Related papers: Affine statistical bundle modeled on a Gaussian Or…

This paper mainly contributes to a classification of statistical Einstein manifolds, namely statistical manifolds at the same time are Einstein manifolds. A statistical manifold is a Riemannian manifold, each of whose points is a…

Recently Ohsawa has studied the Marsden-Weinstein-Meyer quotient of the manifold $T^*\mathbb{R}^n\times T^*\mathbb{R}^{2n^2}$ under a $\operatorname{O}(2n)$-symmetry, and has used this quotient to describe the relationship between two…

We present an intrinsic geometric classification of the supermanifold of maps from $\mathbb{R}^{0|2}$ to any smooth manifold $S$, avoiding auxiliary structures. The key isomorphism relates this space to the pullback of the decomposable…

The Musielak--Orlicz setting unifies the variable exponent, Orlicz, weighted Sobolev, and double-phase spaces. They inherit technical difficulties resulting from general growth and inhomogeneity. In this survey we present an overview of…

The Bures--Wasserstein geometry of covariance matrices provides a canonical distance on the statistical manifold of centred Gaussian measures and lies at the intersection of information geometry, quantum information, and optimal transport.…

We study quotients of quasi-affine schemes by unipotent groups over fields of characteristic 0. To do this, we introduce a notion of stability which allows us to characterize exactly when a principal bundle quotient exists and, together…

We develop a rigorous theoretical framework for principal manifold estimation that recovers a latent low-dimensional manifold from a point cloud observed in a high-dimensional ambient space. Our framework accommodates manifolds with…

The configuration space of a non-linear sigma model is the space of maps from one manifold to another. This paper reviews the authors' work on non-linear sigma models with target a homogeneous space. It begins with a description of the…

Embeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. The equivalence of their Gagliardo-Slobodeckij norms to norms defined via Littlewood-Paley decompostions, via oscillations, or via Besov type…

We prove the Bonnet theorem for statistical manifolds, which states that if a statistical manifold admits tensors satisfying the Gauss--Codazzi--Ricci equations, then it is locally embeddable to a flat statistical manifold (or a Hessian…

Modelling randomness in shape data, for example, the evolution of shapes of organisms in biology, requires stochastic models of shapes. This paper presents a new stochastic shape model based on a description of shapes as functions in a…

This paper studies the geometry of immersions into statistical manifolds. A necessary and sufficient condition is obtained for statistical manifold structures to be dual to each other for a non-degenerate equiaffine immersion. Then we…

In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of Homological…

We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class $C_b^k$ with respect to appropriate reference measures. The case…

Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold…

A rather complete investigation of anisotropic Bessel potential, Besov, and H\"older spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends'…

We introduce a weighted Sobolev space theory for the non-local elliptic equation $$ \Delta^{\alpha/2}u=f, \quad x\in \mathcal{O}\,; \quad r_{\overline{\mathcal{O}}^c}u=g $$ as well as for the non-local parabolic equation $$…

The standard topological approach to indistinguishable particles formulates exchange statistics by using the fundamental group to analyze the connectedness of the configuration space. Although successful in two and more dimensions, this…

We compare the quantisation of linear systems of bosons and fermions. We recall the appearance of projectively flat connection and results on parallel transport in the quantisation of bosons. We then discuss pre-quantisation and…

We propose a new approach to non-parametric density estimation that is based on regularizing a Sobolev norm of the density. This method is statistically consistent, and makes the inductive bias of the model clear and interpretable. While…