  A popular way to study N=1 supersymmetric gauge theories is to realize them
geometrically in string theory, as suspended brane constructions, D-branes
wrapping cycles in Calabi-Yau manifolds, orbifolds, and otherwise. Among the
applications of this idea are simple derivations and generalizations of Seiberg
duality for the theories which can be so realized.
  We abstract from these arguments the idea that Seiberg duality arises because
a configuration of gauge theory can be realized as a bound state of a
collection of branes in more than one way, and we show that different brane
world-volume theories obtained this way have matching moduli spaces, the
primary test of Seiberg duality.
  Furthermore, we do this by defining ``brane'' and all the other ingredients
of such arguments purely algebraically, for a very large class of N=1 quiver
supersymmetric gauge theories, making physical intuitions about brane-antibrane
systems and tachyon condensation precise using the tools of homological
algebra.
  These techniques allow us to compute the spectrum and superpotential of the
dual theory from first principles, and to make contact with geometry and
topological string theory when this is appropriate, but in general provide a
more abstract notion of ``noncommutative geometry'' which is better suited to
these problems. This makes contact with mathematical results in the
representation theory of algebras; in this language, Seiberg duality is a
tilting equivalence between the derived categories of the quiver algebras of
the dual theories.
