Integral forms are characteristic supergeometric objects that allow us to define a meaningful notion of integration on supermanifolds. In this talk, I will introduce a double complex of non-commutative sheaves that relates integral forms to the more customary notion of differential forms. I will then discuss how this framework specializes to so-called cotangent bundle supermanifolds, which are relevant to odd symplectic geometry and BV theory. If time permits, I will explain how the geometry of forms is related to the problem of splitting a complex supermanifold in this particular setting.

A factorization algebra is a cosheaf-like local-to-global object which is meant to model the structure present in the observables of classical and quantum field theories. In the Batalin–Vilkovisky (BV) formalism, one finds that a factorization algebra of classical observables possesses, in addition to its factorization-algebraic structure, a compatible Poisson bracket of cohomological degree +1. Given a "sufficiently nice" such factorization algebra on a manifold N, one may associate to it a factorization algebra on N x [0,∞). The aim of the talk is to explain the sense in which the latter factorization algebra "knows all the classical data" of the former. This is the bulk-boundary correspondence of the title. Time permitting, we will describe how such a correspondence appears in the deformation quantization of Poisson manifolds.

Chern–Simons theories with gauge supergroups appear naturally in string theory and they possess interesting applications in mathematics, e.g. for the construction of knot and link invariants. In my talk I will review the framework for combinatorial quantization of Chern–Simons theory and explain how this framework can be adapted for applications to superalgebras. This will give rise to interesting new observables which can be computed by exploiting the rich representation theory of Lie superalgebras.

Weak higher categories can be difficult to work with algebraically, with the weak structure potentially leading to considerable bureaucracy. Conjecturally, every weak $\infty$-category is equivalent to a "semistrict" one, in which unitors and associators are trivial; such a setting might reduce the burden of constructing large proofs. In this talk, I will present the proof assistant homotopy.io, which allows direct construction of composites in a finitely-generated semistrict $(\infty,\infty)$-category. The terms of the proof assistant have an interpretation as string diagrams, and interaction with the proof assistant is entirely geometrical, by clicking and dragging with the mouse, completely unlike traditional computer algebra systems. I will give an outline of the underlying theoretical foundations, and demonstrate use of the proof assistant to construct some nontrivial homotopies, rendered in 2d, 3d, and in 4d as movies. I will close with some speculations about the possible interaction of such a system with more traditional type-theoretical approaches. (Joint work with Nathan Corbyn, Calin Tataru, Lukas Heidemann, Nick Hu and David Reutter.)

I will sketch the construction of the Fukaya-Morse category of a Riemannian manifold $X$ — an $A_\infty$ category (a category where associativity of composition holds only up-to-homotopy) where the higher composition maps are given in terms of numbers of embedded trees in $X$, with edges following the gradient trajectories of certain Morse functions. I will give simple examples and explain different approaches to understanding the structure and proving the quadratic relations on the structure maps — (1a) via homotopy transfer, (1b) effective field theory approach, (2) topological quantum mechanics approach. The talk is based on a joint work with O. Chekeres, A. Losev and D. Youmans, arXiv:2112.12756.