Ramon van Handel (Princeton University)

Title: Around the Alexandrov-Fenchel inequality

Abstract: The isoperimetric theorem states that the ball minimizes surface area among all bodies of the same volume. For convex bodies, however, volume and surface area are merely two examples of a large family of natural geometric parameters called mixed volumes that arise as coefficients of the volume polynomial. Mixed volumes were discovered by Minkowski in a seminal 1903 paper that laid much of the foundation for modern convex geometry. In particular, Minkowski, Alexandrov and Fenchel discovered a remarkable set of quadratic inequalities between mixed volumes that constitute a far-reaching generalization of the classical isoperimetric theorem. The theory of these inequalities and their applications in characterized by unexpected connections with various questions in geometry, analysis, algebra, and combinatorics, and features some long-standing open problems. My aim in these lectures is to introduce some of the problems, connections, and recent progress on this topic.

Lecture 1: Video

References:
R. Schneider, "Convex Bodies: The Brunn-Minkowski Theory"
T. Bonnesen and W. Fenchel, "Theory of Convex Bodies".

Lecture 2: Video

References:
Y. Shenfeld and R. van Handel, "Mixed volumes and the Bochner method".https://arxiv.org/abs/1811.08710

Lecture 3: Video

References:
A. Kolesnikov and E. Milman, "Local L^p-Brunn-Minkowski inequalities for $p<1$", https://arxiv.org/abs/1711.01089
E. Putterman, "Equivalence of the local and global versions of the L^p-Brunn-Minkowski inequality", https://arxiv.org/abs/1909.03729

Lecture 4: Video

References: 
Y. Shenfeld and R. van Handel: "The extremals of the Alexandrov-Fenchel inequality for convex polytopes", https://arxiv.org/abs/2011.04059
Y. Shenfeld and R. van Handel: "The extremals of Minkowski's quadratic inequality", https://arxiv.org/abs/1902.10029

Lecture 5: Video

References:
V. A. Timorin, "An analogue of the Hodge-Riemann relations for simple convex polytopes", https://doi.org/10.1070/RM1999v054n02ABEH000134

Mark Rudelson (University of Michigan)

Title: On the delocalization of the eigenvectors of random matrices

Abstract: Consider a random matrix with i.i.d. normal entries. Since its distribution is invariant under rotations, any normalized eigenvector is uniformly distributed over the unit sphere. For a general distribution of the entries, this is no longer true. Yet, if the size of the matrix is large, the eigenvectors are distributed approximately uniformly. This property, called delocalization, can quantified in various senses. In these lectures, we will discuss recent results on delocalization for general random matrices.
We will consider two notions of delocalization. First, we will discuss the sup-norm delocalization. It is easy to see that all coordinates of a  vector uniformly distributed over the unit sphere are small with probability close to 1. We strive to extend this fact to the eigenvalues of general random matrices. We will also consider a notion of ``no-gaps'' delocalization. Namely, we will show that with high probability, any relatively large set of coordinates carries a non-negligible portion of the norm of the eigenvector. These two notions are complementary to each other: while the sup-norm delocalization rules out large coordinates of the eigenvector, the no-gaps delocalization rules out the small ones. If time allows, we will consider applications of delocalization to random graphs in which these two properties work in tandem.

Our approach to establishing delocalization will rely in a large part on ideas of high-dimensional convex geometry and measure concentration. The lectures will be self-contained, and all necessary tools will be introduced along the way.

Delocalization surveys:
O'Rourke, Sean; Vu, Van; Wang, Ke Eigenvectors of random matrices: a survey. J. Combin. Theory Ser. A 144 (2016), 361–442.

Rudelson, Mark Delocalization of eigenvectors of random matrices. Random matrices, 303–340, IAS/Park City Math. Ser., 26, Amer. Math. Soc., Providence, RI, 2019.

Lecture 1: Video

References :
General random matrix theory, Stieltjes transform, Semicircular Law:
Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010.
Tao, Terence Topics in random matrix theory. Graduate Studies in Mathematics, 132. American Mathematical Society, Providence, RI, 2012.

Local Semicircular Law:
Erdős, László ; Yau, Horng-Tzer, A dynamical approach to random matrix theory.Courant Lecture Notes in Mathematics, 28. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2017.

Lecture 2 : Video

References:
Hanson-Wright inequality.
Rudelson, Mark; Vershynin, Roman Hanson-Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18 (2013), no. 82, 9 pp.

Lecture 3 : Video

References :
Geometric method for establishing the \ell_\infty delocalization.
Rudelson, Mark; Vershynin, Roman Delocalization of eigenvectors of random matrices with independent entries. Duke Math. J. 164 (2015), no. 13, 2507–2538.

Lecture 4 : Video

References :

Density of a projection of a random vector :
Halász, G. Estimates for the concentration function of combinatorial number theory and probability. Period. Math. Hungar. 8 (1977), no. 3-4, 197–211.
Rudelson, Mark; Vershynin, Roman Small ball probabilities for linear images of high-dimensional distributions. Int. Math. Res. Not. IMRN 2015, no. 19, 9594–9617.
Livshyts, Galyna; Paouris, Grigoris; Pivovarov, Peter On sharp bounds for marginal densities of product measures. Israel J. Math. 216 (2016), no. 2, 877–889.

Brascamp-Lieb inequality :
Brascamp, Herm Jan; Lieb, Elliott H. Best constants in Young's inequality, its converse, and its generalization to more than three functions. Advances in Math. 20 (1976), no. 2, 151–173.
Barthe, Franck Inégalités de Brascamp-Lieb et convexité. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 8, 885–888.
Barthe, Franck On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134 (1998), no. 2, 335–361.

Lecture 5 : Video

References :

No-gaps delocalization:
Rudelson, Mark; Vershynin, Roman No-gaps delocalization for general random matrices. Geom. Funct. Anal. 26 (2016), no. 6, 1716–1776.

Braess' paradox :
Eldan, Ronen; Rácz, Miklós Z.; Schramm, Tselil, Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors. Random Structures Algorithms 50 (2017), no. 4, 584–611.