arXiv research

Researchers develop weighted GJMS operators for smooth metric measure spaces.

Smooth bundles with rough data maintain Hodge kernel isomorphism.

Unified positive mass theorem and Dirac operator study on weighted manifolds.

We consider the operator algebra generated by pseudodifferential operators on a closed smooth surface and shift operator induced by a Morse--Smale diffeomorphism of this surface. Elements in this algebra are considered as operators in the scale of Sobolev spaces and the aim of this paper is to describe how Fredholm pro…

Study differential operators over maps and their applications in supermanifolds.

Confirming operator characterization on smooth manifolds.

Novel method for nowcasting implied volatility using neural operators.

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set o…

We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah-Hirzebruch (AHSS) type, where we provide a filtration by the Cech resolution of smooth manifolds. This allows for systematic st…

The paper studies boundedness of pseudo-differential operators on smooth manifolds.

The purpose of this note is to extend the results of V. Guillemin on elliptic self-adjoint pseudodifferential operators of order one, from operators defined on smooth functions on a closed manifold to operators defined on smooth sections in a vector bundle of Hilbert modules of finite type over a finite von Neumann alg…

Let $M$ be a compact smooth manifold equipped with a positive smooth density $μ$ and $H$ be a smooth distribution endowed with a fiberwise inner product $g$. We define the Laplacian $Δ_H$ associated with $(H,μ,g)$ and prove that it gives rise to an unbounded self-adjoint operator in $L^2(M,μ)$. Then, assuming that $H$ …

Constructs families of Toeplitz operators for symplectic fibrations.

Proves invertibility of layer potentials for generalized Stokes operators on smooth domains.

This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.

Study of singularities in two-dimensional Nijenhuis operators with non-zero trace differential.

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…

Suspensions of manifolds by circle surgeries are key in free action constructions.

Article studies symmetry in smooth vector bundles using advanced operations.

Study BV operators on holomorphic polyvector fields on toric varieties.

Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for …

The three operator splitting scheme was recently proposed by [Davis and Yin, 2015] as a method to optimize composite objective functions with one convex smooth term and two convex (possibly non-smooth) terms for which we have access to their proximity operator. In this short note we provide an alternative proof for the…

Multisections generalize trisections for 4-manifolds, allowing complex operations and explicit diagrams.

New conditions for weighted composition operators in group homomorphisms.

FNOs learn solution operators of dissipative equations efficiently via spectral methods.

Researchers solve the Calderón problem for fractional Dirac operators.

Motivated by the study of Hörmander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized function…

We give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural $\mathbb{R}^\times_+$-action. Specifically, we show that a properly supported semiregular distribution on $M\times M$ is the Schwartz kernel of a classical …

Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space …

First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept o…

The space of differential operators acting on skewsymmetric tensor fields or on smooth forms of a smooth manifold are representations of its Lie algebra of vector fields. We compute the first cohomology spaces of these representations and show how they are related to the cohomology with coefficients in ther space of sm…

Sharp estimates derived for quasilinear equations on metric measure spaces.

The Alesker-Poincare pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pai…

For an arbitrary Riemannian manifold $X$ and Hermitian vector bundles $E$ and $F$ over $X$ we define the notion of the normal symbol of a pseudodifferential operator $P$ from $E$ to $F$. The normal symbol of $P$ is a certain smooth function from the cotangent bundle $T^*X$ to the homomorphism bundle $Hom (E,F)$ and dep…

In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domai…

We conjecture that satellite operations are either constant or have infinite rank in the concordance group. We reduce this to the difficult case of winding number zero satellites, and use $SO(3)$ gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank s…

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that the Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${…

Regularity results for geodesic X-ray transform on nonsmooth manifolds

Introduces slice knots and concordance, linking to exotic smooth structures.

We study differential invariants of linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles over smooth manifolds with respect to groups of authomorphisms.

We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective space blown-up eight times, seven times, and six times. We also show how this operatio…

Cut-DeepONet handles discontinuities and sharp transitions in neural operators.

We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold.

The paper analyzes MACD using operator theory.

Proves rigidity for specific initial data sets under the dominant energy condition.

Unified approach to discrete and smooth isoperimetric inequalities of arbitrary order.

Classifies local boundary conditions for Dirac-type operators on manifolds.

This thesis predicts the distribution of smoothed zeros of random sections on line bundles.