Convex geometric analysis

MATH-564 / 5 crédits

Enseignant: Tiba Marius

Langue: Anglais

Summary

In this course we study the fundamental notion of convexity at the interface of analysis and geometry, an area that has seen remarkable progress in recent years.

Content

The course begins with the classical isoperimetric inequality, dating back to antiquity: among all closed curves in the plane of fixed perimeter, the circle maximizes the enclosed area. Isoperimetric inequalities arise in many familiar settings, including Euclidean and Gaussian spaces, and the sphere, and they lead to the remarkable concentration of measure phenomenon: a Lipschitz function on a high-dimensional sphere concentrates around its mean. These belong to a broad family of fundamental geometric and functional inequalities including Brunn-Minkowski, Prekopa-Leindler and Sobolev, with applications to areas such as PDEs, geometry and probability. An important research theme is the investigation of equality or near-equality cases of these inequalities. We will apply powerful techniques to study them: symmetrization processes, convex localization, and optimal transport.

 

The second part of the course deals with the structure of high-dimensional convex bodies. We begin with John's theorem, a classical result controlling the Banach-Mazur distance between an arbitrary norm and the Euclidean norm. We then prove Dvoretzky's celebrated theorem, which states that every high-dimensional normed space contains an almost Euclidean subspace of large dimension. We will also mention two recent breakthroughs: the resolution of Bourgain's slicing conjecture - every convex body of unit volume has a slice with constant volume, and of the thin shell conjecture. We will derive a central limit theorem for convex bodies: in most directions their marginals are approximately Gaussian. Concentration of measure serves as a central tool.

Learning Prerequisites

Required courses

  • Analysis I - Real Analysis
  • Analysis II - Vector Analysis
  • Probability (Introductitiory)

The first exercise class will be devoted to recalling the necessary background

Recommended courses

  • Analysis IV - Lebesgue measure, Fourier analysis

Important concepts to start the course

  • Multivariable Calculus (Differentiation and Integration in R^n, Volume and Surface area formulae)
  • Basic Concepts in Probability (Probability Density Function, Expectation, Independence, Central Liimit Theorem)

Learning Outcomes

By the end of the course, the student must be able to:

  • Prove classical geometric and functional inequalities
  • Investigate the equality cases (or stability) of these inequalities
  • Use techniques such as symmetrization, convex localization and optimal transport
  • Apply these inequalities to other contexts
  • Quantify high-dimensional phenomena arising from convexity
  • Compute concrete examples
  • Prove main results

Transversal skills

  • Demonstrate a capacity for creativity.
  • Demonstrate the capacity for critical thinking
  • Continue to work through difficulties or initial failure to find optimal solutions.
  • Use a work methodology appropriate to the task.

Teaching methods

Weekly lectures and exercise sessions

Expected student activities

Students are expected to attend the lectures and actively participate in the exercise sessions. In addition, they are expected to attempt the problems on the exercise sheets. The graded homework consists of one exercise per week.

Assessment methods

Weekly graded homeworks (20%) and written final exam (80%)

Supervision

Office hours Yes
Assistant.e.s Yes
Forum Yes

Resources

Bibliography

  • K. Böröczky, A. Figalli, J. Ramos. Isoperimetric inequalities, Brunn-Minkowski theory and Minkowski type Monge-Ampère equations on the sphere. EMS
  • R. Gardner. The Brunn-Minkowski Inequality. Bull. of the AMS, 2002
  • S. Artstein-Avidan, A. Giannopoulos, V. Milman, Asymptotic Geometric Analysis, part I. Mathematical Surveys and Monographs 202 AMS, 2015
  • S. Brazitikos, A. Giannopoulos, P. Valettas, B.-H. Vritsiou, Geometry of Isotropic Convex Bodies. Mathematical Surveys and Monographs 196 AMS, 2014

Notes/Handbook

Lecture notes will be provided

Moodle Link

Dans les plans d'études

  • Semestre: Printemps
  • Forme de l'examen: Ecrit (session d'été)
  • Matière examinée: Convex geometric analysis
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Ecrit (session d'été)
  • Matière examinée: Convex geometric analysis
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Ecrit (session d'été)
  • Matière examinée: Convex geometric analysis
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel

Semaine de référence

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