A natural place where combinatorics and topology meet is in the study of simplicial complexes. These are very natural combinatorial objects which have a straightforward geometric/topological interpretation. It is, therefore, very appealing to consider simplicial complexes from the perspective of modern combinatorics. It is of particular interest to consider simplicial complexes from three perspectives which play a prominent role in modern combinatorics: Extremal, asymptotic and probabilistic. I will survey several results in this spirit and mention a number of intriguing open problems.
The results I will present are from joint papers with Roy Meshulam, Mishael Rosenthal and Lior Aronshtam.
No previous background in topology is required to follow this talk.