Associate Professor, PACM , ORFE

Associated Faculty, Department of Mathematics, Princeton University Web: https://web.math.princeton.edu/~rvan/

will deliver the Ashok Maitra Memorial Lectures 2022-23 at

Time: 3.30 - 4.30 PM

Title: Nonasymptotic random matrix theory

Abstract: What does the spectrum of a random matrix look like when we make no assumption whatsoever about the mean and covariance pattern of its entries? It may appear hopeless that anything useful can be proved at this level of generality, which lies far outside the scope of classical random matrix theory. The aim of my lecture is to describe the basic ingredients of a new theory that provides sharp nonasymptotic information on the spectrum in an extremely general setting. This is made possible by an unexpected phenomenon: the spectra of essentially arbitrarily structured random matrices turn out to be accurately captured by certain models of free probability theory under surprisingly minimal assumptions. (Based on joint work with Afonso Bandeira and March Boedihardjo.)

Link to Join Zoom:

https://us02web.zoom.us/j/2215256633?pwd=TmdNbHVGNDVPdGRnNzhDejhDMUpTZz09

Meeting ID: 221 525 6633

Passcode: SeminarSMU

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https://www.youtube.com/channel/UCvSVZ1dDHlchT3iHGnp2CUQ/

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Time: 2.00 - 3.00 PM

Title: Vector concentration inequalities

Abstract: The concentration of measure phenomenon asserts that in many cases, nonlinear (scalar-valued) functions of high-dimensional distributions have small fluctuations. Nowadays, such phenomena are regarded as basic tools that are widely used in different areas of probability and related fields for the analysis of complex, high-dimensional models. In this lecture, I will discuss a seemingly innocuous question: is there a meaningful concentration of measure phenomenon for functions taking values in normed spaces? This question arises naturally in functional analysis, metric geometry, and probability theory, but it is not even clear a priori how it may be formulated. The "correct" way to think about this problem was discovered by Pisier in the 1980's, who proved a vector concentration principle for Gaussian measures. However, until recently this phenomenon was not known to hold in any other situation. A couple of years ago, the author discovered (with Paata Ivanisvili) a vector concentration principle on the discrete cube, which enabled the resolution of an old question of Enflo in the metric theory of Banach spaces. New developments since then suggest that there is a much more general theory lying behind all currently known results. I will aim to explain this circle of ideas, the applications that motivate them, and a variety of open problems that arise from them.

Time: 3.15 – 4.15 PM

Title: The extremals of the Alexandrov-Fenchel inequality

Abstract: The Alexandrov-Fenchel inequality is a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes. It is one of the central results in convex geometry, and has deep connections with other areas of mathematics. The characterization of its extremal bodies (i.e., its equality cases) is a long-standing open problem that dates back to the original works of Minokwski (1903) and Alexandrov (1937). The known extremals are already numerous and strikingly bizarre, and a fundamental but incomplete conjecture on their general structure, due to Loritz and Schneider, has remained wide open except in some very special cases. Significant new progress on these problems was made in joint works with Yair Shenfeld. In particular, we succeeded to characterize all extremals of the Alexandrov-Fenchel inequality for convex polytopes, which completely settles the combinatorial aspect of the problem. In this talk I will aim to give a flavor of what the extremals look like, and illustrate how they arise in an entirely different context in a combinatorial question due to Stanley. No prior background will be assumed.

Time: 4.20 - 5.20 PM

Venue: NAB-1 (Ground Floor, A.N. Kolmogorov Bhavan), ISI Kolkata

Title: The diabolical bubbles of H. Minkowski

Abstract: In the late 1800s, in the course of his study of classical problems of number theory, the young Hermann Minkowski discovered the importance of a new kind of geometric object that we now call a convex set. He soon developed a rich and beauiful theory for understanding such sets, laying the foundations of convex geometry that are widely used to this day. Among the most surprising observations of Minokwski's theory is that the classical isoperimetric theorem---which states that the ball has the smallest surface area among all bodies of a given volume (a fact known instinctively to any child who has played with soap bubbles)---is just one special case of a much more general phenomenon. When one fixes geometric parameters other than surface area, Minkowski discovered that the resulting "bubbles" can be strikingly bizarre---they may have spikes sticking out of them in any direction. A complete understanding of such objects has remained a long-standing open problem, with major progress being achieved only recently in joint work with Yair Shenfeld. In my lecture, I will aim to trace the history of these ideas from their origin to the present day. In particular, I will describe how they arise in different guises across a surprisingly wide range of mathematical disciplines, including geometry, analysis, algebra, and combinatorics.

Zoom link: https://us02web.zoom.us/j/81256221317?pwd=RGwzMVpsdUUrMzBnNnZpdnRiVHk3QT09

Meeting ID: 812 5622 1317

Passcode: 276011

Time: 4.20 - 5.20 PM

Venue: L-infinity room (5th Floor, A.N. Kolmogorov Bhavan), ISI Kolkata

Title: Universality in nonasymptotic random matrix theory

Abstract: Classical random matrix theory is largely concerned with the asymptotic behavior of special models of random matrices, such as matrices with i.i.d. entries, in high dimension. In this setting, a phenomenon called universality asserts that the behavior of such models is insensitive to the distributions of the entries. In this talk, I will disuss a nonasymptotic universality principle that opens the door to investigating a large class of essentially arbitrarily structured random matrices. When combined with recent developments in nonasymptotic random matrix theory (discussed in the Delhi Colloquium), this principle enables the investigation of nonhomogeneous, non-Gaussian random matrices that lie far outside the scope of classical random matrix theory. I will discuss several applications to random graphs, sample covariance matrices, and free probability. (Based on joint work with Tatiana Brailovskaya.)

Note: while this lecture complements the Delhi Colloquium, their contents are independent and attendance of the Delhi Colloquium is not assumed.

Zoom link: https://us02web.zoom.us/j/81398007181?pwd=ZncweC9EOGVWa0RKcVJ6ZG5oSlhqQT09

Meeting ID: 813 9800 7181

Passcode: 764630

Last modified : August 22, 2022