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Quantization Commutes with Reduction, a Survey
We review the themes relating to the proposition that “quantization commutes with reduction” ([ Q, R ] = 0), from symplectic manifolds to...
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Kostant Prequantization of Symplectic Manifolds with Contact Singularities
The relationship between the Bohr-Sommerfeld quantization condition and the integrality of the symplectic structure in Planck constant units is...
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Conjectures on Reductive Homogeneous Spaces
We address some conjectures and open problems in the ‘analysis of symmetries’ which include the study of non-commutative harmonic analysis and...
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Entanglement of Mixed States in Kähler Quantization
Let M is a product of two integral compact Kähler manifolds. Fix a sufficiently large positive integer N. With a submanifold...
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Souriau-Casimir Lie Groups Thermodynamics and Machine Learning
In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in the framework of Symplectic model of Statistical Mechanics. This Souriau’s...
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A dual pair for the contact group
Generalizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted...
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Berezin–Toeplitz Quantization on Symplectic Manifolds of Bounded Geometry
AbstractThe theory of Berezin–Toeplitz quantization on symplectic manifolds of bounded geometry is developed. The quantization space is a suitable...
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Symmetric spaces as adjoint orbits and their geometries
We realize specific classical symmetric spaces, like the semi-Kähler symmetric spaces discovered by Berger, as cotangent bundles of symmetric flag...
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Toric Degenerations in Symplectic Geometry
Toric degeneration in algebraic geometry is a process of degenerating a given projective variety into a toric one. Then one can obtain information...
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Multiplicity in Restricting Minimal Representations
We discuss the action of a subgroup on small nilpotent orbits, and prove a bounded multiplicity property for the restriction of minimal...
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Symmetries and Conservation Laws
In this final chapter, we study the relation between the presence of symmetries and the existence of conserved quantities in a variational problem....
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Invariant Geometry
This chapter discusses a number of invariant geometric structures on homogeneous spaces. The structures of concern will be defined in terms of...
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The Enhanced Period Map and Equivariant Deformation Quantizations of Nilpotent Orbits
In a previous paper, the author and his collaborator studied the problem of lifting Hamiltonian group actions on symplectic varieties and Lagrangian...
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On the Moduli Space of Wigner Quasiprobability Distributions for N-Dimensional Quantum Systems
A map** between operators on the Hilbert space of an N-dimensional quantum system and Wigner quasiprobability distributions defined on the...
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Complexified Hermitian Symmetric Spaces, Hyperkähler Structures, and Real Group Actions
There is a known hyperkähler structure on any complexified Hermitian symmetric space G / K , whose construction relies on identifying G / K with both a...
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Fusion Frame Homotopy and Tightening Fusion Frames by Gradient Descent
Finite frames, or spanning sets for finite-dimensional Hilbert spaces, are a ubiquitous tool in signal processing. There has been much recent work on...
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ON THE REGULARITY OF D-MODULES GENERATED BY RELATIVE CHARACTERS
Following the ideas of Ginzburg, for a subgroup K of a connected reductive ℝ-group G we introduce the notion of K -admissible D -modules on a...
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Lectures on Poisson Algebras
The notion of a Poisson algebra was probably introduced in the first time by A.M. Vinogradov and J. S. Krasil’shchik in 1975 under the name...
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Weil–Petersson Teichmüller Theory of Surfaces of Infinite Conformal Type
Over the past two decades the theory of the Weil–Petersson metric has been extended to general Teichmüller spaces of infinite type, including for...