Topology and Geometry
Department of Mathematics and Statistics
Texas Tech University
In this talk, we explain the construction of models for twisted Spin(^c) bordism and its Anderson dual, in homotopy-theoretic, geometric, and differential settings. The talk is based on joint work with Yuanchu Li.The easiest source of fully extended TQFTs is found in E_n-algebras: for any symmetric monoidal category V and E_n-algebra A in V, one automatically obtains an n-dimensional TQFT given by factorization homology. Sometimes, however, these TQFTs can be extended to (n+1)-dimensional ones. In this talk, I will explain how to characterize precisely when this extension is possible and when the resulting theory is invertible.Locally presentable (∞, 1) categories admit two popular models: Presentable quasicategories and combinatorial model categories. Building on the foundational work of Dugger and Lurie, Pavlov proved that their associated (∞, 1)-categories are equivalent, confirming a long-standing expectation. He then went on to conjecture that this should also hold multiplicatively, i.e., for presentably symmetric monoidal quasicategories and combinatorial symmetric monoidal model categories. Such an equivalence would be of foundational importance in higher algebra.
In arXiv:2603.23018, I proved this conjecture. The main difficulty is that existing techniques to rigidify quasicategories often break down multiplicatively. In the talk, I will explain how to overcome this. If time permits, I will also explain applications to enriched infinity operads (arXiv:2603.23019).The tangle hypothesis is a variant of the cobordism hypothesis that considers cobordisms embedded in some finite-dimensional Euclidean space (together with framings). Such tangles of codimension k can be organized into an Ek-monoidal d-category, where d is the maximal dimension of the tangles. The tangle hypothesis then asserts that this category of tangles is the free Ek-monoidal d-category with duals generated by a single object.
In this talk, based on joint work in progress with Yonatan Harpaz, I will describe an infinitesimal version of the tangle hypothesis: instead of showing that the Ek-monoidal category of tangles is freely generated by an object, we show that its cotangent complex is free of rank 1. This provides support for the tangle hypothesis (of which it is a direct consequence), but can also be used to reduce the tangle hypothesis to a statement at the level of Ek-monoidal (d+1, d)-categories by means of obstruction theory.