arXiv research

Every countable compact subset of sphere is tame.

For every countable group G we construct a compact path connected subspace K of R^4 whose fundamental group is isomorphic to G. Our construction is much simpler than the one found recently by Virk.

Finite group action on a surface yields a special homology subspace.

Minimal orbits of semi-simple Lie groups are studied and related to invariant subspaces.

Krein's formula for conic Laplacians on compact Riemann surfaces

For an infinite cardinal $κ$ let $\ell_2(κ)$ be the linear hull of the standard othonormal base of the Hilbert space $\ell_2(κ)$ of density $κ$. We prove that a non-separable convex subset $X$ of density $κ$ in a locally convex linear metric space if homeomorphic to the space (i) $\ell_2^f(κ)$ if and only if $X$ can be…

This article deals with two topics: the first, which has a general character, is a variation formula for the the determinant line bundle in non-Kählerian geometry. This formula, which is a consequence of the non-Kählerian version of the Grothendieck-Riemann Roch theorem proved recently by Bismut, gives the variation of…

The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.

The study restricts stable minimal immersions in product spaces to specific configurations.

We discuss an "extrinsic" property of knots in a 3-subspace of the 3-sphere $S^3$ to characterize how the subspace is embedded in $S^3$. Specifically, we show that every knot in a subspace of the 3-sphere is transient if and only if the exterior of the subspace is a disjoint union of handlebodies, i.e. regular neighbor…

Active subspace is a model reduction method widely used in the uncertainty quantification community. In this paper, we propose analyzing the internal structure and vulnerability and deep neural networks using active subspace. Firstly, we employ the active subspace to measure the number of "active neurons" at each inter…

The space of the global sections of chiral de Rham complex on a compact Ricci-flat Kähler manifold is calculated and it is expressed as an invariant subspace of a $βγ-bc$ system under the action of certain Lie algebra.

We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the deco…

Simplicial learning improves classification by generating compact sparse representations.

We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on t…

Hiss and Szczepański proved in 1991 that the holonomy group of any compact flat Riemannian manifold, of dimension at least two, acts reducibly on the rational span of the Euclidean lattice associated with the manifold via the first Bieberbach theorem. Geometrically, their result states that such a manifold must admit a…

In this paper, we propose a novel method for projecting data from multiple modalities to a new subspace optimized for one-class classification. The proposed method iteratively transforms the data from the original feature space of each modality to a new common feature space along with finding a joint compact descriptio…

It was proved in 1998 by Ben-David and Litman that a concept space has a sample compression scheme of size d if and only if every finite subspace has a sample compression scheme of size d. In the compactness theorem, measurability of the hypotheses of the created sample compression scheme is not guaranteed; at the same…

We study the classification performance of Kronecker-structured models in two asymptotic regimes and developed an algorithm for separable, fast and compact K-S dictionary learning for better classification and representation of multidimensional signals by exploiting the structure in the signal. First, we study the clas…

The study of flat manifolds and their reducible holonomy groups.

In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group $G$ which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a $G$-stratified space carries a system of $G$-equivariant control data. As an application, we show that if $A \su…

We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in $\mathbb R^n$ is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant rela…

We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by c…

Study on low-dimensional adversarial perturbations in classification models.

The paper studies affine manifolds with linear foliations and their topological properties.

We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the to…

We develop a method of extending actions of compact transformation groups which is then applied to the problem of preservation of equivariant extensor property by passing to a subspace of given orbit types.

Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.

Extends Kähler metrics theory to symplectic manifolds with toric actions.

This article is written for the Proceedings of the Conference on Current Developments in Mathematics in Harvard University, November 16-17, 2007. It is an exposition of the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type. It lists and discusses the …

The paper finds Koopman invariant subspaces using personalized PageRank.

The high-dimensional data setting, in which p >> n, is a challenging statistical paradigm that appears in many real-world problems. In this setting, learning a compact, low-dimensional representation of the data can substantially help distinguish signal from noise. One way to achieve this goal is to perform subspace le…

We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of t…

Paper proposes a new method for learning compact representations of sequential data.

We prove that for compact, non-contractible, one dimensional geodesic spaces, a version of the marked length spectrum conjecture holds. For a compact one dimensional geodesic space X, we define a subspace Conv(X). When X is non-contractible, we show that X deformation retracts to Conv(X). If two such spaces X, Y have t…

The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. Let X be a symmetric space of noncompact type, and G be its group of isometries. The space X identifies with the subspace of maximal compact subgroups of G : taking the closure gives ri…

Existence of smooth valuations on subspaces is shown for certain conditions.

It is an open question (Pawlikowski) whether every finitely generated group can be realized as a fundamental group of a compact metric space. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of four dimensional Euclidean space. According to theorems of Shela…

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the group-compact coarse structure generalizing several familiar results for discrete group…

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of $sl(2,\mathbb{C})$, generalizing the well known structure on the harmonic f…

This paper classifies Hamiltonian actions by symplectic groupoids using Delzant subspaces.

Study of tangent cones at infinity for algebraic sets.

In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples $(Δ, \mathfrak{h}, G)$ of n…

Constructs a Morse-Bott function on symplectic Grassmannians.

We prove the following Theorem: Let X be a nonempty compact metrizable space, let $l_1 \leq l_2 \leq...$ be a sequence of natural numbers, and let $X_1 \subset X_2 \subset...$ be a sequence of nonempty closed subspaces of X such that for each k in N, $dim_{Z/p} X_k \leq l_k < \infty$. Then there exists a compact metriz…

We consider Schrödinger operators $H=-Δ_{g_\varepsilon} + V$ on a fibre bundle $M\stackrelπ{\to}B$ with compact fibres and a metric $g_\varepsilon$ that blows up directions perpendicular to the fibres by a factor ${\varepsilon^{-1}\gg 1}$. We show that for an eigenvalue $λ$ of the fibre-wise part of $H$, satisfying a l…

We provide an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear e…

Proposes methods for local clustering in attributed graphs.