ABSTRACT: The classical Busemann-Petty Problem from the 1950s asked the following tomographic question:

Assuming you have two origin-symmetric convex bodies K and L in the n-dimensional Euclidean space satisfying the volume inequality

|K \cap \theta^{\perp}| \leq |L \cap \theta^{\perp}| for all \theta \in S^{n-1},

does it follow that |K| \leq |L|? The answer is affirmative for n \leq 4 and negative whenever n \geq 5. However, if K belongs to a certain class of convex bodies, the intersection bodies, then the answer to the Busemann-Petty problem is affirmative in all dimension. Several extensions of this result have been shown in the case of measures on convex bodies, and isomorphic results of the same type have been established. Moreover, the isomorphic Busemann-Petty problem is actually equivalent to the isomorphic slicing problem of Bourgain (1986), which remains open to this day.

In this talk, we will introduce the notion of an intersection function, provide a Fourier analytic characterization for such functions, and show some versions of the Busemann-Petty problem in this setting. In particular, we will show that if you have a pair of continuous, even, integrable functions f,g \colon \R^n \to \R_+ which satisfy [Rf] \leq [Rg], where R denotes the Radon transform, then one has that |f|_{L^{p+1}} \leq |g|_{L^{p+1}} provided that the function f^p is an intersection function.

This is based on a joint work with Alexander Koldobsky and Artem Zvavitch.

ABSTRACT: The study of the class L+2\mathcal{L}^{+\,2} of Hilbert space
operators which are the product of two bounded positive operators first
arose in physics in the early 60s. On finite dimensional Hilbert
spaces, it is not hard to see that an operator is in this class if and
only if it is similar to a positive operator. We extend the exploration
of L+2\mathcal{L}^{+\,2} to separable infinite dimensional Hilbert
spaces, where the structure is much richer, connecting (but not
equivalent) to quasi-similarity and quasi-affinity to a positive
operator. The generalized spectral properties of elements of
L+2\mathcal{L}^{+\,2} are also outlined, as well as membership in
L+2\mathcal{L}^{+\,2} among various special classes of operators,
including algebraic and compact operators.
This is joint work with Maximiliano Contino, Alejandra Maestripieri, and Stefania Marcantognini.

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ABSTRACT: In an economics model analyzed by Mossay and Tabuchi, freeness-of-trades matrices are introduced to represent loss incurred via trade between two entities. The corresponding paper proved that if the freeness-of-trade matrix is positive definite, then the model admits a unique equilibrium. We show that such matrices need not be positive definite by analyzing extreme rays on the cone of pseudo-metrics on finite points. Tangential results are explored.

ABSTRACT: Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial f in d noncommuting arguments, find an nc polynomial p_n, of degree at most n, to minimize c_n:=\|p_nf -1\|^2. (Here, the norm is the \ell^2 norm on coefficients.) We show that c_n\to 0 if and only if f is nonsingular in a certain nc domain (the row ball). As an application, we give a new, elementary proof of a theorem of Jury, Martin, and Shamovich on cyclic vectors for the d-shift.

SLIDES: Approximation_problem_for_free_polynomials

ABSTRACT: A sequence {z_n} in the unit disk is called an interpolating sequence if every interpolation problem f(z_n)=w_n, n=1, 2, 3,…, with {w_n} bounded, can be solved with a bounded analytic function f. Carleson (1958) proved that a sequence in the disk is interpolating if and only if it is separated in the pseudohyperbolic metric and generates a Carleson measure. In a relatively recent breakthrough paper, Aleman, Hartz, McCarthy, and Richter generalized Carleson’s theorem to multiplier algebras of spaces with the complete Pick property. In this talk, we will extend these results to pairs of kernels (k, \ell) such that k is a complete Pick factor of \ell, thus answering a question left open in that paper.

SLIDES: INTERPOLATING_SEQUENCES_FOR_PAIRS_OF_KERNELS_UF_SEMINAR

ABSTRACT: In this talk, we will talk about hyperplane conjecture by Bourgain and its equivalent formulation isotropic constant conjecture, i.e., there exists an absolute constant C > 0 such that L_{k} ≤ C for every n ≥1 and every convex body K in R^n which is verified for several classes of convex sets such as unconditional convex bodies, zonoids, duals to zonoids, etc. but the general case is still open. We will also see the best general bound known to date, which is given by K. Ball.

ABSTRACT: In this talk, I will present a simple mechanism combining log-concavity and majorization in the convex order to derive moments, concentration, and entropy inequalities for random variables that are log-concave with respect to a reference measure.