Rank zero and rank one sets of harmonic maps

Harmonic Functions on Classical Rank One Balls

Zero forcing sets and the minimum rank of graphs ∗

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper introduces a new graph parameter, Z(G), that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank for numerous families of graphs, often e...

Superrigidity in infinite dimension and finite rank via harmonic maps

We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.

Invertibility-preserving Maps of C∗-algebras with Real Rank Zero

In 1996, Harris and Kadison posed the following problem: show that a linear bijection between C∗-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras andΦ : A→ B is a linear map onto B that preserves the spectrum of elements, thenΦ is a Jordan isomorphism if either A or B is a C∗-al...

Fooling-sets and rank

ABSTRACT. An n×n matrix M is called a fooling-set matrix of size n if its diagonal entries are nonzero and Mk,lMl,k = 0 for every k 6= l. Dietzfelbinger, Hromkovič, and Schnitger (1996) showed that n ≤ (rkM)2 , regardless of over which field the rank is computed, and asked whether the exponent on rkM can be improved. We settle this question. In characteristic zero, we construct an infinite fami...