arXiv research

New Heegaard Floer homology for orbifolds with cyclic singularities.

Resolves singular fibers of a 5-manifold with $S^1$ action.

Algorithm classifies saddle-focus singularities in Hamiltonian systems.

Classifies solutions of Toda equations near singularities.

Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family of real isolated suspension singularities from $\mathbb{R}^6$ to $\mathbb{R}^2$…

We give a complete proof of Thurston's Orbifold Theorem for very good 3-orbifolds of cyclic type. An orbifold is said to be very good when it has a finite cover which is a manifold. A 3-orbifold is of cyclic type if the singular set is a non-empty 1-manifold transverse to the boundary.

Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus. Later, the first author applies this to the particular case of coverings of $\mat…

The paper solves a 5-manifold foliation problem using a Sasaki-Ricci flow.

Study on homology groups of cDV singularity links, identifying their topology.

For a smoothing Y of a 2-dimensional cyclic quotient singularity X, we construct a simple handle decomposition of Y by using a particular birational map from Y to the projective plane. The manifold Y is built up from the product of an annulus with a disk by attaching 2-handles in a manner which can be described by mean…

This paper presents a decomposition of the 3-strand Singular Braid Group.

The aim of this paper is to explain the relationship between the (co)homology of the free loop space and the Hochschild homology of its singular cochain algebra. We introduce all the relevant technical tools, namely simplicial and cyclic objects, and we provide the various steps of the proofs, which are scattered aroun…

We prove every oriented compact cyclic $3$-orbifold has a contact structure. There is another proof in the web by Daniel Herr in his uploaded thesis which depends on open book decompositions, ours is independent of that. We define overtwisted contact structures, tight contact structures and Lutz twist on oriented compa…

The paper studies cyclic covers of rational surfaces and their Hodge structures.

Unbraided wiring diagrams for Stein fillings of lens spaces are described.

In this article, we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a …

We verify the conjecture formulated in math.AG/0111298 for suspension singularities of type $g(x,y,z)= f(x,y)+z^n$, where $f$ is an irreducible plane curve singularity. More precisely, we prove that the modified Seiberg-Witten invariant of the link $M$ of $g$, associated with the canonical $spin^c$ structure, equals $-…

This paper proposes an axiomatic for Cyclic Foam Topological Field theories. That is Topological Field theories, corresponding to String theories, where particles are arbitrary graphs. World surfaces in this case are two-manifolds with one-dimensional singularities. We proved that Cyclic Foam Topological Field theories…

This paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity. It proves that a complete hyperkähler manifold asymptotic to a circle fibration over the Euclidean three-space is either the standard $\rl^3 \times \sph^1$ or a multi-Taub-NUT mani…

Simplified presentation of symplectic fillings of lens spaces.

We show that the explicit ALE Ricci-flat Kahler metrics constructed by Eguchi-Hanson, Gibbons-Hawking, Hitchin and Kronheimer, and their free quotients are metrics obtained by Tian-Yau techniques. The proof relies on a construction of good compactifications of Q-Gorenstein deformations of quotient surface singularities…

Paper studies planar extensions in o-minimal structures.

Study lens spaces' definite fillings, classifying those with specific inequalities.

We use the equivariant Yang-Mills moduli space to investigate the relation between the singular set, isotropy representations at fixed points, and permutation modules realized by the induced action on homology for smooth group actions on certain 4-manifolds.

An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with cyclic quotient singularities is proved. We present two applications of this theorem. The first is to compute the dimension of the deformation space of the Calderbank-Singer scalar-flat Kahler toric ALE spaces. A corollary of t…

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we establish its validity for a large class of singularities: some rational and minimally e…

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollár formulated the algebraic version of the Montgomery-Yang problem: every projective surface $S$ with the second Betti number $b_2…

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere ${\mathbb S}^5$ has at most 3 non-free orbits. Using a certain one-to-one correspondence, Kollár formulated the algebraic version of the Montgomery-Yang problem: every projective surface $S$ with quotient sin…

Let C be the contact structure naturally induced on the lens space L(p,q) by the standard contact structure on the three--sphere. We obtain a complete classification of the symplectic fillings of (L(p,q),C) up to orientation-preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorp…

Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize Kronheimer's results on hyperkahler 4-manifolds, by giving an explicit classificatio…

We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blo…

Foundations for free boundary Brakke flows established.

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on 4-manifolds with non-trivial fu…

The aim of this paper is to define a chain level refinement of the Batalin-Vilkovisky (BV) algebra structure on the homology of the free loop space of a closed, oriented $C^\infty$-manifold. For this purpose, we define a (nonsymmetric) cyclic dg operad which consists of "de Rham chains" of free loops with marked points…

We present a construction of complete self-dual Einstein metrics of negative scalar curvature on an uncountable family of manifolds of infinite topological type, which are enumerated by continued fraction expansions of irrational numbers. These manifolds may be regarded as limits of the resolutions of cyclic quotient s…

We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory over certain cyclic orbifolds of the 5-sphere which are Sasaki-Einstein manifolds. We obtain new quiver gauge theories extending those induced via reduction over the leaf spaces of the characteristic foliation of the Sasaki-Einstein structure, whi…

We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schläfli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we present here the concrete calculations. We express the length of the singular locus in t…

Study symplectic 4-orbifolds with vanishing canonical class, finding new structures and resolutions.

Let $M$ be a closed, connected, orientable topological four-manifold with $H_1(M)$ nontrivial and free abelian, $b_2(M)\ne 0, 2$, and $χ(M)\ne 0$. We show that if $G$ is a finite group of 2-rank $\le 1$ which admits a homologically trivial, locally linear, effective action on $M$, then $G$ must be cyclic. With addition…

This paper completes the classification of certain surface singularities with rational homology disk smoothings.

This paper introduces 'General Cyclical Training' for neural networks.

The paper generalizes cyclic metrics in homogeneous Finsler geometry.

If the fundamental group of the complement of a smooth embedding f: S^2 \subset R^4 is a cyclic group, the map can be deformed to the standard embedding by a generic one-parameter family with at most cusp singularities. If two smooth embeddings are connected by such a deformation, they will be called cusp equivalent. W…

By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all non-degenerate. For such Morse functions, Saeki and Yamamoto have previously defined a …

Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual homological notion of linking by taking into account the fundamental group of the ambient …

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…

In this paper we study the connections between cyclic presentations of groups and branched cyclic coverings of (1,1)-knots. In particular, we prove that every n-fold strongly-cyclic branched covering of a (1,1)-knot admits a cyclic presentation for the fundamental group encoded by a Heegaard diagram of genus n.

Characterizes non-degenerate cyclic metric Lie algebras.