A GLIMPSE OF HIGH-DIMENSIONAL COMBINATORICS
Nati Linial, Hebrew Univ., School of Computer Science and Eng.
Have you asked yourself why graphs are so ubiquitous in so many real-world applications of mathematics? My (at least partial) explanation to this phenomenon is that there are numerous situations in which we study some large complex system whose features are determined by the nature of the pairwise relations among its constituents. These can be interacting molecules in a biological systems, two companies doing business together, or a pair of computers that communicate with each other. But what do you do with a large system whose constituents create 3-way (or, in general, k-way) interactions?
Our supply of mathematical tools for dealing with such situations is much more limited. One possible approach to this problem is to try and develop a combinatorial theory of higher-dimensional simplicial complexes. (Recall that a graph is a one-dimensional simplicial complex). There are already some initial successes to report and in particular a fascinating theory of random simplicial complexes is starting to take shape. I will explain some of these recent developments.