Study non-formal pseudo-differential operators over formal ones.
problem Understanding structure of non-formal pseudo-differential operators.
method Diffeological principal bundles, smoothing connections.
result Structure of diffeological bundle of non-formal pseudo-differential operators over formal ones.
New hierarchies derived from KP hierarchy using non-formal operators and Yang-Mills action.
problem Formal solutions of KP hierarchy and their non-formal counterparts.
method Developed new hierarchies of non-linear equations on non-formal pseudo-differential operators.
result Expressed one hierarchy as Yang-Mills action minimization.
Proves well-posedness of KP hierarchy using Frölicher Lie groups and formal pseudo-differential operators.
problem Well-posedness of the Kadomtsev-Petviashvili hierarchy in a smooth category.
method Combining Frölicher Lie groups, formal pseudo-differential operators, and Mulase factorization.
result Two proofs of well-posedness for the KP hierarchy in a smooth category.
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.
Develops global pseudo-differential calculus on homogeneous vector bundles.
problem Global theory of subelliptic pseudo-differential operators on homogeneous vector bundles.
method Global symbolic calculus, complex functional calculus, Hörmander system of vector-fields.
result Global pseudo-differential calculus on homogeneous vector bundles.
Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…
Study pseudo-differential operators on compact Lie groups using symbols and functional calculus.
problem Analytical index of pseudo-differential operators on compact Lie groups.
method Use operator-valued symbols and McKean-Singer index formula with operator-valued functional calculus.
result Developed tools for calculating the index of pseudo-differential operators.
The paper studies boundedness of pseudo-differential operators on smooth manifolds.
problem Boundedness of pseudo-differential operators in Lp-Lq spaces on smooth manifolds. method Using global symbols and extending Hörmander's condition, the paper investigates Lp-boundedness, L∞-BMO estimates, and Lp-Lq boundedness for Fourier multipliers and pseudo-differential operators. result The paper proves Lp-Lq boundedness for the range 1<p≤2≤q<∞. We develop here a concept of deformed algebras and their related groups through two examples. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of deformed algebra used in \cite{Ma2013} is only an example among others, and how th…
In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose-Singer theorem that enlarges the one stated in \cite{Ma2}, and conclude with a differential geometric treatment of KP hierarchy. The example…
Simplified calculus for manifold operators, proving index theorems.
problem Developing calculus for manifold operators and proving index theorems.
method Introducing a simplified pseudo-differential calculus for zero-order operators on manifolds with a tangent Lie structure.
result Proving index theorems for `h-elliptic' operators on manifolds with a tangent Lie structure.
The notion of pseudo-differential operators with coefficients in a continuous trace algebra over a manifold are introduced and their index theory is studied. The algebra of principal symbols in this calculus provides an abstract Poincaré dual to the continuous trace algebra. Index formulas for pseudo-differential opera…
Formulae for Dixmier trace and noncommutative residue on compact manifolds.
problem Calculating traces and residues for pseudo-differential operators on compact manifolds.
method Using global symbols, Fourier analysis, representation theory, and pseudo-differential calculus.
result Formulae for Dixmier trace and noncommutative residue on compact manifolds.
Study perturbs Dirac operator on 4D manifolds, proving Kastler-Kalau-Walze theorems.
problem Analyzing perturbations of Dirac operator on compact manifolds.
method Defining pseudo-differential perturbations and proving Kastler-Kalau-Walze theorems.
result Proved Kastler-Kalau-Walze theorems for 4D compact manifolds with boundary.
Researchers compute Wodzicki residue for pseudo-differential operators on compact Lie groups.
problem Computing the Wodzicki residue for pseudo-differential operators on compact Lie groups.
method Analytic continuation of traces and matrix-valued symbols.
result Main theorem complementary to [2], removing ellipticity hypothesis.
Paper establishes convergence rates for learning elliptic pseudo-differential operators.
problem Learning elliptic pseudo-differential operators in partial differential equations.
method Wavelet-Galerkin framework, structured infinite-dimensional regression problem, sparse estimator, matrix compression, nested-support strategy.
result Obtained convergence rates for the estimator and efficient Galerkin solver.
Researchers extend a groupoid approach to calculate Wodzicki residue and Kontsevich-Vishik trace.
problem Calculating Wodzicki residue and Kontsevich-Vishik trace for pseudo-differential operators of any order.
method Groupoid approach to pseudo-differential operators.
result Extension of van Erp and Yuncken's work to operators of any order.
Extends compactness theory to variable-coefficient pseudo-differential operators on manifolds.
problem Compensated compactness for pseudodifferential operators on vector bundles.
method Establishes a theorem for weakly convergent sequences of sections under a pseudo-differential operator.
result Quadratic form converges in distributional sense under certain conditions.
Study on well-posedness of EPDiff equations with pseudo-differential inertia.
problem Analyzing the EPDiff equations with fractional Sobolev metrics.
method Fractional order Sobolev-type metrics on diffeomorphism groups, proving well-posedness.
result Proves local and global well-posedness for EPDiff equations.
In this paper, we enlarge the space of uniformly supported pseudo-differential operators on some groupoids by considering kernels satisfying certain asymptotic estimates. We show that such enlarged space contains the compact parametrix, and the generalized inverse of uniformly supported operators with Fredholm vector r…
Constructs conformal boundary operators and fractional Laplacians.
problem Developing conformally invariant boundary operators and fractional Laplacians.
method Constructs continuously parametrised families of conformally invariant boundary operators on densities.
result Constructs odd-order conformally invariant fractional Laplacian pseudo-differential operators.
In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the cal…
Global analysis of Dixmier traces and Wodzicki residues on compact Lie groups.
problem Computing Dixmier traces and Wodzicki residues on compact Lie groups.
method Global quantisation approach, using global symbols and representation theory.
result Explicit formulae for Dixmier traces and Wodzicki residues on compact Lie groups.
Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our ma…
Extends pseudo-differential operators theory to compact Lie groups.
problem Global pseudo-differential operators on compact Lie groups.
method Develops a subelliptic pseudo-differential calculus for compact Lie groups.
result Establishes subelliptic versions of Fefferman and Calderón-Vaillancourt theorems.
Introduces a new operator generating higher Koszul brackets on differential forms.
problem Developing a new operator for higher Koszul brackets on differential forms.
method Introducing a formal ℏ-differential operator Δ generating higher Koszul brackets on differential forms. result Established properties of the introduced BV type operator and its inclusion in a one-parameter family.
Proves spectral inequality and null-controllability for elliptic operators on closed manifolds.
problem Proving spectral inequalities and null-controllability for elliptic pseudo-differential operators.
method Periodization approach in time inspired by global pseudo-differential calculus.
result Established spectral inequality and null-controllability for elliptic operators on closed manifolds.
Global calculus for manifolds with boundary, solving evolution problems.
problem Global solvability of evolution problems on manifolds with boundary.
method Established global functional calculus and Gårding inequality for pseudo-differential operators without local coordinates.
result Global solvability for a class of evolution problems.
Research covers geometry, analysis, and integration on infinite-dimensional spaces.
problem Exploring geometric and analytical structures in infinite-dimensional settings.
method Analyzes numerical schemes, Lie groups, connections, and integration theory.
result Developed new methods for integration and analysis on infinite-dimensional manifolds.
We study pseudo-differential operators on a wedge with continuous and variable discrete branching asymptotics.
New method approximates MMD using pseudo-differential operators and singular values.
problem Approximating MMD with pseudo-differential operators and singular values.
method Corresponding pseudo-differential operators to Mercer kernels, approximating p(x,y) with its first r singular values. result The new MMD distance measures the difference of two distributions with respect to r∗ local moments, where r∗ depends on singular values decay rate. Researchers prove Fredholm property for Dirac operator on specific spacetimes.
problem Proving Fredholm property for Dirac operator on asymptotically static spacetimes.
method Combining time-dependent scattering theory and Egorov's theorem for pseudo-differential hyperbolic systems.
result The Dirac operator is Fredholm under Atiyah-Patodi-Singer boundary conditions.
We use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for biharmonic Steklov eigenfunctions.
Unified treatment of two extension problems using heat equation in Heisenberg group.
problem Two extension problems for pseudo-differential operators in Heisenberg group.
method Heat equation, fractional powers, semigroup methods.
result Explicit computation of fundamental solutions for pseudo-differential operators.
For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical {\sf GJMS} operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue…
Study on degeneration of spectral sequence in complex manifolds under deformations.
problem Behavior of spectral sequence degeneration in complex manifolds under small deformations.
method Deformation theory, pseudo-differential operators, Kodaira-Spencer techniques.
result Degeneration at second step is open under certain conditions but not without them.
New equations for rigid body motion on infinite-dimensional spaces of operators.
problem Integrating rigid body dynamics on infinite-dimensional spaces of operators.
method Introducing pseudo-Riemannian metrics and adapting classical integrability theory.
result Existence of geodesics and integrals of motion for the rigid body equations.
We study the index of the G-invariant elliptic pseudo-differential operator acting on a complete Riemannian manifold, where a unimodular, locally compact group G acts properly and cocompactly. An L2-index formula was obtained using the heat kernel method.
We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on S3.
We investigate (pseudo)differential forms in the framework of supergeometry. Definitions, basic properties and Cartan calculus (DeRham differential, Lie derivative, inner product, Hodge operator) are presented; the symplectic supermechanics (even and odd) is formulated; and the question of quantization is discussed. In…
We show that elliptic complexes of (pseudo)differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi and Singer …
New method constructs deformation groupoid for inhomogeneous pseudo-differential calculus.
problem Recovering inhomogeneous pseudo-differential calculus using a deformation groupoid.
method Elementary construction of deformation groupoid for Heisenberg calculus, then generalization to arbitrary filtrations.
result Elementary construction of deformation groupoid for inhomogeneous pseudo-differential calculus.
The article defines conditions for a manifold to be conformal to an Einstein space.
problem Determining when a manifold is conformal to an Einstein space.
method Algorithmic conditions based on the metric tensor and the Weyl endomorphism.
result General necessary and sufficient conditions for a pseudo-Riemannian manifold to be conformal to an Einstein space.
Extends tangent functor to microformal morphisms, creating non-linear pullbacks for forms and cohomology.
problem Generalizing smooth maps to microformal morphisms for new types of mappings.
method Introduces microformal morphisms and shows how they act on functions and forms via non-linear pullbacks.
result Non-linear pullbacks of forms respect de Rham differentials and induce transformations of cohomology.
Abstract: Determinants and formulas for operators on various spaces.
problem Determinants and formulas for operators on different algebras and spaces.
method Use of Poincaré type determinants, invariant operators, and full matrix-symbols.
result Explicit formulas for determinants of elliptic operators and periodic pseudo-differential operators.
The paper addresses the expansion of Berezinian and super exterior powers, revealing new insights into supertraces.
problem The breakdown of classical volume elements in supergeometric settings and the need for generalized forms.
method Introduces and analyzes r∣s-forms, demonstrates the expansion of Ber(E+zA), and identifies supertraces. result Intermediate expansions in annular regions encode supertraces of representations on vector spaces.
Constructs smooth integrable magnetic systems on a two-torus.
problem Creating smooth magnetic systems on a two-torus with specific properties.
method Uses Nash-Moser implicit function theorem to find zeros of an action functional.
result Characterizes Zoll magnetic systems and proves their existence.
Defines a new calculus for cusp pseudodifferential operators and proves index theorems.
problem Developing a calculus for pseudodifferential operators on manifolds with corners.
method Using blowing up technique to generalize existing calculi and proving Fredholm conditions.
result Proves the relative index theorem for non-closed Z/k-manifolds.