**Date & Time:** Wednesday, 29th June, 2011 at 15:30 hrs.

**Venue:** Ramanujan Hall

**Title:** The Hardness of the Noncommutative Determinant

**Speaker:** Prof. Srikanth Srinivasan, IAS Princeton.

**Abstract:**
We study the computational complexity of computing the
noncommutative determinant. We first consider the arithmetic circuit
complexity of computing the noncommutative determinant polynomial. Then,
more generally, we also examine the complexity of computing the determinant
(as a function) over noncommutative domains. Our hardness results are
summarized below:

1. We show that if the noncommutative determinant polynomial has small noncommutative arithmetic circuits then so does the noncommutative permanent, which would imply that the commutative permanent polynomial would have small commutative arithmetic circuits.

2. For any field F we show that computing the n X n permanent over F is polynomial-time reducible to computing the 2n X 2n (noncommutative) determinant whose entries are O(n^2) X O(n^2) matrices over the field F.

3. We also derive as a consequence that computing the n X n permanent over nonnegative rationals is polynomial-time reducible to computing the noncommutative determinant over Clifford algebras of n^{O(1)} dimension.

Our techniques are elementary and use primarily the notion of the Hadamard Product of noncommutative polynomials. Joint work with V Arvind.