kazhdan's property (t) is a representation-theoretic property of groups, which was introduced by kazhdan in 1967, but has found numerous applications in an amazingly large variety of subjects from representation theory and ergodic theory to combinatorics (expanders) and the theory of networks. i will start with a gentle introduction to the emerging subject of "noncommutative real algebraic geometry," a subject which deals with equations and inequalities in noncommutative algebra over the reals, with the help of analytic tools such as representation theory and operator algebras. i will then present a surprisingly simple proof that a group $g$ has property (t) if and only if a certain inequality in the group algebra ${\bf r}[g]$ is satisfied. this inequality is rather amenable to computer-assisted analysis and is useful in finding new examples of property (t) groups or better bounds of the kazhdan constants of known property (t) groups.