**Title:** Random Surfaces, the Gaussian Free Field, and the KPZ formula

**Speaker:** Boris Hanin

**Speaker Info:** Northwestern University

**Brief Description:**

**Special Note**:

**Abstract:**

How could we choose a "random" two dimensional Riemannian manifold and what are its (expected) properties? My talk is an elementary introduction to this question. Measures on the space of two dimensional Riemannian manifolds (and their singular limits) are much studied by physicists for their connections to conformal field theory and string theory (among other things). In string theory, for example, the analog of the path integral, which is an integral over histories of a zero dimensional particle, is an integral over the space of random surfaces, which can be thought of as histories of one dimensional strings.In recent years, mathematicians have understood how to rigirously study the Gaussian Free Field, a surface chosen according to a particular (Gaussian) measure on a particular space of (possibly singular) two dimensional Riemannian manifolds. The purpose of my talk is to introduce the Gaussian Free Field and the KPZ formula, which allows us to compute the expected dimension of a curve lying on the Gassian Free Field.

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