Expander graphs have been studied intensively in the last 40 years. In recent years a theory of high-dimensional expanders is emerging. We will describe some aspects of this theory such as topological expanders, coboundary expanders and the search for a random model.
I will report on the latest results on super-approximation. Roughly super-approximation gives us the right condition in order to get a family of expanders out of the Cayley graphs of the congruence quotients of a group generated by finitely many rational matrices. I will mention a sum-product phenomenon in number fields which is used in the proof of super-approximation. Some of the applications of super-approximation will be mentioned at the end.
I will describe a notion of high dimensional expansion called "agreement expansion", that can be described as a "sheaf cohomology". Agreement expansion captures certain PCP questions and in particular abstracts low degree tests such as plane vs. plane or line vs. line.
I will then describe an agreement question on the finite-field Grassmannian which is a high dimensional version of the Raz-Safra plane vs. plane low degree test. We will discuss a hypothesis regarding agreement expansion on the Grassmannian. This hypothesis, if true, implies NP-hardness of 2:1 games, a variant of the unique games conjecture.
We explain what Ramanujan graphs are, and prove that there exist infinite families of bipartite Ramanujan graphs of every degree. Our proof follows a plan suggested by Bilu and Linial, and exploits a proof of a conjecture of theirs about lifts of graphs. Our proof of their conjecture applies the method of interlacing families of polynomials to Mixed Characteristic Polynomials. A bound on the roots of these polynomials will follow from a bound of Heilmann and Lieb on the roots of the matching polynomials of graphs. We also prove that there exist infinite families of irregular bipartite Ramanujan graphs.
Random graphs and expander graphs can be viewed as sparse approximations of complete graphs, with Ramanujan expanders providing the best possible approximations. We formalize this notion of approximation and ask how well an arbitrary graph can be approximated by a sparse graph. We prove that every graph can be approximated by a sparse graph almost as well as the complete graphs are approximated by the Ramanujan expanders: our approximations employ at most twice as many edges to achieve the same approximation factor. Our algorithms follow from the solution of a problem in linear algebra. Given an expression for a rank-n symmetric matrix A as a sum of rank-1 symmetric matrices, we show that A can be well approximated by a weighted sum of only O(n) of those rank-1 matrices.
A d-regular graph is Ramanujan if its non-trivial eigenvalues in absolute value are bounded by 2√(d-1). Recently Adam Marcus, Daniel Spielman and Nikhil Srivastava gave a positive answer to this question by showing that any bipartite d-regular Ramanujan graph has a 2-fold cover that is also Ramanujan. In this talk we shall discuss their approach and mention similarities with function field towers.
A two-hour course on expanders, thin subgroups of Lie groups, and superstrong approximation.
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