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.)
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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
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.
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.
Title: Universality in nonasymptotic random matrix theory
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.