INTERDISCIPLINARY STOCHASTIC PROCESSES COLLOQUIUM [Contact David Aldous if interested in going to dinner after talk] Tuesday November 16, room 60 Evans, 4.10 - 5.00pm Speaker: Scott Sheffield, UC Berkeley Title: Schramm Loewner evolution and the Gaussian free field: Part I Abstract: The Gaussian free field (called the Euclidean bosonic massless free field in the physics literature) is a higher dimensional analog of Brownian motion. It is also a fundamental object in statistical physics and a starting point for many constructions in quantum field theory. Let D be a planar domain and H(D) the Hilbert space of boundary-vanishing functions defined by the Dirichlet inner product. (The Dirichlet inner product of f and g is the integral over D of the dot product of their gradients.) An instance h of the Gaussian free field is a Gaussian element of H(D) -- i.e., h = SUM a_i f_i where the a_i are i.i.d. one-dimensional Gaussians and the f_i are an orthonormal basis for H(D). This sum diverges pointwise, but h is well defined as a random distribution. We describe several basic properties of discrete and continuum Gaussian free fields. In particular, we define "level sets" of h and show that they look locally like the celebrated Schramm-Loewner evolution, SLE(4). This talk is based on joint work with Oded Schramm. [Part 2 will be given in the Probability Seminar, Wednesday November 17, 334 Evans, 3.10 - 4.00]