### Course 2: New trends in high dimensional expanders based on Ramanujan complexes and their applications

#### Instructor: Ori Parzanchevski

##### Lecture 1: Expanders and Ramanujan graphs

We will recall what are expanders graphs, and review some of their combinatorial properties. From the spectral theory of regular trees we will arrive at the notion of Ramanujan graphs, and then glimpse how number theory appears in the construction of such graphs.

##### Lecture 2: Buildings and Ramanujan complexes.

We will explain the problem of constructing high-dimensional expanders, and in particular the lack of random constructions. Then, we will present Ramanujan complexes as a highdimensional generalization of Ramanujan graphs. For this purpose we will describe what are Bruhat-Tits buildings, which are the high-dimensional analogues of regular trees.

##### Lecture 3: Expansion in Ramanujan complexes

We will study the spectral theory of Ramanujan complexes by two methods: a combinatorial analysis which will allow us to deduce global expansion from local expansion, and representation theory, which will give us optimal global results. No previous knowledge of representation theory will be assumed..

##### Lecture 4: Random walks on Ramanujan digraphs and complexes.

We will describe some notions of random walks on Ramanujan complexes, and show that they converge optimally fast to the uniform distribution. The analysis will involve combinatorics, representation theory, and the notion of Ramanujan digraphs, which will be expained as well.