Topology and Geometry
Department of Mathematics and Statistics
Texas Tech University
In this talk, I will indicate how the sheaf topos of smooth sets serves as a sufficiently powerful and convenient context to host classical (bosonic) Lagrangian field theory. As motivation, I will recall the textbook description of variational Lagrangian field theory, and list desiderata for an ambient category in which this can rigorously be phrased. I will then explain how sheaves over Cartesian spaces naturally satisfy all the desiderata, and furthermore allow to rigorously formalize several more field theoretic concepts. Time permitting, I will indicate how the setting naturally generalizes to include the description of (perturbative) infinitesimal structure, fermionic fields, and (gauge) fields with internal symmetries. This is based on joint work with Hisham Sati. Please email the seminar organizer for Zoom meeting details.
In this talk I will discuss applications of derived differential geometry to study a non-perturbative generalisation of classical Batalin–Vilkovisky (BV-)formalism. First, I will describe the current state of the art of the geometry of perturbative BV-theory. Then, I will introduce a simple model of derived differential geometry, whose geometric objects are formal derived smooth stacks (i.e. stacks on formal derived smooth manifolds), and which is obtained by applying Töen-Vezzosi’s homotopical algebraic geometry to the theory of derived manifolds of Spivak and Carchedi-Steffens. I will show how derived differential geometry is able to capture aspects of non-perturbative BV-theory by means of examples in the cases of scalar field theory and Yang-Mills theory.All sorts of algebro-geometric moduli spaces (of stable curves, stable sheaves on a CY 3-folds, flat bundles, Higgs bundles...) are best understood as objects in derived geometry. Derived enhancements of classical moduli spaces give transparent intrinsic meaning to previously ad-hoc structures pertaining to, for instance, enumerative geometry and are indispensable for more advanced constructions, such as categorification of enumerative invariants and (algebraic) deformation quantization of derived symplectic structures. I will outline how to construct such enhancements for moduli spaces in global analysis and mathematical physics, that is, solution spaces of PDEs in the framework of derived ${\rm C}^\infty$ geometry and discuss the elliptic representability theorem, which guarantees that, for elliptic equations, these derived moduli stacks are bona fide geometric objects (Artin stacks at worst). If time permits some applications to enumerative geometry (symplectic Gromov-Witten and Floer theory) and derived symplectic geometry (the global BV formalism).I will present a new proof of Berwick-Evans, Boavida de Brito, and Pavlov’s theorem that for any smooth manifold A, and any sheaf X on the site of smooth manifolds, the mapping sheaf Hom(A,X) has the correct homotopy type. The talk will focus on the main innovation of this proof, namely the use of test categories to construct homotopical calculi on locally contractible ∞-toposes. With this tool in hand I will explain how a suitable homotopical calculus may be constructed on the ∞-topos of sheaves on the site of smooth manifolds using a new diffeology on the standard simplices due to Kihara. The main theorem follows using a similar argument that for any CW-complex A, and any topological space X the set of continuous maps Hom(A,X) equipped with compact-open topology models the mapping-homotopy-type map(A,X).Classical vector valued paths are widespread across pure and applied mathematics: from stochastic processes in probability to time series data in machine learning. Parallel transport of such paths in principal G-bundles have provided an effective method to characterise such paths. In this talk, we provide a brief overview of these results and their applications. We will then discuss recent work on extending this framework to characterizing random and possibly nonsmooth surfaces using surface holonomy. This is based on joint work with Harald Oberhauser.Using previous work by Bass, Dundas, and Rognes giving a geometric model of the iterated K-theory spectrum K(ku) in terms of bundles of Kapranov-Voevodsky 2-vector spaces, and recent work by Grady and Pavlov providing a rigorous foundation for fully-extended functorial field theories, we construct a model of K(ku) in terms of 2-dimensional functorial field theories valued in KV 2-vector spaces. 