The “generating function” of configuration spaces, as a source for explicit formulas and representation stability

Geometry Topology Seminar
Monday, September 16, 2019 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Nir Gadish – Massachusetts Institute of Technology –
Kevin Kordek

As countless examples show, sequences of complicated objects should be studied all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of (orbit-)configuration spaces: using the notion of twisted commutative algebras, which categorify exponential generating functions. With this idea the configuration space “generating function” factors into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it also gives rise to representation stability - a notion of homological stability for sequences of representations of differing groups.