Topology and Geometry
Department of Mathematics and Statistics
Texas Tech University
In their seminal work, Freed and Hopkins studied the moduli space of topological, reflection positive, invertible, Euclidean field theories, providing a complete classification in terms of certain objects arising in stable homotopy theory. In this work, it was also conjectured that a similar classification holds in the case of nontopological field theories, and this conjecture is already being used in a variety of applications to condensed matter physics. In this talk, I will discuss a recent result which provides an affirmative answer to this conjecture. I will begin by reviewing motivation and background on reflection positive theories. Then I will state the conjecture and sketch of the proof.Partition Lie algebras are sophisticated algebraic objects introduced by Brantner–Mathew to control infinitesimal deformations in positive characteristics. This talk will present a Koszul duality between partition Lie algebras and specific complete filtered derived rings. This duality helps to understand the homotopy operations on partition Lie algebras. Additionally, a “many-object” version of this duality connects partition Lie algebroids with infinitesimal derived foliations in the sense of Toën–Vezzosi.Categorical spectra, developed by Stefanich, are a directed version of spectra where the suspension of pointed ∞-groupoids is replaced by that of pointed ω-categories. They are very useful for capturing stability phenomena in iterated categorifications, or for defining “∞-vector spaces”. In this talk, I will explain that they can be understood as a weak version of Lessard's ℤ-categories, a kind of category with arrows in all negative as well as positive dimensions, which allows for a more direct study of their structure.