arXiv research

Develops method to create non-Abelian Ricci-flat graphs via bundles.

Study orders of canonical bundles over graph configuration spaces.

Optimizes edge coloring in graph bundling for better edge differentiation.

Localized signal representation on graph bundles using Fourier analysis.

Investigates conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.

Develops mixed quantization for graph vector bundles.

The paper proves drilled bundles over graphs are virtually special cubulable.

Surjectivity of Cannon-Thurston map proven for metric graph bundles.

It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minim…

Paper proves stability of flow in cotangent bundle for special Lagrangian submanifolds.

We consider graphs Sigma^n in R^m with prescribed mean curvature and flat normal bundle. Using techniques of Schoen, Simon and Yau, and Ecker-Huisken, we derive an interior curvature estimate of the form |A|^2<=C/R^2 up to dimension n<=5, where C is a constant depending on natural geometric data of Sigma^n only. This g…

A compact 4-dimensional manifold is a non-singular graph-manifold if it can be obtained by the glueing T^2-bundles over compact surfaces (with boundary) of negative Euler characteristics. If none of glueing diffeomorphisms respect the bundle structures, the graph-structure is called reduced. We prove that any homotopy …

We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the e…

Study Higgs sections and flat sections for nonlinear harmonic bundles.

The paper extends Laplacian spectra approximations to vector bundles.

Graph coloring is explained using a topological field theory with defects.

Explicit computation of Kontsevich weights for symplectic Poisson structures.

The paper studies symplectic geometry of cotangent bundles using sheaves theory.

Characterizes lamination spaces of graphs on a pair of pants.

Extends model geometries for surface bundles over graphs using tight trees.

Improve exposition and explain metric bundle equivalence.

Researchers find hypersurfaces in a Riemannian vector bundle with specific curvature.

This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold $N$, the set of mapping degrees $cD(M,N)$ is finite for any 3-manifold $M$, unless $N$ is finitely covered by either a torus bundle, or a trivial circle…

The paper proposes and discusses semiorthogonal decompositions for moduli spaces of vector bundles.

The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then…

We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclid…

VDWs enhance graph neural networks for analyzing complex data.

Revisits and proves a reparametrization theorem for multi-valued graphs in higher codimension.

Graph continuous operators become Riesz continuous after multiplication by unitary operators.

We review recent probabilistic results on covariant Schrödinger operators on vector bundles over (possibly locally infinite) weighted graphs, and explain applications like semiclassical limits. We also clarify the relationship between these results and their formal analogues on smooth (possibly noncompact) Riemannian m…

Diagonal complexes generalize associahedra to surfaces, providing models for ribbon graphs and tautological bundles.

In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, an…

This paper derives formulas for Chern classes of triangulated circle bundles using combinatorial necklaces.

In this paper we aim for a generalisation of the Steenrod Approximation Theorem from, concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalisation is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a c…

Develops a method to construct entire minimal graphs of odd dimensions.

Study $G_2$-manifolds from symplectic $SU(3)$-manifolds with $T^2$-symmetry.

New examples of complex automorphisms with unique graph structures.

Let N be a closed irreducible 3-manifold and assume N is not a graph manifold. We improve for all but finitely many S^1-bundles M over N the adjunction inequality for the minimal complexity of embedded surfaces. This allows us to completely determine the minimal complexity of embedded surfaces in all but finitely many …

The study constructs optimal tori on Fano manifolds and confirms mirror symmetry.

Study special circle bundles over moduli spaces of quadratic differentials.

We introduce a new construction, the isotropy groupoid, to organize the orbit data for split $Γ$-spaces. We show that equivariant principal $G$-bundles over split $Γ$-CW complexes $X$ can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex $A=Γ\backsl…

In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of…

We introduce the concept of Hypoelliptic Diffusion Maps (HDM), a framework generalizing Diffusion Maps in the context of manifold learning and dimensionality reduction. Standard non-linear dimensionality reduction methods (e.g., LLE, ISOMAP, Laplacian Eigenmaps, Diffusion Maps) focus on mining massive data sets using w…

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…

We construct a Legendrian version of Envelope theory. A tangential family is a 1-parameter family of rays emanating tangentially from a smooth plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle $PT^*\R^2$. We study the singular…

We perform two explicit computations of bordered Heegaard Floer invariants. The first is the type D trimodule associated to the trivial S^1 bundle over the pair of pants P. The second is a bimodule that is necessary for self-gluing, when two torus boundary components of a bordered manifold are glued to each other. Usin…

Study asymptotic expansion of graph Laplacian on discretized surfaces, relating spanning trees and cycle-rooted forests.

Let $M=Σ_1\times Σ_2$ be the product of two compact Riemannian manifolds of dimension $n\geq 2$ and two, respectively. Let $Σ$ be the graph of a smooth map $f:Σ_1\mapsto Σ_2$, then $Σ$ is an $n$-dimensional submanifold of $M$. Let ${\frak G}$ be the Grassmannian bundle over $M$ whose fiber at each point is the set of …