arXiv research

Universal spaces for finite topological spaces simplify shape descriptions.

Metric spaces with certain curvature properties are universally infinitesimally Hilbertian.

The study extends kernel universality to Riemannian symmetric spaces.

WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance …

We introduce a class of metrics on gauge theoretic moduli spaces. These metrics are made out of the universal matrix that appears in the universal connection construction of M. S. Narasimhan and S. Ramanan. As an example we construct metrics on the c_{2}=1 SU(2) moduli space of instantons on R^4 for various universal m…

We extend a recently proposed 1-nearest-neighbor based multiclass learning algorithm and prove that our modification is universally strongly Bayes-consistent in all metric spaces admitting any such learner, making it an "optimistically universal" Bayes-consistent learner. This is the first learning algorithm known to e…

Universal geometry organizes heterotic moduli spaces.

In this paper we modify the coordinate construction in our previous paper on the universal moduli space of pair consisting of a Riemann Surfaces and a stable holomorphic bundles on the Riemann Surface, so as to produce a new set of coordinates, which are in fact K\" ahler coordinates on this universal moduli space. Fur…

We study the natural Kähler metrics on moduli spaces of stable oriented pairs in a very general framework, and we prove a universal formula expressing the Kähler class of such a moduli space in terms of characteristic classes of the universal bundle. We use these results to compute explicitly the volumina of certain Qu…

We prove that the universal Teichmuller space T(1) carries a new structure of a complex Hilbert manifold. We show that the connected component of the identity of T(1), the Hilbert submanifold T_{0}(1), is a topological group. We define a Weil-Petersson metric on T(1) by Hilbert space inner products on tangent spaces, c…

Assume that all spaces and maps are localised at a fixed prime $p$. We study the possibility of generating a universal space $U(X)$ from a space $X$ which is universal in the category of homotopy associative, homotopy commutative H-spaces in the sense that any map f:X->Y to a homotopy associative, homotopy commutative …

Constructs universal link invariants from intersections in configuration spaces.

The paper explores universal circles for Anosov foliations and their uniqueness.

Characterizes Legendrian knots in lens spaces.

Develops a new approach to establish universality for any-dimensional machine learning models.

The paper finds universal inequalities for eigenvalues on hyperbolic spaces.

Study of Hitchin moduli spaces over Teichmüller space.

Study shows $k$-NN classifier is not universally consistent on $(0,1)$ but consistent on discrete and specific measure spaces.

The study constructs a dense orbit in the universal commensurability augmented Teichmüller space.

The universal Teichmüller space is an infinitely dimensional generalization of the classical Teichmüller space of Riemann surfaces. It carries a natural Hilbert structure, on which one can define a natural Riemannian metric, the Weil-Petersson metric. In this paper we investigate the Weil-Petersson Riemannian curvature…

The study characterizes quasiperiodic surfaces in pseudo-hyperbolic spaces with curvature conditions.

Universal approximation for stochastic processes using Brownian motion.

We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of the asymptotic dimension with the asymptotic inductive dimensions.

For each $n$, we construct a separable metric space $\mathbb{U}_n$ that is universal in the coarse category of separable metric spaces with asymptotic dimension ($\mathop{asdim}$) at most $n$ and universal in the uniform category of separable metric spaces with uniform dimension ($\mathop{udim}$) at most $n$. Thus, $\m…

TVS-FNNs can approximate any continuous function on expanded input spaces.

The main goal of the paper is to prove the existence of the universal cover for $RCD^*(K,N)$-spaces. This generalizes earlier work of C. Sormani and the second named author on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental gro…

The paper explores simple and relatively simple transformation groups and their universal coverings.

Estimates means in metric spaces using quantization.

We prove that the boundary of a right-angled hyperbolic building is a universal Menger space. Corollary: the 3-dimensional universal Menger space is the boundary of some Gromov-hyperbolic group.

In the first part of the paper we describe the complex geometry of the universal Teichmüller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo Möbius …

The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.

We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements.

Decomposes ultrametric spaces into scaled simplices.

We define and give explicit construction of the universal tree-graded space with a given collection of pieces. We apply that to proving uniqueness of asymptotic cones of relatively hyperbolic groups whose peripheral subgroups have unique asymptotic cones. Modulo the Continuum Hypothesis, we show that if an asymptotic c…

The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on d…

Study properties of balanced hyperbolic compact complex manifolds.

New upper bound for Neumann Laplacian eigenvalues on convex domains.

We consider the group of sense-preserving diffeomorphisms $\Diff S^1$ of the unit circle and its central extension, the Virasoro-Bott group, with their respective horizontal distributions chosen to be Ehresmann connections with respect to a projection to the smooth universal Teichmüller space and the universal Teichmül…

Study of laminations for pseudo-Anosov flows on three-manifolds.

Research shows RCD* spaces are semi-locally simply connected.

We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.

The paper defines a universal Teichmüller space for PGL_d(R) and proves its properties.

Study shows universal circle isomorphic to flow space ideal boundary.

Paper derives Riccati equation for static spaces and proves its applications.

We study the geometry of universal embedding spaces for compact almost complex manifolds of a given dimension. These spaces are complex algebraic analogues of twistor spaces that were introduced by J-P. Demailly and H. Gaussier. Their original goal was the study of a conjecture made by F. Bogomolov, asserting the "tran…

Deep Sets approximates functions on sets with high-dimensional latent space.

New algorithms for regression with adversarial responses on various metric spaces.

New method for reducing dimensions of distributional data.