spectral gap*

David Alonso-Gutiérrez

Departamento de Matemáticas and IUMA,
Universidad de Zaragoza,
Pedro Cerbuna 12, 50009 Zaragoza, Spain
E-mail: [email protected]

Jesús Bastero

Departamento de Matemáticas and IUMA,
Universidad de Zaragoza,
Pedro Cerbuna 12, 50009 Zaragoza, Spain
E-mail: [email protected]

The main idea of these notes is to present the Kannan-Lovász-Simonovits spectral gap conjecture on the correct estimate for the spectral gap of the Laplace-Beltrami operator associated to any log-concave probability on

ⁿ.

Keywords: Log-concave probability, Brunn-Minkowski inequality, concentration phenomena, convex bodies, spectral gap, isoperimetric inequality, Poincaré inequality.

1.Introduction and Notation

These notes are the content of the three lectures explained by the second author in the “VI International Course of Mathematical Analysis in Andalucía”, held in Antequera in September, 2014. The authors want to express their gratitude to the organizers of this meeting for giving to one of them the possibility of presenting this quite new theory to young researchers. The content of this course is included in the reference [2], where a more detailed and complete information appears.

The main idea of these notes is to present the Kannan-Lovász-Simonovits spectral gap conjecture (KLS). This question was originally posed in relation with some problems in theoretical computer science, but it has a wellunderstood analytic-geometrical meaning: give the correct estimate for the spectral gap (first non trivial eigenvalue) of the Laplace-Beltrami operator associated to any log-concave probability in

ⁿ. The KLS conjecture can also be expressed in terms of a type of Cheeger’s isoperimetric inequality and, in this way, is related to Poincare’s inequalities and to the concentration of measure phenomenon. In the meanwhile this conjecture is now one of the central points in asymptotic geometric analysi...