Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.
problem Developing equations for Dirac operators in complex 3D domains.
method First-kind boundary integral equations, generalized Garding inequalities, Fredholm operators, finite dimensional kernels, Betti numbers.
result Finite dimensional kernels equal to the sum of Betti numbers, explaining the bilinear forms.
Extends Calabi operator to Riemannian locally symmetric spaces.
problem Local integrability conditions on Riemannian locally symmetric spaces.
method Generalizes Calabi operator to Riemannian locally symmetric spaces.
result Generalised operator works in irreducible case and fails in products.
GIT-Net uses neural networks to approximate PDE operators efficiently.
problem Approximating PDE operators for complex geometries.
method Parametrizes adaptive generalized integral transforms with deep neural networks.
result GIT-Net outperforms existing neural network operators in multiple areas.
Novel duality theory for operator Frobenius algebras solves long-standing hydrodynamic integrable systems problem.
problem Long-standing Eisenhart-Stäckel problem for non-degenerate integrable systems.
method Introduce duality for operator Frobenius algebras and use mutual symmetry assumption.
result Construct new infinite-dimensional integrable systems of hydrodynamic type.
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.
problem Solving evolution equations on Lie groupoids.
method Developed calculus of Fourier integral operators and used them to study the fundamental solution of the evolution equation.
result Developed a method to find the fundamental solution of the evolution equation on Lie groupoids.
Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.
problem Cohomology of relative Rota-Baxter operators on Lie algebras and groups.
method Cohomology construction, infinitesimal deformations study, differentiation and integration of Rota-Baxter operators.
result Integration of Rota-Baxter operators on Lie groups and Lie algebras.
Right inverse found for Cartan differential in rank-1 symmetric spaces.
problem Finding a right inverse for the Cartan differential in symmetric spaces.
method Integral operator approach to the Cartan differential on exact forms.
result Extension of Gauss linking integral to rank-1 symmetric spaces.
This paper integrates Nijenhuis structures into Lie groupoids.
problem Understanding Nijenhuis structures and their global counterparts.
method Identifying and integrating Lie algebroids and Lie groupoids.
result Nijenhuis structures can be integrated to Lie groupoids and vice versa.
Quantizes Stäckel integrable systems into self-adjoint operators.
problem Quantizing Stäckel integrable systems into self-adjoint operators.
method Constructs commutative self-adjoint operators from quadratic Hamiltonians in involution.
result Proves multiplicative separation of variables for Stäckel integrable systems.
Paper explores RKHS properties for derivative and integral operators.
problem Establishing sufficient conditions for reproducing property in RKHS.
method Establishing reproducing property for combinations of composition operators.
result Provides framework for regularized learning algorithms involving function values, gradients, or operators.
Study YB operators and their deformations, finding integrable and nontrivial cases.
problem Understanding deformations of Yang-Baxter operators and their integrability.
method Relating deformations to Lie algebra deformations, analyzing cohomology groups.
result Existence of integrable YB deformations and nontrivial cases not arising from SD deformations.
Study examines the geometry of a group of Fourier-integral operators related to Diff(S^1).
problem Investigating the geometry of a specific group of Fourier-integral operators.
method Defined a right-invariant pseudo-Riemannian metric on the group using renormalized traces of pseudo-differential operators.
result Extended the Hilbert-Schmidt Riemannian metric to the group.
The paper studies the asymptotic expansion of Gaussian integral operators on Riemannian submanifolds.
problem Analyzing the asymptotic behavior of Gaussian integral operators on Riemannian submanifolds.
method Deriving a full asymptotic expansion of the Gaussian integral operator and computing the first-order correction term.
result Explicit computation of the first-order correction term in terms of mean curvature vector and scalar curvature.
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.
problem Bounding Carleson-Sjölin operators on manifolds with special curvature conditions.
method Two different methods: one using distance function conditions and the other using contact orders of oscillatory integral operators.
result Improved Lp bounds for Carleson-Sjölin operators on manifolds with constant sectional curvature and those satisfying Sogge's chaotic curvature condition. 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…
Extends orbital integral evaluation to center of enveloping algebra.
problem Evaluate semisimple orbital integrals for arbitrary elements in the center of the enveloping algebra.
method Explicit geometric evaluation of Casimir operator to arbitrary elements in the center of the enveloping algebra.
result Extension of orbital integral evaluation to center of enveloping algebra.
New connections found between curvature and Euler characteristic using Schrödinger operators.
problem Establishing relationships between curvature and topological invariants of Riemannian manifolds.
method Using twisted Dirac operators and scaling of potentials to analyze the kernel of these operators.
result Found conditions under which the Euler characteristic of a manifold can be zero or non-zero.
Analyzes string topology operations using Chen's integrals and homotopy transfer.
problem Relating string topology to perturbative Chern-Simons theory.
method Develops integrals over configuration spaces and applies homotopy transfer.
result Intertwines involutive Lie bialgebra structures on homology.
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.
Researchers transform equations and define integral operators on a ball.
problem Transforming equations from half space to ball.
method Identify Poisson kernel, define extension operator, prove inequalities.
result Uniqueness of extremal functions in limit case.
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.
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.
problem Integrating geometric concepts in Riemannian manifolds with foliations.
method Deriving integral formulas involving shape operators and curvature tensor.
result Generalizes results for foliated Riemannian manifolds and includes arbitrary functions.
New method constructs solution operators for PDEs with prescribed support properties.
problem Constructing solution operators for under/overdetermined PDEs with specific support properties.
method Using a recovery on curves condition and taking smooth averages over curves, we obtain integral solution operators and representation formulas.
result Our method leads to integral representation formulas for overdetermined PDEs and solution operators for underdetermined PDEs.
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 P for elasticity satisfying $\mathscr{D}\mathscr{P…
Geometric approach to Dirac operator evolution on spacetimes.
problem Constructing the Cauchy evolution operator for Lorentzian Dirac operators.
method Realizing the operator as a sum of oscillatory integrals, relating to Feynman propagator.
result Relating Cauchy evolution operators to Feynman propagators and constructing Hadamard states.
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…
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 ∫B∗XA^(T∗X)expθ−∫B∗YA^(T∗Y)expθ. Here B∗ stands for the unit coball bundle and θ is a certain characteristic…
Incompatible operations affect Khovanov homology and spectral sequences.
problem Incompatibility between operations and spectral sequences.
method Observation of obstructions to integral lifting and spectrification.
result Lipshitz-Sarkar Steenrod operations are incompatible with Szabo's spectral sequence.
Paper generalizes paracomposition and change of variables for paradifferential operators.
problem Generalizing paracomposition and change of variables for paradifferential operators in low regularity settings.
method Drops diffeomorphism hypothesis, estimates in Sobolev and Zygmund spaces, discusses pull-back of pseudodifferential and paradifferential operators.
result Sharp estimates for composition in Sobolev and Zygmund spaces, change of variables in paradifferential operators.
Clarifies mathematical aspects of Picture Changing Operators.
problem Understanding the geometric properties of PCOs.
method Showed PCOs are chain maps between differential and integral forms on supermanifolds.
result PCOs are chain maps, providing a new perspective on their structure.
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.
problem Microlocal analysis of Lévy processes on manifolds with conjugate points.
method Microlocal analysis, pseudodifferential operators, Fourier integral operators.
result Generator can be expressed as sum of pseudodifferential and Fourier integral operators.
Construct Lax pairs for BKM equations and related integrable hierarchies.
problem Integrability of BKM equations and related systems.
method Construction of Lax pairs for BKM equations and related integrable hierarchies using Sturm-Liouville operators.
result Lax pairs for many integrable hierarchies including previously studied ones.
Inverts rank m symmetric tensor fields using line integrals.
problem Recovering symmetric tensor fields from line integrals.
method Computes normal operator and presents inversion formula.
result Recovering rank m tensor fields from data (Nm0f,…,Nmmf). We give an integral formula for the total Q′-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′-operator.
Scattering theory for harmonic one-forms on Riemann surfaces.
problem Understanding harmonic one-forms on Riemann surfaces.
method Constructing scattering theory through boundary value problems and integral operators.
result Explicit expression for the scattering matrix and proof of unitarity.
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.
problem Defining and understanding Born geometries.
method Using Künneth structures and recursion operators.
result Born connection derived from Künneth connection for integrable 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.
problem Index problems of Wilson-Dirac operators on lattice approximations of manifolds.
method Formulates and proves a K-theoretic formula for an index-type invariant. result Main theorem gives a formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds.
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…