arXiv research

The paper shows infinitely many components in Floer Hessians space.

Study flip graphs for surfaces of infinite type, finding uncountably many connected components.

For any n\ge 2, we give infinitely many unsplittable links of n components in the 3-sphere which admit non-trivial surgery yielding the 3-sphere again and whose components are mutually distinct hyperbolic knots. Berge and Kawauchi gave 2-component hyperbolic links with those two properties. We can also give infinitely …

Study on knot $7_4$ surgeries reveals infinite residue characteristics and infinite order points.

Paper finds first examples of unlinked knots that can't be separated.

Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no longer well-defined and has to be replaced by new concepts. This article discusses the nature of two such concepts, recently introduc…

The paper defines infinite Schottky groups and their applications to infinite type surfaces.

The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.

Study shows infinitely many path components for positive Ricci curvature metrics on certain spin manifolds.

Study mapping class groups of infinite type surfaces with noncompact boundaries.

Defined a new graph type for compact surfaces, proving its connectedness and infinite diameter.

Study shows infinite families of manifolds with nonnegative curvature.

Genus $g$ Torelli space is the moduli space of genus $g$ curves of compact type equipped with a homology framing. The hyperelliptic locus is a closed analytic subvariety consisting of finitely many mutually isomorphic components. We use properties of the hyperelliptic Torelli group to show that when $g\geq 3$ these com…

We give examples of generalized complex four-manifolds whose moduli space has infinitely many components.

We give infinitely many $2$-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any $2$-component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.

The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.

The study finds infinitely many different geometries for odd-dimensional manifolds with positive Ricci curvature.

This paper investigates the strength of the trace field as a commensurability invariant of hyperbolic 3-manifolds. We construct an infinite family of two-component hyperbolic link complements which are pairwise incommensurable and have the same trace field, and infinitely many 1-cusped finite volume hyperbolic 3-manifo…

Study shows infinitely many metrics with nonnegative sectional or positive Ricci curvature on specific 5D quotients.

Study shows infinite volumes of moduli spaces for certain groups.

Study infinite symplectic forms on ruled surfaces.

Researchers create infinite Brunnian links of 3-balls in 4-sphere.

We review the standard Hopf construction of Reeb components with leafwise complex structure and almost determine the group of leafwise holomorphic smooth automorphisms for Reeb components of certain type in the case of complex leaf dimension one. In particular, it contains an infinite dimensional vector space.

The study finds infinite knot exteriors with meridional surfaces of any genus and boundary components.

We give an infinite presentation for the mapping class group of a non-orientable surface with boundary components. The presentation is a generalization of the presentation given by the second author [15].

The paper finds virtual knots with specific unknotting indices.

We investigate great circle links in the three-sphere, the class of links where each component is a great circle. Using the geometry of their complements, we classify such links up to five components. For any two-bridge knot complement, there is a finite cover that is the complement of a link of great circles in $S^3$.…

We show that in each dimension $4n+3$, $n\ge 1$, there exist infinite sequences of closed smooth simply connected manifolds $M$ of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these proper…

GP-PCA reduces infinite-dimensional GP posteriors to a finite space for meta-learning.

Study shows infinitely many nonnegatively curved metrics on quotient spaces.

The paper finds infinite knots with surfaces of any positive genus.

The automorphisms group of the 3-dimensional Reeb component with complex leaves is computed in the case where the component is obtained by the Hopf construction and the holonomy of the boundary leaf is not tangent to the identity to the infinite order. Combined with a previous work, for 3-dimensional Reeb components ob…

Uniform foliations with Reeb components on 3-manifolds.

Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X times Y. This implies that the minimal spanning forest corresponding …

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy ${\mathbb {R}} P^5$ has infinitely many path components. We also show that in each dimension $4k+1$ there are at least $2^{2k}$ homotopy ${\mathbb {R}} P^{4k+1}$s of pairwise distinct oriented diffeomorphism type for which the…

New method finds infinitely many knots in 3-manifolds.

The paper constructs manifolds with infinite holes from a given manifold.

We compute the Kauffman bracket skein module of the complement of a twist knot, finding that it is free and infinite dimensional. The basis consists of cables of a two-component link, one component of which is a meridian of the knot. The cabling of the meridian can be arbitrarily large while the cabling of the other co…

We use the $η$ invariants of spin$^c$ Dirac operators to distinguish connected components of moduli spaces of Riemannian metrics with positive Ricci curvature. We then find infinitely many non-diffeomorphic five dimensional manifolds for which these moduli spaces each have infinitely many components. The manifolds are …

We demonstrate the existence of spherically-symmetric truly naked black holes (TNBH) for which the Kretschmann scalar is finite on the horizon but some curvature components including those responsible for tidal forces as well as the energy density $\barρ$ measured by a free-falling observer are infinite. We choose a ra…

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link L with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on…

We show that in each dimension $n\ge 10$ there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian $n$-manifolds with $0\le \sec\le 1$, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type whic…

Bayesian nonparametric (BNP) models provide elegant methods for discovering underlying latent features within a data set, but inference in such models can be slow. We exploit the fact that completely random measures, which commonly used models like the Dirichlet process and the beta-Bernoulli process can be expressed a…

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

Develops robust methods for infinite-dimensional stochastic processes.

Study shows infinite-dimensional third bounded cohomology for non-orientable surfaces.

We study the group of leafwise holomorphic smooth automorphisms of Reeb components of leafwise complex foliation which are obtained by a certain Hopf construction. In particular, in the case where the boundary holonomy is infinitely tangent to the identity, we determine the structure of the group of leafwise holomorphi…

We show that the moduli space of Ricci positive metrics on certain homotopy spheres has infinitely many connected components.