A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Natural gradient simplification for deep learning networks.
Study the geometry of curves in projective spaces.
VB uses natural gradients in information geometry.
We consider automorphisms of homogeneous parabolic geometries with a fixed point. Parabolic geometries carry the distinguished distributions and we study those automorphisms which enjoy natural actions on the distributions at the fixed points. We describe the sets of such automorphisms on homogeneous parabolic geometri…
Quantum Natural Gradient uses quantum geometry for optimization.
A four-dimensional Walker geometry is a four-dimensional manifold M with a neutral metric g and a parallel distribution of totally null two-planes. This distribution has a natural characterization as a projective spinor field subject to a certain constraint. Spinors therefore provide a natural tool for studying Walker …
Introduces new deformation classes in generalized Kähler geometry.
This note aims to demonstrate that every parabolic geometry has a naturally defined per-Courant algebroïd structure. This structure is a Courant algebroïd if and only if the the curvature of the Cartan connection vanishes. In all other cases, if the parabolic geometry is regular, there does not exist a natural univ…
Study natural foliations in cotangent bundles of Cartan spaces.
We introduce the notion of manifolds of amalgamation geometry and its generalization, split geometry. We show that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map.
This is the second in a series of papers on natural modification of the normal tractor connection in a parabolic geometry, which naturally prolongs an underlying overdetermined system of invariant differential equations. We give a short review of the general procedure developed in [5] and then compute the prolongation …
As a natural application of the {\it theory of geometric averaging} in Finsler geometry and generalized Finsler geometry, a new approach to investigate {\it generalized Finsler geometry}, based on a convex invariance of the average structures, is introduced.
We construct natural Riemannian metrics on Seiberg-Witten moduli spaces and study their geometry.
The note is about some nonlinear curvature conditions which arise naturally in conformal geometry.
Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.
Survey para-Hermitian geometry and its applications in physics.
UMAP connects to Information Geometry principles.
Bi-forms extend contrast functions to handle torsion in information geometry.
New approach connects Finsler geometry's metric and connections.
We compute the curvature tensor of the tangent bundle of a Riemannian manifold endowed with a natural metric and we get some relationships between the geometry of the base manifold and the geometry of the tangent bundle.
The author exposes the metrical multi-time Lagrange geometry of physical fields which naturally generalizes the classical Lagrangian developped by Miron and Anastasiei. In other words, one constructs a natural theory of physical fields on the 1-jet fibre bundle, attached to a Kronecker h-regular multi-time Lagrangian w…
New definition of naturally reductive Finsler manifolds using geodesic graphs.
Study reveals geometric context of second-order superintegrable systems.
We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one.
The paper introduces natural α-embeddings for item representations.
A family of naturally reductive pseudo-Riemannian spaces is constructed out of the representations of Lie algebras with ad-invariant metrics. We exhibit peculiar examples, study their geometry and characterize the corresponding naturally reductive homogeneous structure.
In this lecture I review how a matrix/Azumaya-type noncommutative geometry arises for D-branes in string theory and how such a geometry serves as an origin of the master nature of D-branes; and then highlight an abundance conjecture on D0-brane resolutions of singularities that is extracted and purified from a work of …
Diffeology extends differential geometry to complex spaces.
The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map. This is an exposition of a special case of th…
We interpret the physical -field renormalization group flow in the language of Courant algebroids, clarifying the sense in which this flow is the natural "Ricci flow" for generalized geometry. Next we show that the -field renormalization group flow preserves T-duality in a natural sense. As corollaries we obtain …
We define the notion of sub-Finsler geometry as a natural generalization of sub-Riemannian geometry with applications to optimal control theory. We compute a complete set of local invariants, geodesic equations, and the Jacobi operator for the three-dimensional case and investigate homogeneous examples.
The paper explores how AI systems use information geometry to encode semantic structure.
A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on natural…
The paper examines Bitcoin's nature using fractal geometry and finds it highly persistent, affecting predictability and decentralization.
Heterotic backgrounds described using generalised geometry, preserving minimal supersymmetry.
Study -flows reducing to complex geometry flows, focusing on -anomaly and -Laplacian coflow.
Model structures on multicomplexes help study complex geometry.
Constructs new explicit proper r-harmonic functions on Thurston geometries.
In this paper is proposed a kind of model theory for our axiomatic differential geometry. It is claimed that smooth manifolds, which have occupied the center stage in differential geometry, should be replaced by functors on the category of Weil algebras. Our model theory is geometrically natural and conceptually motiva…
This paper studies lightlike Cartan geometries and their properties.
We introduce and study the notion of a biholomorphic gerbe with connection. The biholomorphic gerbe provides a natural geometrical framework for generalized Kahler geometry in a manner analogous to the way a holomorphic line bundle is related to Kahler geometry. The relation between the gerbe and the generalized Kahler…
This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini-Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian st…
Study of harmonic Riemannian submersions from 3D geometries.
Griffiths extremal metrics solve complex Finsler equations and quantify Kähler geometry.
We present the linearized metrizability problem in the context of parabolic geometries and subriemannian geometry, generalizing the metrizability problem in projective geometry studied by R. Liouville in 1889. We give a general method for linearizability and a classification of all cases with irreducible defining distr…
A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.
A framework for natural gradient with arbitrary similarity measures.
New puzzles from geometry and topology.