arXiv research

Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.

Extends Calabi operator to Riemannian locally symmetric spaces.

GIT-Net uses neural networks to approximate PDE operators efficiently.

Novel duality theory for operator Frobenius algebras solves long-standing hydrodynamic integrable systems problem.

We prove that a local Hamiltonian operator of hydrodynamic type K_1 is compatible with a nondegenerate local Hamiltonian operator of hydrodynamic type K_2 if and only if the operator K_1 is locally the Lie derivative of the operator K_2 along a vector field in the corresponding domain of local coordinates. This result …

Study evolution equations on Lie groupoids using Fourier integral operators.

Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.

Right inverse found for Cartan differential in rank-1 symmetric spaces.

This paper integrates Nijenhuis structures into Lie groupoids.

Quantizes Stäckel integrable systems into self-adjoint operators.

Paper explores RKHS properties for derivative and integral operators.

Study YB operators and their deformations, finding integrable and nontrivial cases.

Study examines the geometry of a group of Fourier-integral operators related to Diff(S^1).

The paper studies the asymptotic expansion of Gaussian integral operators on Riemannian submanifolds.

This paper explains the fundamental relation between Jacobi structures and the classical Spencer operator coming from the theory of PDEs so as to provide a direct and geometric approach to the integrability of Jacobi structures. It uses recent results on the integrability of Spencer operators and multliplicative forms …

As announced in [12], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study convolability and invertibility of Lagrangian conic submanifolds of the symplectic groupoid T * G. We also identify those Lagrangian which correspond to equivariant families parametrized by the …

Improved bounds for Carleson-Sjölin operators on manifolds with specific curvature conditions.

We extend the notions of CR GJMS operators and Q-curvature to the case of partially integrable CR structures. The total integral of the CR Q-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR manifolds equipped with an orientation of the bundle of contact forms, which is nontr…

In the framework of path integral the evolution operator kernel for the Merton-Garman Hamiltonian is constructed. Based on this kernel option formula is obtained, which generalizes the well-known Black-Scholes result. Possible approximation numerical schemes for path integral calculations are proposed.

Using the Arthur-Selberg trace formula we express the index of a Dirac operator on an arithmetic quotient over a totally real field with at least two real embeddings as the integral over the index form plus a sum of orbital integrals. For the Euler operator these orbital integrals are shown to vanish for products of ra…

New connections found between curvature and Euler characteristic using Schrödinger operators.

Analyzes string topology operations using Chen's integrals and homotopy transfer.

We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of `quantum thick morphisms' defined here as particular oscillatory integral operators on functions.

In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility reductions of PDEs.

Researchers transform equations and define integral operators on a ball.

We endow the group of invertible Fourier integral operators on an open}manifold with the structure of an ILH Lie group. This is done by establishing such structures for the groups of invertible pseudodifferential operators and contact transformations on an open manifold of bounded geometry, and gluing those together vi…

Quillen proved that, if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventu…

The article derives integral formulas for foliated sub-Riemannian manifolds.

New method constructs solution operators for PDEs with prescribed support properties.

We propose a general strategy to derive null-homotopy operators for differential complexes based on the Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex. Focusing on the elasticity complex, we derive path integral operators $\mathscr{P}$ for elasticity satisfying $\mathscr{D}\mathscr{P…

Geometric approach to Dirac operator evolution on spacetimes.

We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's pri…

Incompatible operations affect Khovanov homology and spectral sequences.

We show that the index of an elliptic Fourier integral operator associated to a contact diffeomorphism $φ$ of cosphere bundles of two Riemannian manifolds X and Y is given by $\int_{B^*X}\hat{A}(T^*X)\expθ - \int_{B^*Y}\hat{A}(T^*Y)\expθ$. Here $B^*$ stands for the unit coball bundle and $θ$ is a certain characteristic…

Clarifies mathematical aspects of Picture Changing Operators.

The action of Batalin-Vilkovisky Delta-operator on semidensities in an odd symplectic superspace is defined. This is used for the construction of integral invariants on surfaces embedded in an odd symplectic superspace and for more clear interpretation of the Batalin-Vilkovisky formalism geometry.

Study analyzes Lévy process structure on manifolds with conjugate points.

Construct Lax pairs for BKM equations and related integrable hierarchies.

Inverts rank m symmetric tensor fields using line integrals.

We give an integral formula for the total $Q^\prime$-curvature of a three-dimensional CR manifold with positive CR Yamabe constant and nonnegative Paneitz operator. Our derivation includes a relationship between the Green's functions of the CR Laplacian and the $P^\prime$-operator.

Scattering theory for harmonic one-forms on Riemann surfaces.

We give a detailed microlocal study of X-ray transforms over geodesics-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator. We compute the principal symbol of both operators and the canonical relation as…

New definition of Born geometry connects to known geometries.

The Laplace-Beltrami operator (LBO) is a fundamental object associated to Riemannian manifolds, which encodes all intrinsic geometry of the manifolds and has many desirable properties. Recently, we proposed a novel numerical method, Point Integral method (PIM), to discretize the Laplace-Beltrami operator on point cloud…

Proves a lattice version of the Atiyah-Singer index theorem.

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is also equivalent to the description of all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. It is proved that every such Hamiltonian operator (or the submanifold correspondi…

We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodi…

Study normal operators of double fibration transforms with conjugate points.