We consider the computational complexity of some problems
dealing with matrix rank. Let E,S be subsets of a
commutative ring R. Let x_1, x_2, ..., x_t be variables.
Given a matrix M = M(x_1, x_2, ..., x_t) with entries
chosen from E union {x_1, x_2, ..., ... more >>>

We consider the conjecture stating that a matrix with rank
$o(n)$ and ones on the main diagonal must contain nonzero
entries on a $2\times 2$ submatrix with one entry on the main
diagonal. We show that a slightly stronger conjecture implies
that ... more >>>

Several researchers, including Leonid Levin, Gerard 't Hooft, and
Stephen Wolfram, have argued that quantum mechanics will break down
before the factoring of large numbers becomes possible. If this is
true, then there should be a natural "Sure/Shor separator" -- that is,
a set of quantum ... more >>>

In this paper, we investigate the low-depth circuit complexity of Group Isomorphism in the multiplication (Cayley) table model. We prove the first circuit lower bounds for Group Isomorphism: namely, we show that every family of depth-$2$ Boolean circuits deciding Group Isomorphism requires quasipolynomial-size. We complement this with upper bounds of ... more >>>

Given any fixed constant $0<\varepsilon<1$ and a matrix space
$\mathcal{B}=\langle B_1,\ldots,B_m\rangle\le\mathbb{Q}^{n\times n}$,
we give a deterministic $NC^3$ algorithm that outputs a matrix \(A\in\mathcal{B}\) such that $rank(A)\geq (1-\varepsilon) crk(\mathcal{B})$,
where $crk(\mathcal{B})$ denotes the maximum rank of a matrix in $\mathcal{B}$. This complements the recent breakthrough of Chatterjee, Ghosh, Gurjar, Raj, and ... more >>>