a b s t r a c tIn this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldorf from manifolds to stacks.In the process of proving our main result we simplify Schreiber and Waldorf's original definition of a transport functor for principal bundles with connections over manifolds and provide a more direct proof of the correspondence between principal bundles with connections and transport functors.

The purpose of this note is to define sheaves for diffeological spaces and give a construction of their Čech cohomology. As an application, we prove that the first degree Čech cohomology classes for the sheaf of smooth functions to an abelian diffeological group G classify diffeological principal G-bundles.

We consider a variation of the Kuramoto model with dynamic coupling, where the coupling strengths are allowed to evolve in response to the phase difference between the oscillators, a model first considered by Ha, Noh, and Park. We demonstrate that the fixed points of this model, as well as their stability, can be completely expressed in terms of the fixed points and stability of the analogous classical Kuramoto problem where the coupling strengths are fixed to a constant (the same for all edges). In particular, for the "all-to-all" network, where the underlying graph is the complete graph, the problem reduces to the problem of understanding the fixed points and stability of the all-to-all Kuramoto model with equal edge weights, a problem that is well understood.

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Let G be a torus with Lie algebra g. We provide a classification of two types of toric objects: symplectic toric cones and symplectic toric stratified spaces with isolated singularities. Both types of object are classified via orbital moment map and a second degree cohomology class.As symplectic toric stratified spaces with isolated singularities are locally modeled on symplectic toric cones, we first focus on classifying symplectic toric cones. We show that symplectic toric cones have a certain type of map ψ : W → g * (called homogeneous unimodular local embeddings) as orbital moment maps. Conversely, every ψ has a symplectic toric cone for which it is an orbital moment map. We classify the symplectic toric cones with orbital moment map ψ by showing that their isomorphism classes are in bijective correspondence with the first Chern classes H 2 (W ; Z G ) of principal G-bundles over W , for Z G the integral lattice ker(exp : g → G). This generalizes Lerman's classification of compact connected contact toric manifolds.Symplectic toric stratified spaces with isolated singularities are spaces with neighborhoods of singularities modeled on symplectic cones. We first show their quotients W are space stratified by manifolds with corners and their moment maps are a particular type of map ψ : W → g * called stratified unimodular local embeddings. Every stratified unimodular local embedding ψ is the orbital moment map of a symplectic toric stratified space. Finally, we show that, for any stratified unimodular local embedding ψ and for W reg the top stratum of W , the isomorphism classes of symplectic toric stratified spaces with isolated singularities with orbital moment map ψ are in bijective correspondence with the cohomology classes H 2 (W reg ; Z G ) × C, for C ⊂ H 2 (W reg ; R) a subspace dependent on the topology of W . This generalizes Burns, Guillemin, and Lerman's classification of the compact connected symplectic toric stratified spaces with isolated singularities.