Many-vertex triangulations of manifolds

Speaker: Lucy Tobin 
Affiliation: University of Sydney

Abstract

In this talk, I'll discuss an interesting combinatorial puzzle with implications in 4-dimensional topology: take $n$ $d$-dimensional simplices, each with $d+1$ vertices, and glue them together along their faces until there are no faces left unglued. What is the maximum number of distinct vertices you can end up with? And does the answer change if we impose that the space this forms is a manifold, or some particular manifold, or something almost-but-not-quite a manifold? In dimension 4 these turn out to be very hard questions, but with a nice little graph-theoretic counting argument we can answer them in every odd dimension. I'll guide you through the essence of that argument, and discuss some progress in the 4-dimensional case: partial results, interesting phenomena, and how it relates to understanding the complexity of 4-manifolds.