Geometric approach to moment maps in complex geometry.
problem Constructing moment maps in complex geometry.
method Introducing universal families and equivariant differential forms.
result New geometric proofs and equations for moment maps.
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48 results for “geometric moments”
We establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a noncompact symplectic manifold with proper moment map.
Multi-Purpose Binomial Model: Fitting all Moments to the Underlying Geometric Brownian Motionq-fin.PR
We construct a binomial tree model fitting all moments to the approximated geometric Brownian motion. Our construction generalizes the classical Cox-Ross-Rubinstein, the Jarrow-Rudd, and the Tian binomial tree models. The new binomial model is used to resolve a discontinuity problem in option pricing.
The moment-angle complex Z_K is cell complex with a torus action constructed from a finite simplicial complex K. When this construction is applied to a triangulated sphere K or, in particular, to the boundary of a simplicial polytope, the result is a manifold. Moment-angle manifolds and complexes are central objects in…
Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.
problem Uniqueness of smooth structures on real moment-angle manifolds.
method Arguments from calculus applied to results from complex moment-angle manifolds.
result Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.
We study Hamiltonian spaces associated with pairs (E,A), where E is a Courant algebroid and A\subset E is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties are discussed, including a reduction procedure. This set-u…
Researchers extend geometric quantization to complex Abelian Lie supergroups.
problem Quantization of super Kähler structures on complex Abelian Lie supergroups.
method Extended geometric quantization scheme to super Kähler setting, constructed unitary representation.
result Irreducible subrepresentations of the constructed representation are determined by the moment map.
The paper trivializes moment maps for various geometric structures.
problem Trivializing moment maps for different geometric structures.
method General framework of a reductive group acting on a smooth affine variety, using Kempf-Ness theory, Morse theory, and ideas from Nakajima and Kronheimer.
result Locally trivial fibration of moment maps over a regular locus of the center of the Lie algebra of a maximal compact subgroup.
Gaussian Moments as Physically Inspired Molecular Descriptors for Accurate and Scalable Machine Learning Potentialsphysics.comp-ph
New method uses geometric moments for accurate machine learning potentials.
problem Creating high-dimensional potential energy surfaces efficiently.
method Feed-forward neural networks with invariant local molecular descriptors based on geometric moments.
result Accuracy comparable to established models, high efficiency.
Authors create déjà vu links in Legendrian geometry.
problem Understanding déjà vu moments in spacetime.
method Constructing Legendrian links with specific properties.
result Found examples of déjà vu links in various geometric settings.
Heterotic backgrounds described using generalised geometry, preserving minimal supersymmetry.
problem Characterizing heterotic backgrounds preserving minimal supersymmetry in four dimensions.
method Using generalised geometry, characterizing backgrounds by an structure and an involutive subbundle of the generalised tangent bundle.
result The analysis of infinitesimal deformations reproduces known cohomologies of massless moduli.
New algorithm learns HMM parameters on Riemannian manifolds.
problem Learning hidden Markov models on non-Euclidean spaces.
method Geometric method of moments algorithm for Riemannian manifolds.
result Significantly improved speed and accuracy compared to existing methods.
GMMNs model cross-sectional dependence for better option pricing and simulation.
problem Modeling cross-sectional dependence between stochastic processes.
method Generative moment matching networks (GMMNs) for geometric Brownian motions and ARMA-GARCH models.
result GMMNs produce dependent quasi-random samples with variance reduction.
Analysis of Vlasov plasma dynamics using matched pair Lie-Poisson formulation.
problem Understanding the dynamics of Vlasov plasma and its kinetic moments.
method Hamiltonian (Lie-Poisson) analysis and matched pair decomposition.
result Observation of mutual interactions between subdynamics in Vlasov plasma.
New theory for Hamiltonian actions on special geometric structures.
problem Hamiltonian actions on cosymplectic groupoids.
method Developed a moment map theory for 0-shifted cosymplectic structures.
result Established a version of the Kirwan convexity theorem.
New geometric quantisation scheme for hyper-Kähler manifolds.
problem Quantising hyper-Kähler manifolds with symmetry.
method Constructing unitary quantum representations of isometries.
result Decomposition of quantum representations into irreducibles.
We extend the classical Cox-Ross-Rubinstein binomial model in two ways. We first develop a binomial model with time-dependent parameters that equate all moments of the pricing tree increments with the corresponding moments of the increments of the limiting Itô price process. Second, we introduce a new trinomial model i…
Quantizes -symplectic toric manifolds using -modules.
problem Quantization of -symplectic toric manifolds.
method Bohr-Sommerfeld quantization via -modules.
result Dimension of quantization coincides with signed count of integral points in moment polytope.
We investigate special lcs and twisted Hamiltonian torus actions on strict lcs manifolds and characterize them geometrically in terms of the minimal presentation. We prove a convexity theorem for the corresponding twisted moment map, establishing thus an analog of the symplectic convexity theorem of Atiyah and Guillemi…
Study local perturbations of vector bundles with polynomial curvature solutions.
problem Existence and stability of solutions to geometric PDEs under deformations.
method Geometric invariant theory, moment map framework, polystability conditions.
result Existence and uniqueness of solutions under local polystability conditions.
Study nearest-neighbor radii under dependent sampling, finding they remain informative.
problem Analyzing nearest-neighbor radii under dependent sampling.
method Consider strong mixing dependent observations, establish distribution-free almost sure convergence and sharp non-asymptotic moment bounds.
result Nearest-neighbor geometry remains informative under dependence sampling.
Researchers develop a generalised geometric Brownian motion for better asset pricing.
problem Irregularities in simple geometric Brownian motion for asset dynamics.
method Introduce a memory kernel to generalise GBM, derive moments and probability density functions.
result The performance of kernels in pricing options depends on option maturity and moneyness.
Study compares Kähler quotients of torus actions under varying moment maps.
problem Comparing Kähler quotients of torus actions under varying moment maps.
method Analyzes the transformation of Kähler quotients as moment maps change, proving bimeromorphic transformations and desingularizations.
result Each nondegenerate singular Kähler quotient has a partial and rational desingularization.
A new method extracts features and reconstructs moments in dynamical systems using information geometry.
problem Reconstructing moments in dynamical systems efficiently and accurately.
method Information-geometric approach on spaces of probability measures.
result Moments can be expanded in eigenfunctions of a kernel integral operator, enabling nonparametric forecasting.
We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler-Ricci solitons. This naturally enlarges the moduli space of Fano manifolds with Kähler-Einstein metrics, which was constructed by Odaka and Li-Wang-Xu. We discover a moment map picture for Kähler-Ricci solitons, and give comp…
Geometric regularisation improves statistical models by avoiding degeneracy loci.
problem Non-identifiability, singular information, and moment indeterminacy in statistical models.
method Develops the geometric regularisation of distribution-kernel pairs using Whitney, Thom, and Mather theorems.
result Finite-dimensional weak transversality theorem for generic kernels, avoiding degeneracy strata of high codimension.
Motivated by the prediction of cell loads in cellular networks, we formulate the following new, fundamental problem of statistical learning of geometric marks of point processes: An unknown marking function, depending on the geometry of point patterns, produces characteristics (marks) of the points. One aims at learnin…
A higher moment formula for the Siegel--Veech transform over quotients by Hecke triangle groupsmath.GT
We compute higher moments of the Siegel--Veech transform over quotients of by the Hecke triangle groups. After fixing a normalization of the Haar measure on we use geometric results and linear algebra to create explicit integration formulas which give information about densities of…
Geometric quantization and quantum moment maps on coadjoint orbits and Kähler-Einstein manifoldsmath.DG
Geometric quantization shows compatibility of symmetries on coadjoint orbits and Kähler-Einstein manifolds.
problem Compatibility of symmetries in geometric quantization.
method Deformation and geometric quantization on Kähler manifolds, Hamiltonian actions.
result Strict compatibility of symmetries on coadjoint orbits and Kähler-Einstein manifolds.
We study meromorphic actions of unipotent complex Lie groups on compact Kähler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric quotients that carry compactifiable Kähler structures obtained by symplectic reduct…
We apply a local differential geometric framework from Kähler toric geometry to (re)construct Calabi's extremal Kähler metrics on $\bbC\bbP^n$ blown-up at a point from data on the moment polytope.
We prove a generalization of the fundamental inequality of Guivarc'h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of CAT(-1) metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is…
Study explores geometric structure and prior for beta-logistic distribution.
problem Understanding the geometric structure and prior distributions of the beta-logistic distribution.
method Exploring dual geometric structure and uncovering -parallel prior.
result The beta-logistic distribution admits an -parallel prior for any real number .
We study Dirac structures associated with Manin pairs (\d,\g) and give a Dirac geometric approach to Hamiltonian spaces with D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach in terms of quasi-Poisson structures. We explain how these two distinct frameworks are related to each other, pro…
A generalized Calabi-Yau structure is a geometrical structure on a manifold which generalizes both the concept of the Calabi-Yau structure and that of the symplectic one. In view of a result of Lin and Tolman in generalized complex cases, we introduce in this paper the notion of a generalized moment map for a compact L…
Prove asymptotics of geometric flows using algebro-geometric methods.
problem Prove asymptotics of geometric flows
method Algebro-geometric methods
result Prove a conjecture of Haiden-Katzarkov-Kontsevich-Pandit
Asymptotics for the Discrete-Time Average of the Geometric Brownian Motion and Asian Optionsq-fin.PR
The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps.…
The paper studies connections on stable bundles and their continuity under metric variations.
problem Continuity of HYM connections under metric variations for stable bundles.
method Semi-stable perturbation techniques for geometric PDEs with moment map interpretation.
result HYM connections depend continuously on the metric, even for semi-stable bundles.
Gradient descent with random weights in linear regression analyzed for various noise types.
problem Analyzing the impact of random noise on gradient descent in linear regression.
method Gradient descent with randomly weighted data points, various weighting distributions, geometric moment contraction.
result Characterization of implicit regularization and non-asymptotic convergence bounds.
We show that all the small covers which are infra-nilmanifolds are exactly real Bott manifolds. This implies that any small cover which admits a flat Riemannian metric must be a real Bott manifold. In addition, we will study small covers which admit Riemannian metrics with positive or nonnegative Ricci curvature or sec…
We introduce -critical connections for holomorphic vector bundles and prove their existence under stability conditions.
problem Existence of -critical connections for holomorphic vector bundles.
method Associated geometric PDEs to Bridgeland stability conditions and used infinite dimensional moment maps.
result In the large volume limit, a sufficiently smooth holomorphic vector bundle admits a -critical connection if and only if it is asymptotically -stable.
Homotopy moment mapsmath.DG
Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a…
The paper classifies quantizable functions and explores symmetry in quantization methods.
problem Classifying quantizable functions and understanding symmetry in quantization methods.
method Deformation quantization and geometric quantization methods are compared and classified.
result Formal quantizable functions are of a specific form and relate to Hamiltonian Killing vector fields.
The paper proposes estimators for bid-ask spreads with and without serial dependence.
problem Estimating bid-ask spreads in financial markets with and without serial dependence.
method The authors propose moment-based estimators for bid-ask spreads, considering both geometric Brownian motion and geometric fractional Brownian motion for price dynamics, and Ornstein-Uhlenbeck process for microstructure noise.
result The estimators are consistent and asymptotically normal, and perform well compared to existing approaches on simulated data.
The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe…
The general aim of this paper is to study which are the solvable Lie groups admitting an Einstein left invariant metric. The space N of all nilpotent Lie brackets on R^n parametrizes a set of (n+1)-dimensional rank-one solvmanifolds, containing the set of all those which are Einstein in that dimension. The moment map f…
A new model for generating point processes with complex geometries.
problem Difficulties in modeling point processes with large numbers of particles and complex geometries.
method Gradient descent algorithm applied to a phase harmonic operator on wavelet transforms of point patterns.
result The model allows for fast sampling of new configurations that match the statistics of observed point processes.
Introduces q-paths for generalizing geometric annealing paths in machine learning.
problem Limited applicability of existing path methods in machine learning.
method Develops a family of paths derived from a generalized mean, including geometric and arithmetic mixtures.
result Empirical gains in Bayesian inference and generative model evaluation.