Dr. Gerard Ben Arous is a professor of mathematics and director of the Courant Institute of Mathematical Sciences at New York University.

**Title: **The Complexity of Random Functions of Many Variables

**Abstract: ** A function of many variables, when chosen at random, is typically very complex. It has a exponentially large number of local minima or maxima, or critical points. It defines a very complex landscape, the topology of its level lines (for instance their Euler characteristic) is surprisingly complex.

This complex picture is valid even in very simple cases, for for random homogeneous polynomials of degree p larger than 2. This has important consequences. For instance trying to find the minimum value of such a function may thus be very difficult.The mathematical tool suited to understand this complexity is the spectral theory of large random matrices. The classification of the different types of complexity has been understood for a few decades in the statistical physics of disordered media, and in particular spin-glasses, where the random functions may define the energy landscapes. It is also relevant in many other fields, including computer science and Machine learning.

I will review recent work with collaborators in mathematics (A. Auffinger, J. Cerny), statistical physics (C. Cammarota, G. Biroli, Y. Fyodorov, B. Khoruzenko), and computer science (Y. LeCun and his team at Facebook, A. Choromanska, L. Sagun among others), as well as recent work of E. Subag and E.Subag and O.Zeitouni.

**About the series:**

Thomas Stelson was a distinguished civil engineer who served as dean of Georgia Tech's College of Engineering from 1971 to 1974, as vice president for research from 1974 to 1988, and as executive vice president from 1988 to 1990. During the 70's and 80's, he oversaw a vast expansion in Tech's research expenditures during an era when Tech went from being primarily teaching-oriented university to a major research institution.

Stelson helped the School of Mathematics create the Center for Dynamical Systems and Nonlinear Studies, and he endowed the School's Stelson lectures in 1988 in honor of his father, Hugh Stelson, who was a mathematician. Hugh Stelson earned his doctorate from the University of Iowa in 1930 and went on to teach at Kent State University and Michigan State University. He worked on problems related to interest rates, annuities, and numerical analysis.

Some previous Stelson lectures are available to the public from the SMARTech repository, at https://smartech.gatech.edu/handle/1853/31161