3:00–4:00 pm
Eckhart Hall Room 202
Random Surfaces and Liouville Quantum Gravity
Abstract:
What is the most natural way of choosing a random surface (2d Riemannian manifold), say with the topology of the sphere? This question does not have an obvious answer since the space of all Riemannian metric tensors on the sphere is infinite-dimensional. One possible approach is to consider random triangulations of a sphere, with $n$ triangles, and take some sort of limit as $n\rta\infty$. This gives rise to a one-parameter family of random metric measure spaces called Liouville quantum gravity (LQG) surfaces, which can be thought of as ``canonical random surfaces". These surfaces have the same topology as the sphere, but very different geometric properties. For example, their Hausdorff dimension is strictly larger than two and the geodesics started from a fixed point form a tree-like structure. LQG surfaces are also of interest in physics, for example in bosonic string theory and conformal field theory. I will discuss the definition, motivation, and basic properties of LQG surfaces, assuming no background beyond a typical first-year graduate curriculum.
