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This
talk concerns two-variable rational inner functions phi with
singularities on the two-torus T^2, the notion of contact order (and
related quantities), and its various uses. Intuitively, contact order is
the rate at which phi’s zero set approaches T^2 along a coordinate
direction, but it can also be defined via phi's well-behaved unimodular
level sets. Quantities like contact order are important because they
encode information about the numerical stability of phi, for example
when it belongs to Dirichlet-type spaces and when its partial
The centerpiece of the subject of integral geometry, as conceived originally by Blaschke in the 1930s, is the principal kinematic formula (PKF). In rough terms, this expresses the average Euler characteristic of two objects A, B in general position in Euclidean space in terms of their individual curvature integrals. One of the interesting features of the PKF is that it makes sense even if A and B are not smooth enough to admit curvatures in the classical sense.
In 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved.Their fifth problem asks the following.Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). (proportional to the volume of the cone spanned by the secion and the support point).
We will discuss several open problems concerning unique determination of convex bodies in the n-dimensional Euclidean space given some information about their projections or sectionson all sub-spaces of dimension n-1. We will also present some related results.
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.
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,