arXiv research

Finite-type surfaces have a topological Hopf property.

Constructs algorithms to recognize and classify 2D surfaces.

Study topological properties of foliations induced by closed 1-forms on orbifolds.

This thesis introduces big mapping class groups and their structure.

Characterizes curves with short representatives on hyperbolic surfaces.

Observable structures of a topological field theory of AKSZ type are analyzed. From a double (or multiple) complex structure of observable algebras, new topological invariants are constructed. Especially, Donaldson polynomial invariants and their generalizations are constructed from a topological field theory of AKSZ t…

The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.

New integrable deformations for topological hierarchies from Frobenius manifolds.

Study characterizes compact Einstein-type manifolds with boundary.

Suppose M is a noncompact connected smooth 2-manifold without boundary and let D(M)_0 denote the identity component of the diffeomorphism group of M with the compact-open C^infty-topology. In this paper we investigate the topological type of D(M)_0 and show that D(M)_0 is a topological ell_2-manifold and it has the hom…

New method confirms conjectures about specific Legendrian knots.

We construct pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume. Topologically these orbifolds are mapping tori of pseudo-Anosov maps of the surface of genus 2, with singular locus a fibred (hyperbolic) link with five components.

Overview of infinite surface mapping class groups.

Two results on end spaces of infinite type surfaces, answering questions about their topology and equivalence.

We classify Legendrian knots of topological type $7_6$ having maximal Thurston--Bennequin number confirming the corresponding conjectures of Chongchitmate--Ng.

This is a survey on two closely related subjects. First, we review the study of topological structure of `finite type' components of spaces of Bridgeland's stability conditions on triangulated categories. The key is to understand Happel-Reiten-Smalo tilting as tiling of cells. Second, we review topological realizations…

Topological twists for 4d N=2 theories depend on spacetime type, gerbe connections, and generalized spin-c structures.

Graph potentials link to topological QFTs, with computational methods.

A polynomial knot in $\mathbb{R}^n$ is a smooth embedding of $\mathbb{R}$ in $\mathbb{R}^n$ such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space $\mathcal{P}$ of polynomial knots in $\mathbb{R}^3$ with the inductive limit topology coming from the spaces $\m…

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-Émery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvat…

New categorical actions link topological and algebraic structures.

Finite type invariants separate PL links in 3D space.

We examine the topology of various spaces of locally homogeneous affine manifolds which arise from the classification result of Opozda [B. Opozda, A classification of locally homogeneous connections on 2-dimensional manifolds, Differential Geom. Appl. 21 (2004), 173-198.] as orbits of the action of $GL(2,\mathbb{R})$ (…

Proves simple type conjecture for mod 2 Seiberg-Witten invariants.

Unlike the case of surfaces of topologically finite type, there are several different Teichmüller spaces that are associated to a surface of topological infinite type. These Teichmüller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, …

Study classifies 3D Hessian manifolds, proving their topology.

Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.

We present and prove a topological characterization of geodesic laminations on hyperbolic surfaces of finite type.

In this paper, we study stability and instability problem for type-II partitioning problem. First, we make a complete classification of stable type-II stationary hypersurfaces in a ball in a space form as totally geodesic $n$-balls. Second, for general ambient spaces and convex domains, we give some topological restric…

Study monopole h-invariants from a topological viewpoint.

It is known that all but finitely many leaves of a measured foliated 2-complex of thin type are quasi-isometric to an infinite tree with at most two topological ends. We show that if the foliation is cooriented, and the associated R-tree is self-similar, then a typical leaf has exactly one topological end. We also cons…

We combine Freedman's topology with Eliashberg's holomorphic theory to construct Stein neighborhood systems in complex surfaces, and use these to study various notions of convexity and concavity. Every tame, topologically embedded 2-complex K in a complex surface, after C^0-small topological ambient isotopy, is the int…

Study complex structures of hyperkähler manifolds with infinite type.

We study the volume growth of hyperkaehler manifolds of type $A_{\infty}$ constructed by Anderson-Kronheimer-LeBrun and Goto. These are noncompact complete 4-dimensional hyperkaehler manifolds of infinite topological type. These manifolds have the same topology but the hyperkaehler metrics are depends on the choice of …

The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves in real toric surfaces is a classical subject that has undergone considerable evolution. On the other hand, not much is …

We show that the length spectrum metric on Teichmüller spaces of surfaces of infinite topological type is complete. We also give related results and examples that compare the length spectrum Teichmüller space with quasiconformal and the Fenchel-Nielsen Teichmüller spaces on such surfaces

The paper explores gaps in curvature-related metrics and rigidity.

Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$, denoted $\mathrm{PMap}(S)$, detects the homeomorphism type of $S$. As a corolla…

New combinatorial type helps distinguish plane curve topologies.

New topologies for star-shaped sets without boundedness.

On $(X,ω)$ compact Kähler manifold, given a model type envelope $ψ\in PSH(X,ω)$ (i.e. a singularity type) we prove that the Monge-Ampère operator is an homeomorphism between the set of $ψ$-relative finite energy potentials and the set of $ψ$-relative energy measures endowed with their strong topologies given as the coa…

Study calculates integral cohomology of non-orientable infinite type surfaces.

Category theory generalizes finite type invariants using diagrams systems.

Develops persistent Khovanov homology for tangles.

In this paper, we construct a complete n-dim Riemannian manifold with positive Ricci curvature, quadratically nonnegatively curved infinity and infinite topological type. This gives a negative answer to a conjecture by Jiping Sha and Zhongmin Shen in the case of n greater than or equal to 6.

Notes on Khovanov and knot Floer theories' stable homotopy types.

The notion of covering type was recently introduced by Karoubi and Weibel to measure the complexity of a topological space by means of good coverings. When X has the homotopy type of a finite CW-complex, its covering type coincides with the minimum possible number of vertices of a simplicial complex homotopy equivalent…

We recently established a Toponogov type triangle comparison theorem for a certain class of Finsler manifolds whose radial flag curvatures are bounded below by that of a von Mangoldt surface of revolution (arXiv:1205.3913). In this article, as its applications, we prove the finiteness of topological type and a diffeomo…