Coisotropic Submanifolds in b-symplectic Geometry

We study coisotropic submanifolds of b-symplectic manifolds. We prove that b-coisotropic submanifolds (those transverse to the degeneracy locus) determine the b-symplectic structure in a neighborhood, and provide a normal form theorem. This extends Gotay’s theorem in symplectic geometry. Further, we introduce strong b-coisotropic submanifolds and show that their coisotropic quotient, which locally is always smooth, inherits a reduced b-symplectic structure.

Deformation spaces and normal forms around transversals

Given a manifold $M$ with a submanifold $N$, the deformation space ${\mathcal{D}}(M,N)$ is a manifold with a submersion to $\mathbb{R}$ whose zero fiber is the normal bundle $\unicode[STIX]{x1D708}(M,N)$, and all other fibers are equal to $M$. This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with $N$ a submanifold transverse to the foliation. New examples include $L_{\infty }$-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around $N$, in terms of a model structure over $\unicode[STIX]{x1D708}(M,N)$.

Frobenius Structures

A Frobenius structure is a monoid together with a comonoid, which satisfies an interaction law. Frobenius structures have a powerful graphical calculus and we prove a normal form theorem that makes them easy to work with. The Frobenius law itself is justified as a coherence property between daggers and closure of a category. We prove classification theorems for dagger Frobenius structures: in Hilb in terms of operator algebras and in Rel in terms of groupoids. Of special interest is the commutative case—as for Hilbert spaces this corresponds to a choice of basis—and provides a powerful tool to model classical information. We discuss phase gates and the state transfer protocol—as well as modules for Frobenius structures—and show how we can use these to model measurement, controlled operations and quantum teleportation.

Normal forms for perturbations of systems possessing a Diophantine invariant torus

We give a new proof of Moser’s 1967 normal-form theorem for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. The proposed approach, based on an inverse function theorem in analytic class, is flexible and can be adapted to several contexts. This allows us to prove in a unified framework the persistence, up to finitely many parameters, of Diophantine quasi-periodic normally hyperbolic reducible invariant tori for vector fields originating from dissipative generalizations of Hamiltonian mechanics. As a byproduct, generalizations of Herman’s twist theorem and Rüssmann’s translated curve theorem are proved.