Department Mathematik



Mathematisches Kolloquium

Am Freitag, 06. Mai 2011, um 16 Uhr c.t. spricht

Prof. Dr. Gerard Ben Arous
(New York University)

im Hörsaal A027 über das Thema

Random Matrices and Morse Theory for random functions of many variables

Zusammenfassung: What is the typical complexity of a random Morse function on a high dimensional manifold? How many critical points of given index are there in a given level set? What is the Euler characteristic of a given level set? I will survey the recent progress of an on-going joint work on these questions, with Antonio Auffinger (Courant) and Jiri Cerny (ETH Zurich). We compute the mean number of critical points of given index, and the mean Euler characteristic of the level sets, for general Gaussian random smooth functions on the N-dimensional sphere. These functions correspond to Hamiltonians of well-known models of statistical physics, i.e general spherical spin glasses. We show that this counting boils down to a problem in Random Matrix Theory. This surprising link allows for a very detailed answer, showing that the complexity of these functions is exponentially large in the dimensions, and much more. It gives a rich picture of the geometry of the level sets of these functions, which will be related to the so-called Parisi replica symmetry breaking scheme, given in the physics literature. This talk will not assume much, I will describe the statistical mechanics (or geometric) model and motivations, as well as the necessary tools from Random Matrix Theory. Many questions remain open, I will try to detail a few.
Alle Interessierten sind hiermit herzlich eingeladen. Eine halbe Stunde vor dem Vortrag gibt es Kaffee und Tee im Sozialraum (Raum 448) im 4. Stock.
Treffpunkt zum Abendessen um 18.00 Uhr wird noch bekannt gegeben.