for this workshop
Beyond Kadison-Singer: paving and consequences
American Institute of Mathematics, Palo Alto, California
Bernhard Bodmann, Pete Casazza, Adam Marcus, and Daniel Spielman
This workshop, sponsored by AIM and the NSF, 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:
- Find concrete implications of the positive solution to the Kadison Singer
problem in the various areas of research where it has equivalent formulations.
- 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.
- 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.
- Fine-tune the proof mechanism developed by MSS and make it as constructive
- 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 $1-\sqrt 2/2-\epsilon.$
- 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.
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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