Hiro Lee Tanaka, Texas State University

Abstract: One of the great applications of infinity-categories is the ability to compute localizations without model categories. We apply this to prove a surprising result in symplectic geometry: A certain 1-category of symplectic manifolds (really, Liouville sectors) localizes to an infinity-category that computes the homotopy type of the correct geometric mapping spaces! Even better, one can construct a further localization that (conjecturally) allows for purely symplectic constructions of localizations of the stable homotopy category. This is based on joint work with Oleg Lazarev and Zack Sylvan. If I have time (which I will not), I hope to talk about applications and some interesting consequences of some of our techniques -- for example, a proof that stabilized manifolds are the same thing as spaces over BO; that wrapped Fukaya categories are functorially sensitive to the homotopy type of embedding spaces; and that (geometric) flexibilization is a (categorical) localization.