Alladi Ramakrishnan Hall

A proof of the Kadison-Singer problem using mixed determinants and three combinatorial conjectures.

Mohan Ravichandran

Minar-Sinan Fine Arts University

The solution of the Kadison-Singer problem by
Marcus, Spielman and Srivastava proceeded by first proving Weaver's
KS_r conjecture, which was known to imply Anderson's paving
conjecture,which in turn implies Kadison-Singer. In this talk, I'll
explain how the two celebrated innovations of MSS, the method of
interlacing polynomials and the multivariate barrier method can be
used to prove the paving conjecture directly without using Weaver's
KS_r. The
relevant expected characteristic polynomials are then succinctly given
by expressions related to the so called Mixed determinant, in the same
way as the Mixed characteristic polynomial of MSS is related to the
mixed discriminant.

This slight change in perspective allows us to make some modest
improvements in the estimates in Kadison-Singer and restricted
invertibility. For instance, one can show that projections with
diagonal 1/2 can be 4 paved (it is expected that they can be 3 paved,
but this method breaks down in a curious fashion at the last step).
This new approach also yields some remarkable combinatorial results,
including a natural, hitherto undiscovered one parameter deformation
of the characteristic polynomial of a matrix and a new convolution on
polynomials, (distinct from the three convolutions introduced by MSS
in 2015)
that preserves real rootedness. I will end with three natural and (to
my mind) fundamental combinatorial questions that are both intriguing
and are likely to have potential applications.