The Differential Structure of an Orbifold

We prove that the underlying set of an orbifold equipped with the ring of smooth real-valued functions completely determines the orbifold atlas. Consequently, we obtain an essentially injective functor from orbifolds to differential spaces.Comment: V3: Final version. Removed a superfluous "2" from some example

Basic Forms and Orbit Spaces: a Diffeological Approach

If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the identity component acts properly, where on the quotient space we take differential forms in the diffeological sense

Tame Circle Actions

In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases.Comment: 25 page

Diffeological Coarse Moduli Spaces of Stacks over Manifolds

In this paper, we consider diffeological spaces as stacks over the site of smooth manifolds, as well as the "underlying" diffeological space of any stack. More precisely, we consider diffeological spaces as so-called concrete sheaves and show that the Grothendieck construction sending these sheaves to stacks has a left adjoint: the functor sending any stack to its diffeological coarse moduli space. As an application, we restrict our attention to differentiable stacks and examine the geometry behind the coarse moduli space construction in terms of Lie groupoids and their principal bundles. Additionally, we define basic differential forms for stacks and confirm in the differentiable case that these agree (under certain conditions) with basic differential forms on a representative Lie groupoid. These basic differentiable forms in turn match the diffeological forms on the orbit space.Comment: 16 pages; v4: besides some minor changes, the abstract was rewritten and we added an application to the differentiable stack section. v3: condensed the presentation and added more information to the introduction. (Also changed the title.

Diffeologies, Differential Spaces, and Symplectic Geometry

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Fr\"olicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential spaces that do not fall into this category. We apply the theory of diffeological spaces to differential forms on a geometric quotient of a compact Lie group. We show that the subcomplex of basic forms is isomorphic to the complex of diffeological forms on the geometric quotient. We apply this to symplectic quotients coming from a regular value of the momentum map, and show that diffeological forms on this quotient are isomorphic as a complex to Sjamaar differential forms. We also compare diffeological forms to those on orbifolds, and show that they are isomorphic complexes as well. We apply the theory of differential spaces to subcartesian spaces equipped with families of vector fields. We use this theory to show that smooth stratified spaces form a full subcategory of subcartesian spaces equipped with families of vector fields. We give families of vector fields that induce the orbit-type stratifications induced by a Lie group action, as well as the orbit-type stratifications induced by a Hamiltonian group action.Comment: V1 is the official PhD thesis - University of Toronto. V2: Some minor changes, including some acknowledgements for two important lemmas and a corollary. 108 page

Case Analyses in Financial Accounting

The following thesis explores topics in the profession of public accounting, a diverse and ever-evolving field. As the global business environment and economy develop, so must accounting standards and ideas in order to protect the interests of the masses who invest and take part in the larger economy. The following cases expound on important concepts in the field of accountancy and provide careful consideration of standards utilized and debated worldwide. Each case is explored within the context of a different company or situation, allowing for a diverse palette of research topics from which to view the business world through an accounting lens. The case studies were completed under the direction of Victoria Dickinson in fulfillment of requirements for the University of Mississippi’s Sally McDonnell Barksdale Honors College and Patterson School of Accountancy ACCY 420 course in the 2017-2018 academic year