arXiv research

Study evolution equations on Lie groupoids using Fourier integral operators.

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…

Study normal operators of double fibration transforms with conjugate points.

Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic …

This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using $KK$-theory.

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 …

We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and …

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…

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

New proof of Zelditch's generalization using Riemannian geometry.

Extends pseudo-differential operators theory to compact Lie groups.

Researchers found a Calabi-Yau structure and constructed a Bargmann type transformation on the Cayley projective plane.

Study analyzes Lévy process structure on manifolds with conjugate points.

Let $M$ be a $Spin$-manifold with $S^1$-action and let $σ\in S^1$ be of finite order. We show that the indices of certain twisted Dirac operators vanish if the action of $σ$ has sufficiently large fixed point codimension. These indices occur in the Fourier expansion of the elliptic genus of $M$ in one of its cusps. As …

X-ray transform on H-type groups solved, revealing function injectivity.

The paper bounds Fourier integral operators on Hardy spaces with specific conditions.

Study solves optimal portfolio selection using HJB equation.

Markov Chain Monte Carlo methods become increasingly popular in applied mathematics as a tool for numerical integration with respect to complex and high-dimensional distributions. However, application of MCMC methods to heavy tailed distributions and distributions with analytically intractable densities turns out to be…

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formu…

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…

Paper generalizes paracomposition and change of variables for paradifferential operators.

Study examines boundedness of oscillating singular integrals on specific Lie groups.

Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.

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…

Develops support theorem for analytic transforms in tomography.

Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.

DAFNO learns surrogates for complex systems on irregular geometries.

In this expository paper, we explain a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applicatio…

We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attache…

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier component…

Constructs smooth integrable magnetic systems on a two-torus.

A graphical calculus for microformal morphisms simplifies complex operations in classical and quantum physics.

Paper calculates indices for group actions using cocycles.

We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic…

Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.

We introduce and study {\it new} relative spectral invariants of {\it two} elliptic partial differential operators of Laplace and Dirac type on compact smooth manifolds without boundary that depend on both the eigenvalues and the eigensections of these operators and contain much more information about geometry. We prov…

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

We give an overview of the generalized Calderón-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. $L^p-L^q$ of…

Study identifies and analyzes three types of errors in learning Fourier operators.

We define a Fourier-Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic…

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…

EFiGP uses Fourier and eigen-decomposition for efficient ODE parameter estimation.

In this paper we give formulae for the Dixmier trace and the noncommutative residue (also called Wodzicki's residue) of pseudo-differential operators by using the notion of global symbol. We consider both cases, compact manifolds with or without boundary. Our analysis on the Dixmier trace of invariant pseudo-differenti…

We study the weighted light ray transform $L$ of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze $L$ as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike singularities of a function $f$ from its the weighted light ray transform …

We study the geometry and topology of (filtered) algebra-bundles ${\bfΨ}^{\mathbb Z}$ over a smooth manifold $X$ with typical fibre $Ψ^{\mathbb Z}(Z; V)$, the algebra of classical pseudodifferential operators of integral order on the compact manifold $Z$ acting on smooth sections of a vector bundle $V$. First a theorem…

Quantizes the relationship between Koszul and Schouten brackets in Poisson geometry.

The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.