Title: Landscape of random functions in many dimensions.
Speaker: Antonio Auffinger
Speaker Info: University of Chicago
How many critical values a "typical" Morse function have on a high dimensional manifold? Could we say anything about the topology of its level sets? In this talk we will address these type of questions in a particular but fundamental example. We investigate the landscape of a general Gaussian random smooth function on the N-dimensional sphere. These corresponds to Hamiltonians of well-known models of statistical physics, i.e spherical spin glasses. We will show that an interesting picture emerge for the bottom landscape of these random Hamiltonians that in particular implies a strong correlation between the index and the critical value. We also propose a new invariant for the possible transition between the so-called 1-step replica symmetry breaking and a Full Replica symmetry breaking scheme in statistical physics and show how the this counting problem is related to the Parisi functional. This talk is based on joint works with G. Ben Arous (Courant) and J. Cerny (ETHZ).Date: Monday, January 23, 2012