arXiv research

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

Study non-formal pseudo-differential operators over formal ones.

Develops global pseudo-differential calculus on homogeneous vector bundles.

New hierarchies derived from KP hierarchy using non-formal operators and Yang-Mills action.

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…

Simplified calculus for manifold operators, proving index theorems.

Study perturbs Dirac operator on 4D manifolds, proving Kastler-Kalau-Walze theorems.

Researchers compute Wodzicki residue for pseudo-differential operators on compact Lie groups.

Paper establishes convergence rates for learning elliptic pseudo-differential operators.

Researchers extend a groupoid approach to calculate Wodzicki residue and Kontsevich-Vishik trace.

Extends compactness theory to variable-coefficient pseudo-differential operators on manifolds.

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…

In this note we study the analytical index of pseudo-differential operators by using the notion of (infinite dimensional) operator-valued symbols (in the sense of Ruzhansky and Turunen). Our main tools will be the McKean-Singer index formula together with the operator-valued functional calculus developed here.

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…

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…

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

Global analysis of 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.

Proves spectral inequality and null-controllability for elliptic operators on closed manifolds.

We recall the notions of Frölicher and diffeological spaces and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of the Mulase factorization of infinite dimensional groups based on formal pseud…

Global calculus for manifolds with boundary, solving evolution problems.

We study pseudo-differential operators on a wedge with continuous and variable discrete branching asymptotics.

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…

In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M. Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical ph…

New method approximates MMD using pseudo-differential operators and singular values.

Researchers prove Fredholm property for Dirac operator on specific spacetimes.

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.

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…

New equations for rigid body motion on infinite-dimensional spaces of operators.

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 $L^2$-index formula was obtained using the heat kernel method.

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…

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 $S^3$.

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…

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 …

The article defines conditions for a manifold to be conformal to an Einstein space.

Abstract: Determinants and formulas for operators on various spaces.

The paper addresses the expansion of Berezinian and super exterior powers, revealing new insights into supertraces.

Constructs smooth integrable magnetic systems on a two-torus.

We consider two principal bundles of embeddings with total space $Emb(M,N),$ with structure groups $Diff(M)$ and $Diff_+(M),$ where $Diff_+(M)$ is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: $$ B(M,N) = E…

These notes form the next episode in a series of articles dedicated to a detailed proof of a cohomological index formula for transversally elliptic pseudo-differential operators and applications. The first two chapters are already available as math.DG/0702575 and arXiv:0711.3898. In this episode, we construct the relat…

Here shape space is either the manifold of simple closed smooth unparameterized curves in $\mathbb R^2$ or is the orbifold of immersions from $S^1$ to $\mathbb R^2$ modulo the group of diffeomorphisms of $S^1$. We investige several Riemannian metrics on shape space: $L^2$-metrics weighted by expressions in length and c…

Study compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.

New method calculates eta invariant without analytic continuation.

We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $δ$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values $η(δ_t,0)$ and $tζ(δ_t,0)$ are smooth functions of …

We show that the classical Szasz analytic function $S_N(f)(x)$ is obtained by applying the pseudo-differential operator $f(N^{-1}D_θ)$ to the Bergman kernels for the Bargmann-Fock space. The expression generalizes immediately to any smooth polarized noncompact complete toric \kahler manifold, defining the generalized S…