arXiv research

The paper solves the Integration Problem for principal connections.

Solves smoothing problem in Chow's connectivity theorem.

New algorithm for Coxeter connections with maximally ramified singularities.

Fractional Laplacian inverse problem solved for connection Laplacians.

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

We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a…

Solves Deligne-Simpson problem for special connections on Gm.

Infinite fractal tree solves shortest connection problem.

Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.

A Lie algebroid classifies G-structures with connections.

Proves rotational symmetry for Serrin-type problems in doubly connected domains.

Uniqueness proof for Calderón's problem on real-analytic vector bundles.

Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.

Geometric framework for inverse problems using foliations and dual connections.

Solves probabilistic Lambert problem connecting astrodynamics with optimal mass transport.

Introduces canonical connections for sub-Riemannian manifolds with constant symbol.

Neural networks can learn optimal auction mechanisms and satisfy mode connectivity.

Given a central extension of Lie groups, we study the classification problem of lifting the structure group together with a given connection. For reductive structure groups we introduce a new connective structure on the lifting gerbe associated to this problem. Our main result classifies all connections on the central …

The paper explores the connection between 3d gravity and Chern-Simons theory using affine group connections.

We will study a linear first order system, a connection $\db$ problem, on a vector bundle equipped with a connection, over a Riemann surface. We show optimal conditions on the connection forms which allow one to find a holomorphic frame, or in other words to prove the optimal regularity of our solution. The underlying …

Study proves inequalities for eigenvalues of symmetric domains in space forms.

Tutorial on information bottleneck problems with connections to coding and learning.

The paper explores Finsler-type objects and their variational problems on spacetimes.

Several problems such as network intrusion, community detection, and disease outbreak can be described by observations attributed to nodes or edges of a graph. In these applications presence of intrusion, community or disease outbreak is characterized by novel observations on some unknown connected subgraph. These prob…

The paper shows connections can be uniquely determined by their boundary data.

Paper proves stability for recovering connections from holonomy traces.

Solves open problems on curved projective varieties.

In this paper we consider the problem of identifying a connection $\nabla$ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian $\nabla^*\nabla$ over conformally transversally anisotropic (CTA) manifolds. This was proved in \cite{LCW} for line bundles in the case of t…

We solve the local equivalence problem for sub-Riemannian structures on (2n + 1)-dimensional manifolds. We show that two sub-Riemannian structures are locally equivalent if and only if? their corresponding canonical linear connections are equivalent. When n = 1, these connections coincide with the generalized Tanaka-We…

Study improves Calderón problem for systems, uniquely determining connections and potentials.

A simple thresholding technique improves graph selection in neural connectivity studies.

In this paper we establish a connection between free boundary minimal surfaces in a ball in $\mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.

Consumers with low demand, like households, are generally supplied single-phase power by connecting their service mains to one of the phases of a distribution transformer. The distribution companies face the problem of keeping a record of consumer connectivity to a phase due to uninformed changes that happen. The exact…

New heuristics for predicting links in multiplex networks.

Researchers find conditions for autoparallels to be Finsler geodesics.

We consider the nonlinear problem of determining a connection and a Higgs field from the corresponding parallel transport along geodesics on a Riemannian manifold with boundary, in any dimension. The problem can be reduced to an integral geometry question of some attenuated geodesic ray transform through a pseudolinear…

A long-standing obstacle to progress in deep learning is the problem of vanishing and exploding gradients. Although, the problem has largely been overcome via carefully constructed initializations and batch normalization, architectures incorporating skip-connections such as highway and resnets perform much better than …

We will describe some results regarding the algorithmic nature of homeomorphism problems for manifolds; in particular, the following theorem. Theorem 1: Every PL or smooth simply connected manifold M^n of dimension n at least 5 can be recognized among simply connected manifolds. That is, there is an algorithm to decide…

We consider the problem of identifying a unitary Yang-Mills connection $\nabla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $\nabla^*\nabla$ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the cas…

We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension $n\geq 5$. We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also…

The article classifies liftings of connections on differential manifolds for geodesic modeling.

Study variational problem on manifold with special distributions.

The purpose of this paper is to use the framework of Lie algebroids to study optimal control problems for affine connection control systems on Lie groups. In this context, the equations for critical trajectories of the problem are geometrically characterized as a Hamiltonian vector field.

We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection a…

Paper finds non-positive Weyl connections on Lie groups, confirming a conjecture.

In this paper we consider existence and multiplicity results concerning affine connections on $C^{k}$-manifolds $M$ whose coefficients are as regular as one needs, following the regularity theory introduced in arXiv:1908.04442. We show that if $M$ admits a $B_{α,β}^{k}$-structure, then the existence of such regular con…

Machine learning predicts flight connections for airline crew scheduling.

The aim of this paper is to generalize the theory of nonlinear connections of Grifone ([3] and [4]). We adopt the point of view of Anona [1] and continue developing the approach established by the first author in [10]. The first part of the work is devoted to the problem of associating to each $L$-regular linear connec…