Fall 2022 Schedule

Date: Nov 21st

Speaker: Ben Goodman

Title: Forcing, Forcing Axioms, and their Generalizations

Abstract: Forcing, a central method of modern set theory, allows universes of sets to be enlarged in a controlled way by adjoining a new object that is "generic" in some sense, then adding all the sets which can be constructed from the new object, in a manner analogous to field extensions in algebra. Forcing axioms are proposed strengthenings of the standard axiomatization of set theory (ZFC) which assert that there already exist pseudogeneric objects which approximate objects that could be added by certain kinds of forcing. I will give a gentle introduction to forcing and forcing axioms, then briefly explain my own work on enhanced forcing axioms, which assert also that sets constructed from pseudogeneric objects should have some of the same properties that sets constructed in the same way from true generic objects would have.

Date: Nov 28th

Speaker: Zhihao Mu

Title: Cubulation of Hulls in Hierarchically Hyperbolic Spaces

Abstract: In hyperbolic spaces, a Gromov’s theorem states that the convex hull of a finite set can be approximated by a tree. The hierarchically hyperbolic space/group(HHS/HHG), introduced by Behrstock, Hegan and Sisto, is a powerful tool to study many interesting groups in geometric group theory, including mapping class groups, right-angled Artin groups and many 3-manifold groups. Analogous to Gromov’s theorem, the hulls in HHS can be cubulated by CAT(0) cube complexes. In this talk, I will introduce HHS briefly, explain the construction of quasi-isometry between hulls in HHS and CAT(0) cube complexes, as well as its applications.

Date: Dec 5th

Speaker: Emilio Minichiello

Title: Diffeological Spaces and Higher Stacks

Abstract: A diffeological space consists of a set X together with a collection D of set functions U -> X where U is a cartesian space, that satisfy three simple axioms. In this talk we will describe how this simple definition provides a new, powerful framework for differential geometry. Namely, every finite dimensional smooth manifold is a diffeological space, as are many infinite dimensional ones, orbifolds, and many other objects of interest in differential geometry. An incredible theorem of Baez and Hoffnung showed that the category of diffeological spaces is equivalent to the category of concrete sheaves on cartesian spaces. This theorem opens the floodgates to study diffeological spaces with the full power of higher sheaf theory. In this talk I will give an introduction to diffeological spaces and if time permits discuss some results from my preprint “Diffeological Principal Bundles and Principal Infinity Bundles.”