Optimal expansion in groups and complexes Part 1: Ori Parzanchevski
The main theme of these talks is the relations between explicit constructions of Ramanujan complexes and results on optimal expansion in finite groups and compact Lie groups. The plan of the talks is:
1. Explicit constructions — We will explain how specially chosen arithmetic lattices give rise to explicit constructions of Ramanujan (and non-Ramanujan) complexes, as clique complexes of finite Cayley graphs.
2. Optimal expansion — We will survey the study of random walks on Ramanujan complexes, and show that the total-variation mixing time of certain walks is optimal in an appropriate sense, and that these walks exhibit Diaconis' cutoff phenomenon. An interesting notion which will arise in this study is that of Ramanujan digraphs.
3. Expansion in groups — From the perspective of group theory, the Lubotzky-Phillips-Sarnak (LPS) construction gives an optimal expansion result for (certain Cayley graphs of) the finite group PGL_2(q). It is natural to seek similar connections between PGL_d(q) and Ramanujan complexes, and we shall explain such a connection, which is more involved than the graph case. In addition to Ramanujan graphs, the work of LPS produced optimal topological generators for the Lie group PU(2), which have recently found an application to quantum computations on a single qubit (so-called “Golden Gates”). Explicit constructions of Ramanujan complexes can again be connected to optimal generators of PU(d), which correspond to computations on more qubits, but again the story becomes more complicated, and much is still unknown.
Exercise: https://algebra.ugent.be/hdx/files/Ex...