arXiv research

Survey of recent results on homogeneous finite-dimensional spaces.

Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.

Finite-dimensional spaces of biharmonic functions on manifolds are explored.

Finite spaces can be or not coproducts of subspaces.

Solves classical problem with Kähler-Einstein metrics in complex projective spaces.

In this paper we consider the complex vector spaces of holomorphic cross-sections of homogeneous holomorphic vector bundles over elliptic adjoint orbits, and provide a sufficient condition for the vector spaces to be finite dimensional in view of root systems.

Outer space for RAAGs is a contractible finite-dimensional space for automorphisms.

It is known that the only finite-dimensional diffeological vector space that admits a diffeologically smooth scalar product is the standard space of appropriate dimension. In this note we consider a way to circumnavigate this issue, by introducing a notion of pseudo-metric, which, said informally, is the least-degenera…

The study connects Kato bounds to finite-dimensional RCD spaces.

A homological selection theorem for C-spaces, as well as, a finite-dimensional homological selection theorem is established. We apply the finite-dimensional homological selection theorem to obtain fixed-point theorems for usco homologically UV^n set-valued maps.

We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson's group T and various generalizations of Thompson's group V have global fixed points when they act semi-simply on finite-d…

Unified formula for higher traces of linear maps on finite-dimensional normed spaces.

Upper bound for Hausdorff distance between hyperbolic space and its medianization.

The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.

Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.

Classifies vector field algebras in complex space.

We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is …

Study local properties of homogeneous ANR-spaces, proving dimension full-valuedness.

We show in this note that the Sobolev Discrepancy introduced in Mroueh et al in the context of generative adversarial networks, is actually the weighted negative Sobolev norm $||.||_{\dot{H}^{-1}(ν_q)}$, that is known to linearize the Wasserstein $W_2$ distance and plays a fundamental role in the dynamic formulation of…

We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An amalgamated product of asymptotically finite dimensional groups has finite asympt…

We give a complete classification in canonical forms on finite-dimensional vector spaces over the real numbers.

Since the work of Henri Cartan finite dimensional Riemannian symmetric spaces are an important subject of mathematical interest. They are related in a natural way to semisimple Lie groups. In this work we introduce and study their infinite dimensional generalization: Affine Kac-Moody symmetric spaces. Affine Kac-Moody …

Study of tautological forms on curve moduli spaces.

Analyzes geometric structures on profinite diffeological spaces.

Extends SGM to functional spaces for multimodal data.

Researchers find a list of non-isometric toric para-Kaehler-Einstein manifolds.

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L2 norm, and study their approximation properties over Hilbert subspaces of L2 . The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in reproduci…

I give a construction of compact group action on a finite dimensional space Y, whose orbit space is infinite dimensional.

The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $\sum_j(Y_j - μ(t_j))^2 + λ\int_a^b [μ"(t)]^2 dt$, where the data are $t_j,Y_j$, $j=1,..., n$. The minimization is taken over an infinite-dimensional function space, the space of all functions wi…

New metrics on curve spaces improve shape analysis.

Consistency of k-NN rule proven in sigma-finite dimensional metric spaces.

In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, we prove that the space of harmonic functions with fixed polynomial growth degree is finite dimensional. We a…

We prove some infinitesimal analogs of classical results of Menger, Schoenberg and Blumenthal giving the existence conditions for isometric embeddings of metric spaces in the finite-dimensional Euclidean spaces.

This paper studies isometric immersions of space forms by means of a hierarchy of finite dimensional integrable systems in Lax form on loop algebras.

Study improves learning algorithms for convex polyhedra in Hilbert spaces.

The paper constructs chaotic solutions to the Euler equations on high-dimensional manifolds.

Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic …

Let $Y=Γ\backslash H^n$ be a quotient of the hyperbolic space by the action of a discrete convex-cocompact group of isometries. We describe certain spaces of $Γ$-invariant currents on the sphere at infinity of $H^n$ with support on the limit set of $Γ$. These spaces are finite-dimensional. The main result identifies th…

We prove that a monomorphic functor $F:Comp\to Comp$ with finite supports is epimorphic, continuous, and its maximal $\emptyset$-modification $F^\circ$ preserves intersections. This implies that a monomorphic functor $F:Comp\to Comp$ of finite degree $deg F\le n$ preserves (finite-dimensional) compact ANR's if the spac…

We show that a Moore space M(Z_m,1) is an absolute extensor for finite dimensional metrizable spaces of cohomological dimension dim_{Z_m} \leq 1.

In a noncompact harmonic manifold $M$ we establish finite dimensionality of the eigenspaces $V_λ$ generated by radial eigenfunctions of the form $\cosh r + c$. As a consequence, for such harmonic manifolds, we give an isometric imbedding of $M$ into $(V_λ,B)$, where $B$ is a nondegenerate symmetric bilinear indefinite …

Proof shows local convexity implies global convexity in special geometric spaces.

It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional at compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.

We prove that every open subset of a euclidean building is a finite dimensional absolute neighborhood retract. This implies in particular that such a set has the homotopy type of a finite dimensional simplicial complex. We also include a proof for the rigidity of homeomorphisms of euclidean buildings. A key step in our…

Survey on finite dimensional Lie groups over real numbers.

We associate certain probability measures on $\R$ to geodesics in the space $\H_L$ of positively curved metrics on a line bundle $L$, and to geodesics in the finite dimensional symmetric space of hermitian norms on $H^0(X, kL)$. We prove that the measures associated to the finite dimensional spaces converge weakly to t…

We prove that direct limits of finite dimensional Lie algebroids and their prolongations can be endowed with structures of convenient spaces.

We give a finite dimensional approach to the Chas-Sullivan product on the free loop space of a manifold, orientable or not.