<p> "Scattering amplitudes in gauge theories and gravity
have extraordinary properties that are completely invisible in the textbook
formulation of quantum field theory using Feynman diagrams. In this usual
approach, space-time locality and quantum-mechanical unitarity are made
manifest at the cost of introducing huge gauge redundancies in our description
of physics. As a consequence, apart from the very simplest processes, Feynman
diagram calculations are enormously complicated, while the final results turn
out to be amazingly simple, exhibiting hidden infinite-dimensional symmetries. This
strongly suggests the existence of a new formulation of quantum field theory
where locality and unitarity are derived concepts, while other physical
principles are made more manifest. The past few years have seen rapid advances
towards uncovering this new picture, especially for the maximally
supersymmetric gauge theory in four dimensions.</p>
<p>These developments have interwoven and exposed
connections between a remarkable collection of ideas from string theory,
twistor theory and integrable systems, as well as a number of new mathematical
structures in algebraic geometry. In this talk I will review the current state
of this subject and describe a number of ongoing directions of
research."</p>