In this talk we present results related to the recent solution of the Kadison–Singer problem by Marcus, Spielman, and Srivastava. We sharpen the constant in the KS₂ conjecture of Weaver that plays a key role in this solution. We then apply this result to prove optimal asymptotic bounds on the size of partitions in the Feichtinger conjecture. The talk is based on a joint work with Casazza, Marcus, and Speegle.

Nuclear C*-algebras enjoy a number of approximation properties, perhaps most famously the completely positive approximation property. The CPAP was sharpened by Hirshberg, Kirchberg and White, who related it to noncommutative covering dimension. We discuss a further improvement, tying together various notions of finite dimensional approximation. This is joint work with Nate Brown and Stuart White.

This talk focuses on the maximum of relative projection constants over all m-dimensional subspaces of the N-dimensional coordinate space equipped with the max-norm. This quantity, called maximal relative projection constant, is studied in parallel with a lower bound, dubbed quasimaximal relative projection constant. Exploiting elegant expressions for these quantities, we show how they can be computed when N is small, before concentrating on reversing the Kadec–Snobar inequality when N does not tend to infinity. Precisely, we first prove that the (quasi)maximal relative projection constant can be lower-bounded by cm^1/2, with c arbitrarily close to one, when N is superlinear in m. The main ingredient is a connection with equiangular tight frames. We then prove that the lower bound cm^1/2 holds with c<1 when N is linear in m. The main ingredient is the semicircle law.

In this talk we revisit a result of Muckenhoupt and Wheeden, which gives a weighted version of the classical John–Nirenberg Theorem (specifically for Ap weights). We will discuss a modern proof of this result, using the recent machinery of sparse operators.

Classical interpolation deals with the problem of reconstructing a function from its values on a set of sample points. In the 1930’s, H. Whitney studied this problem and proved a theorem characterizing the existence of an interpolant with a prescribed degree of smoothness. In practice, due to measurement error, the function values will not be known exactly, but will instead be prescribed to belong to a set of possible values. This leads us to study the following variant of Whitney’s problem: Given a convex set K(x) in ℝ^D associated to each point x of ℝⁿ, under what circumstances does there exist a C^m,1-smooth function F:ℝⁿ→ℝ^D satisfying F(x)∈K(x)? We prove a “finiteness principle” which gives a set of necessary and sufficient conditions for the existence of such a smooth selection. Our results may be interpreted as a generalization of the classical Helly theorem in convex geometry. The proof is based on a stabilization lemma for a certain dynamical process on the space of convex sets.

We present some simplifications to the Gour–Friedman proof of local additivity and recast it using the relative modular operator. We also discuss implications for relative entropy and superadditivity of the Holevo capacity. Joint work with Jon Yard.

If a Banach space coarsely embeds into a superstable Banach space, then it must contain a basic sequence with an ℓ_p spreading model for some p∈[1,∞). The proof will be discussed, with comparisons made to an analogous result proved by Y. Raynaud, which says that if a Banach space uniformly embeds into a superstable Banach space, then it must contain a subspace isomorphic to ℓ_p for some p∈[1,∞). The result obtained implies that not every reflexive Banach space is coarsely embeddable into a superstable Banach space. This is joint work with B. M. Braga.

We shall derive a Khintchine inequality with sharp constants for linear forms of random vectors uniformly distributed on the unit ball of finite dimensional ℓ_p spaces. Joint work with A. Eskenazis and P. Nayar.