It has long been conjectured that the crossing number cr(Km,n) of the complete bipartite graph Km,n is equal to Z(m,n) := bm−1 2 cb2 cbn−1 2 cb2 c. Another long-standing conjecture is that the crossing number cr(Kn) of the complete graph Kn is equal to Z(n) := 4b2 cbn−1 2 cbn−2 2 cbn−3 2 c. In this talk, I will outline a new method that improves the asymptotic lower bounds to 0.83Z(m, n) and 0....

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We study a generalization of the classical Marcinkiewicz-Zygmund inequalities. We relate this problem to the sampling sequences in the Paley-Wiener space and by using this analogy we give sharp necessary and sufficient computable conditions for a family of points to satisfy the Marcinkiewicz-Zygmund inequalities.

Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.

We consider the reduction of problems on general noncommutative Lp-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is an unpublished result of the first-named author which approximates any noncommutative Lpspace by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products...

Although Rogerian reflective listening is considered a fundamental therapeutic practice, it is widely misunderstood. This article endeavors to dispel myths about Rogers’ reflective approach through detailed readings of his work, while also opening up a central problematic in Rogers’ thinking. Rogers struggled repeatedly with the dilemma of how the therapist can faithfully reflect the client’s e...