Topological Combinatorics of Random Graphs and Complexes

Random graphs have become a very popular object of study in both topology and combinatorics, for both their interesting structural properties and their many applications to scientific problems. This walk will discuss some major results of random graph theory involving threshold functions of graph properties, including graph density, graph evolution, and k-connectivity. Random graphs provide a natural bridge to studying higher dimensional objects called "random complexes" which have interesting structural properties in their own right similar to those of random graphs. This talk will delve into the topological combinatorics of random complexes as well, in both the simplicial and cubical settings.