Topology and Geometry
Department of Mathematics and Statistics
Texas Tech University
 | Wednesday
26
3:30 PM
MA 115
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Severin Bunk
Mathematical Institute, University of Oxford
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-->We approach manifold topology by examining configurations of finite subsets of manifolds within the homotopy-theoretic context of $\infty$-categories by way of stratified spaces. Through these higher categorical means, we identify the homotopy types of such configuration spaces in the case of Euclidean space in terms of the category $\Theta_n$.Vertex algebras and factorization algebras are two approaches to chiral conformal field theory. Costello and Gwilliam describe how every holomorphic factorization algebra on the plane of complex numbers satisfying certain assumptions gives rise to a Z-graded vertex algebra. They construct some models of chiral conformal theory as factorization algebras. We attach a factorization algebra to every Z-graded vertex algebra.Bordism is an equivalence relation on manifolds and has been a powerful tool in algebraic topology for the last 60 years. A genus is a $\mathbb{Q}$-valued function on equivalence classes of manifolds (actually it's a ring homomorphism); genera have played an important role in 4-manifold topology, index theory, and homotopy theory. In this talk, I will recall the cobordism ring and discuss generators in terms of explicit manifolds. Next, I will introduce some examples of genera and then recall a general construction due to Hirzebruch. Finally, I will present an interpretation of genera through the lens of quantum field theory and share some recent computations.Algebraic quantum field theory (AQFT) is a time-honoured axiomatic approach to describe and study QFTs on Lorentzian manifolds. In this talk I will try to summarize some of our main insights and results about “levelling up” traditional AQFT to the higher categorical world, which leads to a refined framework that I believe is suitable for quantum gauge theories. I will focus on both the underlying higher algebraic structures, i.e., the question “What does an ∞-AQFT assign? And why?”, and on concrete examples (mostly toy-models) that one can construct through methods from derived algebraic geometry. This talk is based on a long-term research program with Marco Benini.Configuration space integrals and graph complexes have been used to study a number of problems in topology. We will discuss the application to cohomology of spaces of embeddings, which directly generalizes their application to Vassiliev knot invariants. In joint work with Komendarczyk and Volić, we showed that such integrals describe the whole cohomology of spaces of 1-dimensional pure braids in any Euclidean space of dimension at least 4. We related them to Chen’s iterated integrals and showed that the inclusion of 1-dimensional pure braids into 1-dimensional long links induces a surjection in cohomology. This motivated further work of ours on the relationship between higher-dimensional pure braids and string links. In work in progress, I have been using these integrals to relate classes of long links to classes of long knots in various dimensions.We give a gentle introduction to supermanifolds and the splitting problem. Supermanifolds are a mildly noncommutative geometry where the coordinate functions either commute or anticommute. We recount Batchelor's theorem: every supermanifold in the smooth category is noncanonically isomorphic to a vector bundle. Such an isomorphism is called a splitting. Splittings may not exist at all in the holomorphic category, which turns out to be deeply significant to string theory in a way we will sketch. We close with basic examples of splittings and nonsplittings.