# Realization of sign matrices in Euclidean spaces

## Noga Alon, Tel Aviv University and IAS, Princeton

## October 20, 2015

The *signrank* of an $N$ by $N$ real matrix $A$ with nonzero entries is the
minimum dimension $s$ so that there are hyperplanes $h_i$ through the
origin
and points $p_j$ in $R^s$ such that $p_j$ lies in the positive side of $h_i$
if
$A_{ij}=+1$, and in its negative side if $A_{ij}=-1$.

The study of this notion combines combinatorial, algebraic, geometric
and probabilistic techniques with tools from real algebraic geometry,
and
is related to questions in Discrete Geometry, Communication
Complexity,
Computational Learning and Asymptotic Enumeration. I will discuss the
topic focusing on recent results in joint work with Moran and
Yehudayoff,
and mention some intriguing open problems.