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The Exact Renormalization Group (ERG) is a technique which can be fruitfully applied to systems with local interactions that exhibit a large number of degrees of freedom per correlation length. In the first part of the talk I will give a very general overview of the ERG, focussing on its applications in quantum field theory (QFT) and critical phenomena. In the second part I will discuss how a particular extension of the formalism suggests a new understanding of correlation functions in QFTs, in general, and gauge theories in particular.

A system of spins with complicated interactions between them can have many possible configurations. Many configurations will be local minima of the energy, and to get from one local minimum to another requires changing the state of very many spins. A system like this is called a spin glass, and at low temperatures tends to get caught for very long times at a local minimum of energy, rather than reaching its true ground state. Indeed, in many cases, finding the ground state energy of a spin glass is a computationally hard problem, too hard to be solved on a classical computer or even a quantum computer in any reasonable amount of time. Which types of interactions give us computationally hard problems and spin glasses? I will survey what is known as we close in on finding the simplest complex spin systems.