1 citations · 1 across the 13 of their papers we have counts for
14 papers
Isometric Embeddings and Hyperkähler Geometry of the Cotangent Bundle of Complex Projective Space via the Scheme of Rank-1 Projections
Joshua Lackman
We show that the hyperkahler geometry of can be described algebraically by the affine scheme of rank-1 projections, and that this description simultaneously…
A Geometric Definition of the Integral and Applications
Joshua Lackman
The standard definition of integration of differential forms is based on local coordinates and partitions of unity. This definition is mostly a formality and not used used in expli…
A Simple Description of the Hyperkähler Structure of the Cotangent Bundle of Projective Space via Quantization
Joshua Lackman
Quantization identifies the cotangent bundle of projective space with the (non-Hermitian) rank- projections of a Hilbert space. We use this identification to study the natural g…
Quantization of Holomorphic Symplectic Manifolds: Analytic Continuation of Path Integrals and Coherent States
Joshua Lackman
We extend Berezin's quantization to holomorphic symplectic manifolds, which involves replacing the state space with its comple…
On an Axiomatization of Path Integral Quantization and its Equivalence to Berezin's Quantization
Joshua Lackman
We axiomatize path integral quantization of symplectic manifolds. We prove that this path integral formulation of quantization is equivalent to an abstract operator formulation, ie…
A Mathematical Definition of Path Integrals on Symplectic Manifolds
Joshua Lackman
We give a mathematical definition of some path integrals, emphasizing those relevant to the quantization of symplectic manifolds (and more generally, Poisson manifolds) $\unicode{x…
A Derivation of Geometric Quantization via Feynman's Path Integral on Phase Space
Joshua Lackman
We derive the geometric quantization program of symplectic manifolds, in the sense of both Kostant-Souriau and Weinstein, from Feynman's path integral formulation on phase space. T…
Geometric Quantization Without Polarizations
Joshua Lackman
We derive the quantization map in geometric quantization of symplectic manifolds via the Poisson sigma model. This gives a polarization-free (path integral) definition of quantizat…
A Canonical Quantization of Poisson Manifolds: a 2-Groupoid Scheme
Joshua Lackman
We canonically quantize a Poisson manifold to a Lie 2-groupoid, complete with a quantization map, and show that it relates geometric and deformation quantization: the perturbative…
A Groupoid Construction of Functional Integrals: Brownian Motion and Some TQFTs
Joshua Lackman
We formalize Feynman's construction of the quantum mechanical path integral. To do this, we shift the emphasis in differential geometry from the tangent bundle onto the pair groupo…
A Groupoid Approach to the Riemann Integral (and Path Integral Quantization of the Poisson Sigma Model)
Joshua Lackman
We use groupoids and the van Est map to define Riemann sums on compact manifolds (with boundary), in a coordinate-free way. These Riemann sums converge to the usual integral after…
A Formal Equivalence of Deformation Quantization and Geometric Quantization (of Higher Groupoids) and Non-Perturbative Sigma Models
Joshua Lackman
Based on work done by Bonechi, Cattaneo, Felder and Zabzine on Poisson sigma models, we formally show that Kontsevich's star product can be obtained from the twisted convolution al…
The van Est Map on Geometric Stacks
Joshua Lackman
We generalize the van Est map and isomorphism theorem in three ways, and we discuss conjectured connections with homotopy theory, including a proposal of a category which unifies d…
Cohomology of Lie Groupoid Modules and the Generalized van Est Map
Joshua Lackman
The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allo…