Miami University

Polycyclic groups form an interesting and well-studied class of groups that properly contain the finitely generated nilpotent groups. I will discuss the C*-algebras associated with virtually polycyclic groups, their maximal quotients and recent work with Jianchao Wu showing that they have finite nuclear dimension.

Many extremal results on cycles use what may be called BFS method, where a breath first search tree is used as a skeleton to build desired structures. A well-known example is the Bondy-Simonovits theorem that every n-vertex graph with more than 100kn^{1+1/k} edges contains an even cycle of length 2k. The standard BFS method, however, is not easily applicable for supersaturation problems where one wishes to show the existence of many copies of a given  subgraph. The method is also not easily applicable in the hypergraph setting.

In this talk, we focus on some variants of the standard BFS method. We use one of these in conjunction with some useful general reduction theorems that we develop to establish the supersaturation of loose (linear) even cycles in linear hypergraphs. This extends Simonovits' supersaturation theorem on even cycles in graphs. This is joint work with Liana Yepremyan.

If time allows, we will also discuss another variant (joint with Jie Ma) used in the study of Berge cycles of consecutive lengths in hypergraphs.