A glimpse of high-dimensional combinatorics
There are numerous situations where we study a large complex system whose behavior is determined by pairwise interactions among its constituents. These can be interacting molecules, companies involved in some business, humans etc. This explains, at least partially, why graph theory is so ubiquitous wherever mathematics is applied in the real world. However, there are numerous interesting and important situations where the underlying interactions involve more than two constituents. This applies in all the above-mentioned scenarios. The obvious place to look for a relevant mathematical theory is hypergraph theory. Unfortunately, this part of combinatorics is not nearly as well understood as graph theory. In this talk I will speak about a special class of hypergraphs, namely, simplicial complexes. It turns out that there is a fascinating combinatorial theory of simplicial complexes that is rapidly developing in recent years. In this lecture I explain some of the exciting recent findings in this area.
My coworkers in these investigations are (historical order): Roy Meshulam, Mishael Rosenthal, Lior Aronshtam, Tomasz Luczak, Yuval Peled, Yuri Rabinovich and Ilan Newman.