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鉁掞笍 TITRE / TITLE

Quantum Fields, Gravity, and Mathematics

馃搫 R脡SUM脡 / ABSTRACT

I will describe recent progress on the our understanding quantum field theory and quantum gravity. In particular, I will review the theoretical evidence that classical space-time geometry emerges from the entanglement of more fundamental quantum mechanical degrees of freedom, and that -- in a sense -- space-time *is* entanglement. This has led to the idea that Einstein鈥檚 theory of general relativity, which describes the fluctuating geometry of space-time, is a chaotic quantum mechanical system much like a random matrix theory. In making this precise, we will discover some interesting mathematics. For example, the Siegel-Weil formula of number theory allows us to interpret the average over a space of quantum mechanical systems as a the sum over hyperbolic three-manifolds (essentially, states of a black hole) in quantum gravity. This comes from an analogy - and in some cases an equivalence - between general relativity and the theory of random lattices (or sphere packings) in high dimension. Similarly, the Hawking鈥檚 famous formula for the entropy of a black hole leads to a new conjecture for the number of holomorphic sections of line bundles over the moduli space of Riemann surfaces. This talk is intended to be accessible to mathematicians without a specialized physics background.