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Quantum field theory (QFT) is a unified language that describes many diverse
phenomena in particle and condensed matter physics, cosmology, quantum gravity, and string theory. Much
past and recent progress in QFT has been catalyzed by a simple question: what is the right way to count
the number of degrees of freedom described by a given quantum field theory? In this talk, I will explain
the challenges involved in constructing a physically acceptable counting function and describe some of its
uses. A crucial role is played by the intuitive idea that the effective number of degrees of freedom
should decrease as we examine a physical system at longer and longer distances. While such a counting
function has long been known for QFTs in two spacetime dimensions, progress in higher dimensions has been
much more recent. I will review these developments and highlight open problems as well as avenues of
current research. This talk should be accessible to beginning graduate students -- no background in quantum
field theory is assumed.
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