arXiv research

Develops a method to compute Morse homology for clean but not necessarily transverse intersections.

Essential obstruction found for gluing $G_{2}-$instantons with singularities.

This paper constructs Seiberg-Witten monopoles from harmonic spinors on 3-manifolds.

New Morse theory techniques glue nontransverse flowlines.

Solves C^3 null gluing problem for Einstein vacuum equations.

This paper circulated previously in a draft version. Now, upon general request, it is about time to distribute the more detailed (and much longer) version. The main technical issues revolve around the fine structure of the compactification of the moduli spaces of flow lines and the obstruction bundle technique, with re…

New approach removes obstructions in gluing spacelike and null hypersurfaces in Einstein equations.

An $n$-dimensional manifold $M$ ($n\ge 3$) is called {\it generalized graph manifold} if it is glued of blocks that are trivial bundles of $(n-2)$-tori over compact surfaces (of negative Euler characteristic) with boundary. In this paper two obstructions for generalized graph manifold to be nonpositively curved are des…

Study surgeries on Klein bottle knots in 3-manifolds using Heegaard Floer homology.

In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing t…

Solves Einstein vacuum equations gluing problem for close Minkowski data.

We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assu…

Although our main interest here is developing an appropriate analog, for diffeological vector pseudo-bundles, of a Riemannian metric, a significant portion is dedicated to continued study of the gluing operation for pseudo-bundles introduced in arXiv:1509.03023. We give more details regarding the behavior of this opera…

Paper detects duality obstruction in smooth calibrations.

We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action…

We obtain necessary and sufficient conditions for the existence of "conservation laws" on null hypersurfaces for the wave equation on general four-dimensional Lorentzian manifolds. Examples of null hypersurfaces exhibiting such conservation laws include the standard null cones of Minkowski spacetime and the degenerate …

We study a finite rank bundle $\mathbf{F}$ over a neighborhood of $J$-Holomorphic map Moduli Spaces, prove the exponential decay of the derivative of the gluing maps for $\mathbf{F}$ with respect to the gluing parameter.

Griffiths' first obstruction formula for vector bundles is derived.

Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.

Paper solves Einstein vacuum equations gluing problem with applications.

We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for defor…

The paper proves conditions for Kähler-Einstein metrics on certain bundles.

The paper studies Kähler-Einstein metrics on circle bundles and their obstruction flatness.

Proves Massey's theorems on complex structure obstructions.

Analytic surgery and gluing formula for torsion forms in fiber bundles.

In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat Kähler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with $16$ orbifold points, and resolves the orbifold singularities by gluing in $16$ small Eguchi-Hanson manifolds which all h…

The goal of this paper is the construction of a compact manifold with G$_2$ holonomy and nodal singularities along circles using twisted connected sum method. This paper finds matching building blocks by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by …

In this paper, we explore the theme of orbifold stratified spaces and establish a general criterion for them to be smooth orbifolds. This criterion utilizes the notion of linear stratification on the gluing bundles for the orbifold stratified spaces. We introduce a concept of good gluing structure to ensure a smooth st…

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their …

Non-trivial Clifford bundle from loop space tangent bundle.

We demonstrate an obstruction to finding certain splittings of four-manifolds along sufficiently twisted circle bundles over Riemann surfaces, arising from Seiberg-Witten theory. These obstructions are used to show a non-splitting result for algebraic surfaces of general type.

This paper aims to describe the behavior of diffeological differential forms under the operation of gluing of diffeological spaces along a smooth map. In the diffeological context, two ways of looking at diffeological forms are available, that of the vector space of all diffeological forms on a given space, and that of…

Contradicts claims about Poincaré complexes and homology manifolds.

Computes infinitesimal automorphisms for $L$-valued Higgs bundles, leading to DM stacks.

Theory of smooth relative connections on quiver bundles developed.

Study elliptic operators on glued manifolds, reducing to finite-dimensional systems.

In this paper we define a $\mathbf{QP}^1$-valued class function on the mapping class group $\mathcal{M}_{g,2}$ of a surface $Σ_{g,2}$ of genus $g$ with two boundary components. Let $E$ be a $Σ_{g,2}$ bundle over a pair of pants $P$. Gluing to $E$ the product of an annulus and $P$ along the boundaries of each fiber, we …

In this paper we extend first the Bismut-Lott's analytic torsion form for flat vector bundles to the boundary case, then we establish its gluing formula on a smooth fibration under the assumption that a fiberwise Morse function exists. We assume that the metrics have product structures near the cutting hypersurface.

In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted …

We consider a diffeological counterpart of the notion of a vector bundle (we call this counterpart a pseudo-bundle, although in the other works it is called differently; among the existing terms there are a "regular vector bundle" of Vincent and "diffeological vector space over X" of Christensen-Wu). The main differenc…

We prove a gluing theorem for solutions of Hitchin's self-duality equations with logarithmic singularities on a rank-2 vector bundle over a noded Riemann surface representing a boundary point of Teichmüller moduli space.

We study the obstruction to the exactness of the variational complex for a field theory on an affine bundle.

The paper explores properties of CR hypersurfaces and their flatness.

Study integrability of generalized almost complex structures on S^6.

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston's gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in te…

We give new and rather general gluing theorems for anti-self-dual (ASD) conformal structures, following the method suggested by Floer. The main result is a gluing theorem for pairs of conformally ASD manifolds `joined' across a common piece (union of connected components) of their boundaries. This theorem genuinely ope…

The paper examines obstacles to extending deformation quantization of vector bundles.

In this article we use the adiabatic method to prove the gluing formula of real analytic torsion forms for a flat vector bundle on a smooth fibration under the assumption that the fiberwise twisted cohomology groups associated to the fibration of the cutting hypersurface are vanished. In this paper we assume that the m…