Conjectures from SampTA

Back in May, I attended this year’s SampTA at American University. I spoke in a special session on phase retrieval, and as luck would have it, Cynthia Vinzant spoke in the same session about her recent solution of the 4M-4 conjecture. As you might expect, I took a moment during my talk to present the award I promised for the solution:

award

Recall that Cynthia (and coauthors) first proved part (a) of the conjecture, and then recently disproved part (b). During her talk, she also provided a refinement of part (b). Before stating the conjecture, recall that injectivity of the mapping x\bmod\mathbb{T}\mapsto |Ax|^2 is a property of the column space \mathrm{im}(A).

Vinzant’s Refined Injectivity Conjecture. Draw \mathrm{im}(A) uniformly from the Grassmannian of d-dimensional subspaces of \mathbb{C}^{4d-5}. Let p_d denote the probability that the mapping x\bmod\mathbb{T}\mapsto |Ax|^2 is injective.

(a) p_d < 1 for all d.

(b) \displaystyle{\lim_{d\rightarrow \infty}p_d=0}.

In words, she conjectures that part (b) of the original conjecture is asymptotically true in a probabilistic sense. My favorite part is that Cynthia offered to pay forward the prize she won: She will give a can of Coca-Cola for a proof of part (a), and US$100 for a proof of part (b).

Later in the week, Matt Fickus spoke about some recent work we did on equianglar tight frames (ETFs) with John Jasper and Jesse Peterson. During his talk, he mentioned how disappointing it is that we currently lack necessary conditions on the existence of complex ETFs. In response to this void, Matt offered the following conjecture:

The Fickus Conjecture. Consider the three quantities M, N-M, and N-1. An equiangular tight frame of N vectors in \mathbb{C}^M exists only if one of these quantities divides the product of the other two.

This simple conjecture coincides with everything we currently know about ETFs. To keep things interesting, Matt also offered a prize: He will give US$200 for a proof, and US$100 for a disproof.

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