The Richard E. Phillips Distinguished Lecture Series was established in 1997 with generous endowment from the family of our late distinguished colleague Richard E. Phillips. The purpose of the Phillips Lectures is to advertise the utility and power of mathematics within the university, and to stimulate the interest of graduate students, postdocs and faculty. Interaction with graduate students and postdoctoral fellows is an integral part of the visit. Each series consists of three lectures delivered over a period of 4-5 days. The first lecture is targeted to a broad audience with diverse mathematical background and displays the utility of the subfield of mathematics. The second lecture is at the level of a mathematical colloquium, while the third is more focused and highlights technical aspects of the domain.

**March 12-14, 2018**

Alex Lubotzky

Einstein Institute of Mathematics

Hebrew University of Jerusalem

**Lecture 1: Real applications of non-real numbers: Ramanujan graphs**

Monday, March 12: Kellogg Center Auditorium 5:30 - 6:30 PM

The real numbers form a completion of the field of rational numbers. We will describe the fields of p-adic numbers which are different completions of the rationals. Once they are defined, one can study analysis and geometry over them. While being very abstract, the main motivation for studying them came from number theory. Developments in the last 2-3 decades shows various applications to the real world: communication networks, etc. This is done via expander graphs and Ramanujna grpahs which are "Riemann surfaces over these p-adic fields". All notions will be explained.

**Lecture 2: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes**

Tuesday, March 13: 115 International Center 4:00 – 5:00 PM

Expander graphs in general, and Ramanujan graphs, in particular, have played a major role in combinatorics and computer science in the last 4 decades and more recently also in pure math. Approximately 10 years ago, a theory of Ramanujan complexes was developed by Li, Lubotzky-Samuels-Vishne and others. In recent years a high dimensional theory of expanders is emerging. The notions of geometric and topological expanders were defined by Gromov in 2010 who proved that the complete d-dimensional simplicial complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. Ramanujan complexes were shown to be geometric expanders by Fox-Gromov-Lafforgue-Naor-Pach in 2013, but it was left open if they are also topological expanders. By developing new isoperimetric methods for "locally minimal small" F_2- co-chains, it was shown recently by Kaufman- Kazdhan- Lubotzky for small dimensions and Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov's original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders. We will describe these developments and the general area of high dimensional expanders and some of its open problems.

**Lecture 3: Groups' approximation, stability and high dimensional expanders**

Wednesday, March 14: C304 Wells Hall 10:00 – 11:00 AM

Several well-known open questions (such as: are all groups sofic or hyperlinear?) have a common form: can all
groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or
the unitary groups U(n) (in the hyperlinear case)?

In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for
the first time, one of these versions, showing that there exist fintely presented groups which are not
approximated by U(n) with respect to the Frobenius (=L_2)norm.

The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is
proven to imply stability and using higher dimensional expanders, it is shown that some non-residually
finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence
cannot be Frobenius approximated.

All notions will be explained. Joint work with M. De Chiffre, L. Glebsky and A. Thom.

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