Study uses Amari functors to investigate metric structures in gauge theories.
problem Investigating whether a gauge structure in a vector bundle is metric.
method Introduces generalized Amari functors and differential equations to analyze gauge structures.
result Links new index functions to the main concerns of metric structures in gauge theories.
We study the family of α-connections of Amari-Chentsov on the homogeneous space D(M)/Dμ(M) of diffeomorphisms modulo volume-preserving diffeomorphims of a compact manifold M. We show that in some cases their geodesic equations yield completely integrable Hamiltonian systems.
Characterizes connections on multivariate normal distributions.
problem Characterizing connections on statistical manifold of multivariate normal distributions.
method Analyzes statistical manifold (N,gF,ablaA,ablaA∗) of multivariate normal distributions. result The Amari-Chentsov connection ablaA is characterized by conjugate symmetry. Defines vector Laplacian on statistical manifolds.
problem No specific problem stated; focuses on mathematical definition.
method Defines and derives vector Laplacian formula.
result Derives formula for vector Laplacian.
Study curvature and torsion in Gaussian distribution's dual coordinate system.
problem Characterize geometric invariants of Gaussian distribution.
method Investigate Riemannian curvature and torsion in a dual coordinate system of Gaussian distribution.
result Explicitly give Amari formulas in the new coordinate system.
Dual affine connections on Riemannian manifolds have played a central role in the field of information geometry since their introduction by Amari. Here I would like to extend the notion of dual connections to general vector bundles with an inner product, in the same way as a unitary connection generalizes a metric affi…
A new Weyl prior is proposed for Bayesian statistics, offering a more canonical choice for parameter α.
problem Choosing a prior distribution for Bayesian inference.
method Proposed a new Weyl prior based on the Weyl structure on a statistical manifold.
result The Weyl prior is a special case of the α-parallel prior with α = -n, where n is the dimension of the statistical manifold.
We review basic notions in the field of information geometry such as Fisher metric on statistical manifold, α-connection and corresponding curvature following Amari's work . We show application of information geometry to asymptotic statistical inference.
New connections found on zero-mean multivariate normal distributions.
problem Characterizing statistical connections on zero-mean multivariate normal distributions.
method Investigating invariant conjugate symmetric statistical connections on the submanifold of zero-mean multivariate normal distributions.
result Invariant connections on zero-mean multivariate normal distributions are not uniquely characterized by invariance under the general linear group action.
Introduces new Finsler metrics and connects them to information geometry.
problem Generalizing Fisher-Rao metrics and studying their geometric properties.
method Introduces Lp-Fisher-Rao metrics and studies their relations to Amari-Cencov α-connections. result Geodesics of Fp and ∇(α) coincide on Dens+(M) for p=2/(1−α). New geometric interpretation of Amari-Cencov α-connections on probability densities.
problem Geometric interpretation of Amari-Cencov α-connections on probability densities.
method Riemannian metrics and Levi-Civita connections.
result Geodesics of α-connections are energy-minimizing curves.
Manifold calculus of functors, due to M. Weiss, studies contravariant functors from the poset of open subsets of a smooth manifold to topological spaces. We introduce "multivariable" manifold calculus of functors which is a generalization of this theory to functors whose domain is a product of categories of open sets. …
Study on polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
problem Understanding polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
method Analyzing polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
result Results generalize previous work by Katada and study polynomiality and outer nature of these functors.
Develops manifolds of differentiable densities for statistical analysis.
problem Statistical modeling of probability measures with smooth densities.
method Defines infinite-dimensional manifolds of probability measures on Banach spaces with specific smoothness properties, embedded in finite measure manifolds.
result Statistical manifolds are dually flat and admit mixture and exponential representations, with derived curvatures.
A modular functor is constructed from non-semisimple 3d TFTs.
problem Constructing modular functors from non-semisimple 3d topological field theories.
method Using a 3d TFT defined in [arXiv:1912.02063], a symmetric monoidal 2-functor is constructed from a 2-category of bordisms to a 2-category of finite linear categories.
result A modular functor is explicitly described as a symmetric monoidal 2-functor.
Paper constructs infinitely many tangent functors on diffeological spaces.
problem Tangent spaces in diffeological spaces are not uniquely defined.
method Introduced and constructed infinitely many non-isomorphic tangent functors.
result The choice of tangent functor is not unique outside smooth manifolds.
The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study …
Three functors link Lorentzian geometry concepts.
problem Finiteness results, singularity theorems, boundary constructions.
method Review of three functors from Lorentzian categories.
result Novel functor from ordered measure spaces to Lorentzian pre-length spaces.
Improved Bayesian inference using power priors with historical data.
problem Improving Bayesian inference with historical data.
method Generalized power priors that adapt to the α parameter of Amari's α-divergence. result Improved performance through appropriate choices of the α parameter. New insights into Markov chain geometry via positive transition measures.
problem Lack of statistical meaning in the space of transition probabilities.
method Constructing an extension of the space of transition probabilities using Amari's theory of positive measures.
result Introduction of a new dually flat structure for the space of positive transition measures.
The study explores how different Grothendieck topologies and functors between categories preserve locality.
problem Exploring relationships between different Grothendieck topologies and functors.
method Using Grothendieck topologies and functors to relate categories and geometric objects.
result Objects like sheaves, groupoids, and functors are invariant under equivalences of Grothendieck topologies and certain functors.
We study the functor of points and the local functor of points (here called the Weil--Berezin functor) for smooth and holomorphic supermanifolds, providing characterization theorems and fully discussing the representability issues. In the end we examine applications to differential calculus including the transitivity t…
In terms of category theory, the Gromov homotopy principle for a set valued functor F asserts that the functor F can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor F holds if the functor F can be induced from a (co)homology functor. We examin…
Parity functors assign labels to knot diagrams based on crossing parity.
problem Assigning consistent labels to knot diagrams.
method Define parity functors for knot diagrams and surfaces.
result Universal oriented parity functors for free knots and fixed surface knots.
The theory of product preserving functors and Weil functors is partly extended to infinite dimensional manifolds, using the theory of C∞-algebras.
Characterizes connections on normal distributions manifold.
problem Geometric characterization of connections on normal distributions.
method Homogeneous statistical manifold structure and Lie group analysis.
result Geometric characterization of α-connections on Lie group. The paper extends gradient flow and relaxation studies to non-flat Riemannian manifolds.
problem Understanding gradient flows and relaxation in non-flat Riemannian manifolds.
method Developed a criterion for comparing relaxation along gradient descent curves using non-metricity tensor.
result Revealed a universal asymmetry: warming up is faster than cooling down.
A new geometric structure for singular models is introduced.
problem Degenerate metrics in dually flat structures.
method Introducing quasi-Hessian manifolds with degenerate metrics and symmetric cubic tensors.
result Established Amari-Nagaoka's extended Pythagorean and projection theorems for singular models.
We construct the Weil functor TA corresponding to a general Weil algebra A=K⊕N: this is a functor from the category of manifolds over a general topological base field or ring K (of arbitrary characteristic) to the category of manifolds over A. This result simultaneously generalizes results known for o…
Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari-Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. Thi…
Gauge theory connects principal bundles and functors.
problem Understanding the relationship between principal bundles and functors.
method Characterized association functors and established an equivalence between principal bundles and functors.
result Established an equivalence between principal bundles and functors using association functors.
In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive- a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an ass…
In this paper, we extend the notion of modular functor and fusion category to what we called G equivariant modular functor and G equivariant fusion category, where G is a finite group, and establish a correspondence between between these notions.
Proves Hodge structures on modular functors, providing formulas for Hodge numbers.
problem Existence and uniqueness of Hodge structures on modular functors.
method Non-Abelian Hodge correspondence and Ocneanu rigidity.
result Explicit formulas for Hodge numbers in modular functors of level 2 times an odd number.
Cheptea, Habiro and Massuyeau constructed the LMO functor, which is defined on a certain category of cobordisms between two surfaces with at most one boundary component. In this paper, we extend the LMO functor to the case of any number of boundary components, and our functor reflects relations among the parts correspo…
New jet functors generalize classical notions in noncommutative geometry.
problem Defining and understanding jet functors in noncommutative settings.
method Constructing and proving properties of jet functors Jd(n), Jd[n], and Jdn. result Holonomic jet functor Jdn satisfies jet exact sequence under specific conditions. The extension functors between categories of Cartan geometries can be used to define different categories of Cartan geometries with additional morphisms. The Cartan geometries modeled on skeletons can be used for the description of such categories of Cartan geometries and therefore we develop the theory of Cartan geome…
Proves modular functors for SO(3) have integral Hodge structures.
problem Proving modular functors have integral Hodge structures.
method Based on homological models and geometric identification.
result Geometric construction of Hodge structures on SO(3) modular functors.
The Morse complex is shown to be an infinite functor.
problem Understanding the structure of Morse complexes as infinite functors.
method Showed the Morse complex of a compact Lie monoid can be given the structure of an f-bialgebra and defined an ∞-functor.
result Obtained two other ∞-functors mapping manifolds and actions to their Morse complexes.
Functors from web categories differ despite similar definitions.
problem Distinguishing between combinatorial and gauge-theoretic evaluations of webs.
method Exhibited a counterexample showing J♯ restricted to planar webs is not J♭. result Restriction of J♯ to planar webs is distinct from J♭. Defines a transgression functor for higher-dimensional Courant algebroids.
problem None explicitly stated; focuses on definition and properties.
method Definition of transgression functor for Courant algebroids.
result Established a connection between Courant algebroids and Lie algebroids.
Functor decomposes Khovanov spectra for non-alternating diagrams.
problem Computing Khovanov spectra for diagrams without alternating pairs.
method Functor from cube to Burnside 2-category, decomposition into simplicial complexes.
result Homotopy type of almost-extreme Khovanov spectra computed.
Constructs a functor for equivariant smooth h-cobordisms.
problem Defines a functor for equivariant smooth h-cobordisms.
method Constructs an (∞,1)-functor mapping smooth G-manifolds to spaces of equivariant h-cobordisms. result The functor structure is subtle and relies on new ideas.
Researchers create functors to match colored homologies of knots and links.
problem Equivalence of colored HOMFLYPT homologies for links and knots.
method Constructing functors on singular Soergel bimodules to identify homologies.
result Established parity results for intrinsic column-colored homology of positive torus knots.
The paper extends bubble concept to other functors.
problem Generalizing bubble concept to different functors.
method Defined and investigated Hˇn−bubbles, then generalized. result Formulated open questions for further research.
Develops manifold calculus for simplicial complexes.
problem Approximating functors from simplicial complexes to topological spaces.
method Adapting manifold calculus to simplicial complexes and proving an approximation theorem.
result Functors can be approximated by polynomial functors under certain conditions.
The article is devoted to the qR-conformal modular functors, which being ``deformations'' of the conformal modular functor (the projective representation of the category Train(Diff+(S1)), the train of the group Diff+(S1) of all orientation preserving diffeomorphisms of a circle) in the class of all projectiv…
Generalizes string-net modular functors to non-spherical categories.
problem Extending string-net models to non-spherical categories.
method Using non-semisimple string-nets and Drinfeld centers.
result Equivalence between string-net and Lyubashenko modular functors.