# Your search: "author:"WEINSTEIN, ALAN""

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## Scholarly Works (14 results)

This dissertation explores two instances of C^{0} rigidity in symplectic geometry: First, we prove that continuous Hamiltonian flows as defined by Oh and M\"uller have unique generators. Second, we study the behavior of certain Floer theoretic invariants of Hamiltonian flows, called spectral invariants, under C^{0} perturbations of Hamiltonian flows.

Motivated by an attempt to better understand the notion of a symplectic stack, we introduce the notion of a \emph{symplectic hopfoid}, which should be thought of as the analog of a groupoid in the so-called symplectic category. After reviewing some foundational material on canonical relations and this category, we show that symplectic hopfoids provide a characterization of symplectic double groupoids in these terms. Then, we show how such structures may be used to produce examples of symplectic orbifolds, and conjecture that all symplectic orbifolds arise via a similar construction. The symplectic structures on the orbifolds produced arise naturally from the use of canonical relations.

The characterization of symplectic double groupoids mentioned above is made possible by an observation which provides various ways of realizing the core of a symplectic double groupoid as a symplectic quotient of the total space, and includes as a special case a result of Zakrzewski concerning Hopf algebra objects in the symplectic category. This point of view also leads to a new proof that the core of a symplectic double groupoid itself inherits the structure of a symplectic groupoid. Similar constructions work more generally for any double Lie groupoid---producing what we call a \emph{Lie hopfoid}---and we describe the sense in which a version of the ``cotangent functor'' relates such hopfoid structures.

In category theory, monads, which are monoid objects on endofunctors, play a central role closely related to adjunctions. Monads have been studied mostly in algebraic situations. In this dissertation, we study this concept in some categories of smooth manifolds.

Namely, the tangent functor in the category of smooth manifolds is the functor part of a unique monad, which is the main character of this dissertation.

After its construction and the study of uniqueness properties in related categories, we study its algebras, which are to this monad what representations are to a group. We give some examples of algebras, and general conditions that they should satisfy. We characterize them in the category of affine manifolds. We also study an analog of the tangent functor monad and its algebras in algebraic geometry.

We then prove our main theorem: algebras over the tangent functor monad induce foliations on the manifold on which they are defined.

This result links the study of these algebras to the study of foliations.

A natural question is then to characterize foliations which arise this way.

We give some restrictions in terms of the holonomy of such foliations. Finally, we study in greater detail algebras on surfaces, where they take a very simple form, and we nearly characterize them.

In the thesis, we initial first steps in understanding Quantum Mirror Symmetry and noncommutative compactification of moduli spaces of tori. To obtain a global invariant of noncommutative torus bundle, we study the monodromy of Gauss-Manin connection on periodic cyclic homology groups of Heisenberg group. Then the global monodromy map is developed, and provides a important criterion to detect when a noncommutative torus bundle is dequantizable. A process to construct the dequantizing Poisson manifold is given, when the dequantization criterion is satisfied. Naively, it seems that the Morita theory for noncommutative torus bundles can be developed naturally as in Rieffel theory for rotation algebras. However, this assumption turns out to be wrong; the Morita class of a non-dequantizable noncommutative torus bundle is not a classical object in the category of C*- algebras, and we call them C*- stacks. Even with the extended notion of C*- stacks, the Morita theory is still incompatible with noncommutative torus bundles with the infinite Poisson limit. There is no rotation algebra that can be used to compactify the moduli space of rotation algebras, even though we know that the "infinite Poisson rotation algebra" is strongly Morita equivalent to the classical torus. This subtlety is completely solved by a new mathematical structure hidden behind the quantization of constant Dirac structures on the tori.

We develop a new theory of quantum spaces called spatial structure to give a better understanding of quantum spaces and torus fibrations. We clarify some examples of spatial algebras and develop a rigorous way to construct a monoidal structure from the spatial structure. Using Hilsum-Skandalis maps between groupoids, we find that a groupoid presentation of a C* - algebra implies a monoidal structure on the category of representations. We decompose the spatial product of the cyclic modules over the rotation algebras as an example, and propose a conjecture that a quantum mirror symmetry lies behind the spatial structure and the Hopfish structure in the sense of Tang and Weinstein.