In this talk we will discuss some some extremal problems for polynomials. Applications to the problems in discrete dynamical systems as well as in the geometric complex analysis will be suggested.
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Dynamical sampling is a new area in
sampling theory that deals with signals that evolve over time under the
action of a linear operator. There are lots of studies on various
aspects of the dynamical sampling problem. However, they all focus on
uniform
discrete time-sets $\mathcal T\subset\{0,1,2,\ldots, \}$. In our study,
we concentrate on the case $\mathcal T=[0,L]$. The goal of the
present work is to study the frame property of the systems
$\{A^tg:g\in\mathcal G, t\in[0,L] \}$. To this end, we also
In this talk I will discuss the Mikhlin-H\"ormander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. I will show that this theorem does not hold in the limiting case $|1/p - 1/2|=s/n$. I will also present a sharp variant of this theorem involving a space of Lorentz-Sobolev type. Some of the results presented in this talk were obtained in collaboration with Loukas Grafakos.
The n-dimensional L^p Brunn-Minkowski inequality for p<1 , in particular the log-Brunn-Minkowski inequality, is proposed by Boroczky-Lutwak-Yang-Zhang in 2013, based on previous work of Firey and Lutwak . When it came out, it promptly became the major problem in convex geometry. Although some partial results on some specific convex sets are shown to be true, the general case stays wide open. In this talk I will present a breakthrough on this conjecture due to E.
Abstract: Let $(M,g)$ be a compact Riemannian n-manifold without boundary. Consider
the corresponding $L^2$-normalized Laplace-Beltrami eigenfunctions. Eigenfunctions
of this type arise in physics as modes of periodic vibration
of drums and membranes. They also represent stationary states of a free
quantum particle on a Riemannian manifold. In the first part of the
lecture, I will give a survey of results which demonstrate how the
geometry of $M$ affects the behaviour of these special
functions, particularly their “size” which can be quantified by
In this talk we shall explore some of the consequences of the solution
to the Kadison-Singer problem. In the first part of the talk we present
results from a joint work with Itay Londner. We show that every subset $S$ of the torus of positive Lebesgue measure admits a Riesz sequence of
exponentials $\{ e^{i\lambda x}\} _{\lambda \in \Lambda}$ in $L^2(S)$
such that $\Lambda\subset\mathbb{Z}$ is a set with gaps between
consecutive elements bounded by $C/|S|$. In the second part of the talk
we shall explore a higher rank extension of the main result of Marcus,
Koldobsky showed that for an arbitrary measure on R^n, the measure of the largest section of a symmetric convex body can be estimated from below by 1/sqrt{n}, in with the appropriate scaling. He conjectured that a much better result must hold, however it was recemtly shown by Koldobsky and Klartag that 1/sqrt{n} is best possible, up to a logarithmic error. In this talk we will discuss how to remove the said logarithmic error and obtain the sharp estimate from below for Koldobsky's slicing problem.
I will discuss some free probability inequalities on the circle which can be seen in two different ways, one is via random matrix approximation, and another one by itself. I will show what I believe to be the key of these new forms, namely the fact that the circle acts on itself. For instance the Poincare inequality has a certain form which reflects this aspect. I will also briefly show how a transportation inequality can be discussed and how the standard Wasserstein distance can be modified to introduce this interesting phenomena.
Spherical averages, in the continuous and discrete setting, are a canonical example of averages over lower dimensional varieties. We demonstrate here a new approach to proving the sparse bounds for these opertators. This approach is a modification of an old technique of Bourgain.
I shall tell about some background and known results in regards to the celebrated and fascinating Log-Brunn-Minkowski inequality, setting the stage for Xingyu to discuss connections with elliptiic operators a week later.
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