# Beyond Kadison-Singer: paving and consequences

December 1 to December 5, 2014

at the

American Institute of Mathematics, San Jose, California

organized by

Bernhard Bodmann, Pete Casazza, and Adam Marcus

## Original Announcement

This workshop will be devoted to broadening the recent proof of the Kadison-Singer Problem and to exploring its consequences. Many experts in the field agree that the Kadison-Singer Problem has been resolved by the work of Marcus, Spielman and Srivastava (MSS) through verifying Weaver's equivalent $KS_2$ conjecture which was shown to be equivalent to Anderson's paving conjecture. As a consequence, other equivalent formulations in different fields of mathematics are now also known to be true. A strong interaction between researchers in the areas connected by this problem and by its proof promises to enlarge our understanding and to address further open problems.

Particular topics envisioned for the workshop are the following:

1. Find concrete implications of the positive solution to the Kadison Singer problem in the various areas of research where it has equivalent formulations.
1. Find the Anderson paving number for finite Parseval frames with Gram matrices whose diagonal is bounded by 1/2. From the proof of Kadison-Singer it is known that there is a universal number r of sets in the partition and it is also known that r > 2, but MSS currently does not give any direct information on r.
2. Verify the Feichtinger Conjecture directly. This requires generalizing the MSS proof to infinite systems, including the partitioning of each unit-norm Gabor frame into a universal number of Riesz basic sequences.
2. Fine-tune the proof mechanism developed by MSS and make it as constructive as possible.
1. Improve the required norm bound in the proof by MSS. If possible, show that given $\epsilon > 0,$ the Gram matrix for any finite Parseval frame whose diagonal is bounded by $1/2-\epsilon$ can be two-paved. It is known that two-paving with $\epsilon = 0$ fails. Currently, the proof of MSS requires the stronger condition that the norms are bounded by $1-\sqrt 2/2-\epsilon.$
2. Find an algorithm that constructs the partition which results from the MSS proof or show that such a selection would not be achievable with a polynomial time algorithm.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Improved bounds in Weaver and Feichtinger Conjectures
by  Marcin Bownik, Peter G. Casazza, Adam W. Marcus and Darrin Speegle
Restricted invertibility revisited
by  Assaf Naor and Pierre Youssef,  A journey through discrete mathematics, 657–691, Springer, Cham, 2017  MR3726618
Detecting Fourier subspaces
by  Charles A. Akemann and Nik Weaver,  J. Fourier Anal. Appl. 22 (2016), no. 4, 878-886  MR3528403