#### 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.

Done