arXiv research

Proves uniqueness of embedding complex manifold into infinite-dimensional space.

Existence of balanced embedding proved for complex manifold into infinite-dimensional space.

The paper proves infinite-dimensional rational homotopy groups for a specific embedding space.

We construct using Lefschetz fibrations a large family of contact manifolds with the following properties: Any bounding contact embedding into an exact symplectic manifold satisfying a mild topological assumption is non-displaceable and generically has infinitely many leaf-wise intersection points. Moreover, any Stein …

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Frechet manifold (if it is not the empty set). The moduli space of all connected compact orientable metric Lagrangian embedded surfaces…

We present several principal bundles of embeddings of compact manifolds (with or without boundary) whose base manifolds are nonlinear Grassmannians. We study their infinite dimensional differential manifold structure in the Fréchet category. This study is motivated by the occurrence of such objects in the geometric Lag…

An embedding of the group $\Diff(S^{1})$ of orientation preserving diffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplectic group, $\Sp(\infty)$, is studied. The authors prove that this embedding is not surjective. A Brownian motion is constructed on $\Sp(\infty)$. This study is motivated by rece…

Minimal hypertori found in 4D sphere, solving Bernstein conjecture.

Study confirms asymptotic behavior of logarithmic balanced metric near infinity.

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobeniu…

Totally geodesically embeddings of infinitely many closed 7-manifolds into 13-dimensional positively curved closed Riemannian manifolds are constructed. The problems of computing pinching constants and existence of other totally geodesical embeddings are discussed.

We describe a method for learning word embeddings with data-dependent dimensionality. Our Stochastic Dimensionality Skip-Gram (SD-SG) and Stochastic Dimensionality Continuous Bag-of-Words (SD-CBOW) are nonparametric analogs of Mikolov et al.'s (2013) well-known 'word2vec' models. Vector dimensionality is made dynamic b…

The study embeds infinite-dimensional geometric structures in Cayley graphs.

We prove the existence of a one parameter family of minimal embedded hypersurfaces in $R^{n+1}$, for $n \geq 3$, which generalize the well known 2 dimensional "Riemann minimal surfaces". The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar end…

Study extends JB-algebra structure group results to infinite dimensions.

In this note it is shown that every 7-dimensional Eschenburg space can be totally geodesically embedded into infinitely many topologically distinct 13-dimensional Bazaikin spaces. Furthermore, examples are given which show that, under the known construction, it is not always possible to totally geodesically embed a pos…

The paper proves isometric embeddings for smooth manifolds.

In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space $\mathbb{R}^d$. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the sol…

LASE improves local network structure visualization by targeting locally low-dimensional regions.

New mechanism for pure differential privacy on functional summaries using Laplace-like process.

The study connects ECH capacities to Anosov flows, proving infinite capacities and obstructions.

Study on high-dimensional solid tori reveals infinite generation in their diffeomorphism groups.

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we …

4-manifolds have special topological properties which can be used to get a different view on quantum mechanics. One important property (connected with exotic smoothness) is the natural appearance of 3-manifold wild embeddings (Alexanders horned sphere) which can be interpreted as quantum states. This relation can be co…

We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case $\Emb(\Bbb R,\Bbb R)$ which turns out to be Burgers' equation. Then we derive the geodesic equation, the curvature, and the Jacobi equat…

Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree $N$ to be thos…

In Grauert's paper [G] it is noted that finite dimensionality of cohomology groups sometimes implies vanishing of these cohomomogy groups. Later on Laufer formulated a zero or infinity law for the cohomology groups of domains in Stein manifolds. In this paper we generalize Laufer's Theorem in [L] and its version for sm…

The Wythoff construction takes a $d$-dimensional polytope $P$, a subset $S$ of $\{0,..., d\}$ and returns another $d$-dimensional polytope $P(S)$. If $P$ is a regular polytope, then $P(S)$ is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We want …

Aim of this note is to gain cohomological information about the infinite-dimensional manifold $\rm{Emb}_0 (M,N)$ of asymptotically fixed embeddings of M into N from the topology of the target manifold N. This paper has been withdrawn by the author due a conceptual mistake.

The paper proposes a Gaussian mixture model for Hilbert-space-valued data.

Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(F_n) for n>1). We prove that, in degree 3, the bounded cohomology of G with real coefficients is infi…

New learning rates for embeddings in RKHSs, even when the target is not Hilbert-Schmidt.

Let $V$ be a regular neighborhood of a negative chain of $2$-spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let $B_{p,q}$ be a rational homology ball which is smoothly embedded in $V$. Assume that the embedding is simple, i.e. the corresponding rational blow-up can be obtained by just a sequen…

Many features of dimensional reduction schemes are determined by the breaking of higher dimensional general covariance associated with the selection of a particular subset of coordinates. By investigating residual covariance we introduce lower dimensional tensors --generalizing to one side Kaluza-Klein gauge fields and…

The random graph is an infinite graph with the universal property that any embedding of $G-v$ extends to an embedding of $G$, for any finite graph. In this paper we show that this graph embeds in the curve graph of a surface $Σ$ if and only if $Σ$ has infinite genus, showing that the curve system on an infinite genus s…

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order q, the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when q < 3 is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core cu…

The loop space of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to the sphere is an infinite dimensional complex manifold. We compute the Picard group of holomorphic line bundles on this loop space as an infinite dimensional complex Lie group with Lie algebra the first Dolbe…

We define the concordance crosscap number of a knot as the minimum crosscap number among all the knots concordant to the knot. The four-dimensional crosscap number is the minimum first Betti number of non-orientable surfaces smoothly embedded in 4-dimensional ball, bounding the knot. Clearly the 4-dimensional crosscap …

Paper constructs an infinite 3-7 surface in 3D space.

Hermite polynomials improve private data generation by reducing feature count.

Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {\em diagram distances}, most of the recent attempts at using persistence diagrams in machine learning have been done through kernel methods, i.e., embeddings of …

We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric.…

We present a theory for Euclidean dimensionality reduction with subgaussian matrices which unifies several restricted isometry property and Johnson-Lindenstrauss type results obtained earlier for specific data sets. In particular, we recover and, in several cases, improve results for sets of sparse and structured spars…

The paper defines a critical value for a magnetic system and extends solutions beyond blow-up.

Using standard analysis only, we present an extension ${^\bullet\R}$ of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesim…

We investigate certain $4$-dimensional analogues of the classical $3$-dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential $2$-sphere $S$ in the boundary of a simply connected $4$-manifold $W$ such that $S$ i…

Study on exotic smooth embeddings of surfaces in 4-manifolds, revealing different properties and complexities.

Develops a new algebraic framework for differential geometry of infinite dimensional spaces.