Researchers create a parametrix for resolvents on manifolds with ends.
problem Essential self-adjointness of elliptic symmetric differential operators on manifolds with ends.
method Introduced semiclasical pseudodifferential operators compatible with the end structure.
result Essential self-adjointness of elliptic symmetric differential operators proved.
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
Study resolvents of Bochner Laplacians on compact manifolds.
problem Analyzing the resolvents of Bochner Laplacians in the semiclassical limit.
method Introducing Heisenberg semiclassical pseudodifferential operators to study sections of line bundles.
result Resolvents and spectral projections of Bochner Laplacians are studied in the large power limit.
The paper provides formulas for Hadamard coefficients using Green's operators.
problem Calculating Hadamard coefficients from Green's operators.
method Various methods including resolvents, powers of Green's operators, and product with the real line.
result Formulas for Hadamard coefficients in terms of Green's operators.
Characterizes low energy behavior of fibered Dirac operators.
problem Understanding the behavior of fibered Dirac operators near zero energy.
method Pseudodifferential characterization of the resolvent's low energy limit.
result Pseudodifferential characterization of the inverse of a suspended Dirac operator.
Study low energy resolvent behavior on fibred boundary metrics.
problem Analyze the resolvent of Hodge Laplacian on manifolds with fibred boundary metrics.
method Develop a 'split' pseudodifferential calculus to handle different asymptotic behaviors.
result Precise asymptotic behavior of resolvent as a fibred boundary pseudodifferential operator.
We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy exampl…
Study analyzes low-energy behavior of Schrödinger operators with Coulomb potentials.
problem Analyzing the limiting resolvent of Schrödinger operators at low energies.
method Using Vasy's second microlocal approach (Lagrangian approach), uniformly analyzing the resolvent from E=0. result Obtained oscillatory asymptotics for the resolvent output at low energy, differing from short-range cases.
Study on Dirac operator spectrum on shrinking surfaces with cusps.
problem Behavior of Dirac operator spectrum on degenerating Riemannian surfaces.
method Adapted pseudodifferential calculus, including Dirac operators and their resolvents.
result Smoothness of spectral projectors and t2logt regularity for the cusp-surgery trace. Let $\X\simeq G/K$ be a Riemannian symmetric space of non-compact type, $\widetilde \X$ its Oshima compactification, and $(π,\mathrm{C}(\widetilde \X))$ the regular representation of G on $\widetilde \X$. We study integral operators on $\widetilde \X$ of the form π(f), where f is a rapidly falling function on G…
In this paper we consider certain asymptotically Euclidean spaces, namely compact manifolds with boundary X equipped with a scattering metric g, as defined by Melrose. We then consider Hamiltonians H which are `short-range' self-adjoint perturbations of the Laplacian of g. Melrose and Zworski have given a detailed desc…
For geometrically finite hyperbolic manifolds Γ\Hn+1, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of Γ in large balls of Hn+1 in terms of t…
Novel approach to Nash equilibrium in mean-field stochastic games with operator resolvents.
problem Finding Nash equilibrium in mean-field stochastic games with mean-field interaction.
method Proposed a novel approach to derive Nash equilibrium semi-explicitly using operator resolvents and stochastic Fredholm equations.
result Equilibrium of the N-player game converges to mean-field equilibrium, and ε-Nash equilibrium derived as a by-product. The paper proves wellposedness of flows on manifolds with bounded geometry.
problem Analyzing wellposedness of nonlinear flows on manifolds of bounded geometry.
method Establishing conditions for the operator to generate an analytic semigroup, proving existence of resolvent, and using geometric microlocal calculus.
result Wellposedness of nonlinear flows on manifolds of bounded geometry is proven.
Effective quantum dynamics on a thin Möbius strip approximated by a flat model.
problem Quantum dynamics on a Möbius strip with zero width.
method Norm-resolvent convergence to an unconventional flat model with explicit spectrum.
result Spectral properties of the curved Möbius strip are well approximated by a flat model.
We study families of Dirac-type operators, with compatible perturbations, associated to wedge metrics on stratified spaces. We define a closed domain and, under an assumption of invertible boundary families, prove that the operators are self-adjoint and Fredholm with compact resolvents and trace-class heat kernels. We …
We completely resolve the boundary value problem for differential forms and conformally Einstein infinity in terms of the dual Hahn polynomials. Consequently, we produce explicit formulas for the Branson-Gover operators on Einstein manifolds and prove their representation as a product of second order operators. This le…
Formula for Hadamard coefficients from Green's operators on spacetimes.
problem Calculating Hadamard coefficients from Green's operators on spacetimes.
method Developed formulas for diagonal values and integrals over the diagonal of Hadamard coefficients.
result Formulated analogues of Hadamard expansions and resolvents for Green's operators.
Research resolves sign conventions in Floer theory for Morse-Bott case.
problem Sign conventions in filtered A∞-operations for Lagrangian Floer theory. method Defined filtered A∞-operations and verified formulae using de Rham model. result Resolved sign issues in Bott-Morse setting.
For an asymptotically hyperbolic metric on the interior of a compact manifold with boundary, we prove that the resolvent and scattering operators are continuous functions of the metric in the appropriate topologies.
Study of resonances and residue operators for hyperbolic spaces.
problem Understanding resonances and residue operators for pseudo-Riemannian hyperbolic spaces.
method Analyzing the resolvent of the Laplace-Beltrami operator on pseudo-Riemannian hyperbolic spaces.
result Explicit determination of resonances and identification of residue representations.
The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.
problem Spaces of unbounded Fredholm operators and their properties.
method Analyzing the spaces and proving homotopy equivalences.
result Natural maps between four spaces of unbounded Fredholm operators are homotopy equivalences.
Graph continuous operators become Riesz continuous after multiplication by unitary operators.
problem Characterizing Riesz continuity of graph continuous operators.
method Multiplication by unitary operators to transform graph continuity to Riesz continuity.
result The index of graph continuous families of Fredholm operators coincides with N. Ivanov's index.
Paper solves CR positive mass and Yamabe problems on weighted spaces.
problem CR positive mass and Yamabe problems on weighted spaces.
method Analyzes sub-Laplacian on Folland-Stein spaces.
result CR positive mass and Yamabe problems resolved.
Uniform elliptic theory for Dirac operators on orbifold resolutions.
problem Analyzing Dirac operators on orbifold resolutions.
method Viewing orbifolds as conically fibred singular spaces and resolving them by gluing asymptotically conical fibrations.
result Uniform index formula for Dirac operators on orbifold resolutions.
The purpose of this paper is to study the property of the resolvent of the Laplace-Beltrami operator on a noncompact complete Riemannian manifold with various ends each of which has a different limit of the growth rate of the Riemannian measure at infinity, in particular, focusing on the limiting absorption principle. …
This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.
problem Determining the RLCT of sum-of-products polynomials through blow-up.
method Investigates a specific blow-up algorithm for sop polynomials to resolve their singularities.
result It is possible to resolve the singularities of sop polynomials using a specific blow-up algorithm.
Researchers resolve string theory ambiguities and define a new metric for massless spectrum.
problem Ambiguities in string theory regarding spin connection and Hodge decomposition.
method Constructing a vector bundle Q and operators D and D† to define a metric and gauge fixing.
result Massless spectrum are harmonic representatives of the operator D, resolving previous complications.
Analytic convex bodies' Poincaré series extended holomorphically.
problem Analytic continuation of Poincaré series for convex bodies.
method Analytic continuation of Laplace transforms, holomorphic functions, and resolvent of multiplication operators.
result Poincaré series continues holomorphically to a conical neighborhood of the right half-plane, removing countable cuts and points.
Let P be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various P-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the reso…
We show the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recentl…
We analyze the resolvent R(k)=(P+k2)−1 of Schrödinger operators P=Δ+V with short range potential V on asymptotically conic manifolds (M,g) (this setting includes asymptotically Euclidean manifolds) near k=0. We make the assumption that the dimension is greater or equal to 3 and that P has no L2 null …
The article constructs differential operators for parabolic geometries.
problem Developing a machinery for differential operators in parabolic geometries.
method Starting from a relative tractor bundle, constructs a sequence of differential operators.
result Provides a resolution of a sheaf for many geometries.
Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple descripti…
To quantify the operational risk capital charge under the current regulatory framework for banking supervision, referred to as Basel II, many banks adopt the Loss Distribution Approach. There are many modeling issues that should be resolved to use the approach in practice. In this paper we review the quantitative metho…
A new uncertainty principle helps traders better understand market activity.
problem Understanding high-frequency market activity and correlation.
method Integrates market activity, order-flow overlap, and response time into a clock-dependent uncertainty principle.
result Six rules of thumb for traders operating at market-making frequencies.
Optimal portfolio choice with cross-impact propagators, solving complex equations.
problem Maximizing revenue-risk in a continuous-time portfolio choice problem with cross-impact.
method Formulated as a maximization problem, solved explicitly using operator resolvents and stochastic Fredholm equations.
result Sufficient conditions for the absence of price manipulation, providing financial insights.
We resolve Spin(7)-orbifolds using algebraic and symplectic techniques.
problem Resolving Spin(7)-orbifolds in exceptional holonomy metrics.
method Combines algebraic and symplectic techniques with analytic gluing methods.
result Existence of torsion-free Spin(7)-structures extending Joyce's theorem.
We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part −NμNμ. Our objective is to obtain information on the asymptotic expansions of the corresponding r…
On an asymptotically hyperbolic manifold (X,g), we show that the resolvent resonances coincide, with multiplicities, with the poles of the renormalized scattering operator, except for the special points n/2-k (with k>0 integer) where an additional term appears: this is the dimension of the kernel of the k-conformal Lap…
It is established that the existence of non-isotropic vector field which Jacobi operator of maximal rank is an obstacle for the existence of non-trivial second-order symmetric parallel tensor field. In turns out that presence of such obstacle follows that manifold as pseudo-Riemannian manifold is locally non-reducible.…
The study examines spectral properties of the Laplacian on forms for open Riemannian manifolds.
problem Investigating spectral properties of the Laplacian on forms for open Riemannian manifolds.
method Finding sufficient conditions for the Weyl criterion to hold for the Lp-spectrum of the Laplacian on k-forms, proving the decomposition of the Lp-spectrum, and analyzing the resolvent set of the Laplacian. result The Lp-spectrum of the Laplacian on k-forms over hyperbolic space is described in detail. In this paper, we first establish an S1-equivariant index theorem for Spinc Dirac operators on Z/k manifolds, then combining with the methods developed by Taubes \cite{MR998662} and Liu-Ma-Zhang \cite{MR1870666,MR2016198}, we extend Witten's rigidity theorem to the case of Z/k Spinc manif…
The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
Study adiabatic limits of Calderon projector on manifolds with cylindrical ends.
problem Constructing Calderon projector in a specific geometric setting.
method Using resolvent estimates and b-calculus framework.
result Extend a result on adiabatic limits of Cauchy data spaces.
One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the Φ-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Mel…
Study shows deterministic equivalent for neural network kernel convergence.
problem Understanding convergence of neural network kernels.
method Analyzes empirical spectral distribution of Conjugate Kernel, proving convergence to a deterministic limit.
result Obtains a deterministic equivalent for the Stieltjes transform and resolvent of the Conjugate Kernel.
Study uses CNNs to upscale wind speed data from 100 km to 3 km, improving subgrid-scale variability.
problem Recovering fine-scale wind speed information from coarse data.
method Convolutional neural networks (CNNs) with different input configurations (coarse wind speed, fine-scale topography, diurnal cycle) were tested.
result CNN models with coarse wind and fine topography inputs perform best in generalizing to unseen regions.