Non-commutative rank

Computer Science/Discrete Mathematics Seminar II
Topic:Non-commutative rank
Speaker:Visu Makam
Affiliation:University of Michigan; Member, School of Mathematics
Date:Tuesday, February 5
Time/Room:10:30am - 12:30pm/Simonyi Hall 101
Video Link:https://video.ias.edu/csdm/2019/0205-VisuMakam

A linear matrix is a matrix whose entries are linear forms in some indeterminates $t_1,\dots, t_m$ with coefficients in some field $F$. The commutative rank of a linear matrix is obtained by interpreting it as a matrix with entries in the function field $F(t_1,\dots,t_m)$, and is directly related to the central PIT (polynomial identity testing) problem. The non-commutative rank of a linear matrix is obtained by interpreting it as a matrix with entries over the free skew field. The non-commutative rank could be larger than the commutative rank. Many incarnations and applications of non-commutative rank across areas of math, physics and CS have been found, some recently, and there is scope for more. 

In this talk, I will discuss several interesting structural and algorithmic aspects of non-commutative rank, as well as the role it plays in non-commutative rational identity testing and tensor rank lower bounds. Time permitting, I will indicate the connections to invariant theory for quivers, operator scaling and Brascamp--Lieb inequalities. 

No special background beyond standard linear algebra will be assumed.