arXiv research

This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham…

The paper extends affine connection results to singular warped and twisted products.

This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.

We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …

Solves Deligne-Simpson problem for special connections on Gm.

Method counts connected 2D stratifolds with singular curves and components.

New algorithm for Coxeter connections with maximally ramified singularities.

The paper studies complex affine structures near irregular singularities.

New minimal surfaces found using a modified metric connection.

Develops connections and curvature concepts for non-smooth spaces.

In previous work, the authors have developed a geometric theory of fundamental strata to study connections on the projective line with irregular singularities of parahoric formal type. In this paper, the moduli space of connections that contain regular fundamental strata with fixed combinatorics at each singular point …

Paper finds essential regularity in singular connections.

In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In add…

Extends foliation results to singular cases.

Construct opers with apparent singularities from λ-connections on Riemann surfaces.

Bounds on saddle connections on flat spheres with conical singularities.

We establish a connection between Morin singularities and stable homotopy groups of spheres. This connection allows us to describe how the images of singularity strata behave around the image of a more complicated stratum.

The paper proves ideal triangulations and disk unfolding for singular flat surfaces.

We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on $μ$-polystable holomorphic bundles over $\mathbf{P}^{n-1}$.

In dimension 7, we establish a Fredholm theory for a Dirac-type operator associated to a connection with point singularities. There are two applications. $1$. over a closed 7-manifold, under some natural conditions, a $G_{2}-$instanton and its point singularities can still be "seen" when the $G_{2}-$structure is proper…

Study of minimal surfaces in 3D space with special connections.

The paper studies affine connections on singular warped products and their curvature.

Survey connects singularity invariants to link pairings.

We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the …

We proved a uniqueness theorem of tangent connections for a Yang-Mills connection with an isolated singularity with a quadratic growth of the curvature at the singularity. We also obtained controls over the rate of the asymptotic convergence of the connection to the tangent connection under assumptions that the connect…

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 …

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

We construct local examples of singular Hermitian Yang-Mills connections over $B_1\subset \mathbb{C}^3$ with uniformly bounded $L^2$-energy, but the number of essential singular points can be arbitrarily large.

We prove that the introduction of the class of geometrically atomic bundle maps by Harvey and Lawson in their theory of singular connections is not necessary because an arbitrary map satisfies the conditions of geometric atomicity.

We describe all connected components of the space of hyperbolic Gorenstein quasi-homogeneous surface singularities. We prove that any connected component is homeomorphic to a quotient of R^d by a discrete group.

We provide a complete classification of hexagonal singular 3-web germs in the complex plane, satisfying the following two conditions: 1) the Chern connection remains holomorphic at the singular point, 2) the web admits at least one infinitesimal symmetry at this point. As a by-product, a classification of hexagonal wei…

The paper studies stability and singularities of a two-convex level set flow.

We prove a conjecture due to M. Kazarian, connecting two classifying spaces in singularity theory. These spaces are: - Kazarian's space (generalizing Vassiliev's algebraic complex and) showing which cohomology classes are represented by singularity strata. - Author's space $X_τ$ giving homotopy representation of cobord…

Modeling wormhole creation without singularities in relativity.

Introduces Carrollian Lie algebroids to handle singular Carrollian geometries.

Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up to the four exceptional cases in low genera the strata of meromorphic quadratic …

Study families of Lie algebroids on complex spaces, introducing unfoldings.

Study flat connections with logarithmic singularities on complex plane curves.

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: $\mathbb{S}^4$, $\mathbb{CP}^2$, $\mathbb{S}^2\times\mathbb{S}^2$, or $\mathbb{CP}^2\#\pm \mathbb{CP}^2$. As an…

The paper studies deformations and confluences of singularities in meromorphic connections and quadratic differentials.

Paper presents new cobordism sequences for singular maps.

Classifies meromorphic affine connections on complex surfaces.

Study conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.

On a complex curve, we establish a correspondence between integrable connections with irregular singularities, and Higgs bundles such that the Higgs field is meromorphic with poles of any order. The moduli spaces of these objects are obtained by fixing at each singularity the polar part of the connection. We prove that…

Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.

New Heegaard Floer homology for orbifolds with cyclic singularities.

The singular set of a foliation is always connected under certain conditions.

Abstract: New geometric incarnation of isomonodromy functors.