The paper calculates indices for families of Fredholm operators and their extensions.
problem Calculating indices for families of Fredholm operators and their extensions.
method Passing from a Fredholm operator to its graph, deforming the horizontal subspace.
result Index formulas for families of Fredholm realizations and self-adjoint extensions.
Notes on continuity of discrete-spectrum Fredholm operators.
problem Continuity properties of discrete-spectrum families of Fredholm operators.
method Relates recent work on discrete-spectrum families to classical continuity properties.
result Establishes connections between new and classical concepts.
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.
New Fredholm criteria for pseudodifferential operators on manifolds.
problem Characterizing the Fredholm property of G-pseudodifferential operators. method General Simonenko principle applied to G-pseudodifferential operators on Sobolev spaces of sections of vector bundles. result Derivation of novel Fredholm criteria and further characterization for finite groups.
The spectral flow theorem is applied to operators on finite intervals.
problem Operators on finite intervals without boundary conditions are not Fredholm.
method Interpolation theory is used to define boundary conditions making the operators Fredholm. The spectral flow theorem is applied to find the Fredholm index.
result The Fredholm index is given by the spectral flow of the operator path.
Study perturbs APS boundary conditions for Lorentzian Dirac operators.
problem Maintaining Fredholmness of Dirac operators under perturbations of APS boundary conditions.
method Develop criteria for perturbing compact pairs of projections to remain Fredholm.
result Criteria for perturbing APS boundary conditions without losing Fredholmness.
Study shows Fredholmness of a Lorentzian Dirac operator on spacetimes.
problem Analyzing Fredholmness of a Lorentzian Dirac operator on spacetimes.
method Investigates Fredholmness of a (spatial) Γ-invariant Lorentzian Dirac operator under (anti) Atiyah-Patodi-Singer boundary conditions.
result Demonstrates Fredholmness of the operator in the von Neumann sense.
Generalizes Novikov conjecture results to infinite-dimensional bundles.
problem Proving special cases of the Strong Novikov Conjecture.
method Introduces asymptotically flat Fredholm bundles and proves index theorem.
result Relates index of asymptotic Fredholm bundle to asymptotic index of representation.
Derives Fredholm criteria for isotypical components from a Simonenko principle.
problem Finding Fredholm conditions for isotypical components of invariant pseudodifferential operators.
method General Simonenko's local principle and equivariant local principle for restriction to isotypical components.
result Full proof of equivariant local principle and extension of results.
Study Dirac operators on incomplete cusp edge spaces, proving self-adjointness and Fredholm properties.
problem Analyzing Dirac operators on complex geometric spaces.
method Construct heat kernel, prove self-adjointness and Fredholm properties, establish index formula.
result Proved Dirac operators are essentially self-adjoint and Fredholm.
We prove some Fredholm conditions for many algebras of differential operators on particular classes of open manifolds, which include asymptotically Euclidean or asymptotically hyperbolic manifolds. Our typical result is that an operator P is Fredholm if, and only if, it is elliptic and some limit operators $(P_α)_{α\…
Researchers create a Fredholm module on fractal shapes like the Cantor set.
problem Constructing Fredholm modules on complex fractal structures.
method Combining combinatorial techniques with higher-dimensional analogues.
result Calculated Dixmier trace of operators induced by the module.
We describe two topologies on the space of unbounded Fredholm operators and we explain their K-theoretic relevance. In the process we also prove a very general result concerning the continuity of families of first order, elliptic boundary value problems.
Fredholm conditions for invariant operators on compact manifolds.
problem Characterizing operators with Fredholm maps induced by their action on manifolds.
method Defining transversally α-elliptic operators and proving Fredholmness based on their principal symbols and group actions. result Operators are Fredholm if and only if they are transversally α-elliptic. 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.
Paper defines and proves a new analytic index for Fredholm operators.
problem Defining the analytic index rigorously for families of Fredholm operators.
method Based on Segal's ideas, new definition under weaker continuity assumptions.
result New definition agrees with Atiyah-Singer index when applicable.
The paper studies elliptic operators on manifolds with boundary.
problem Characterizing boundary conditions for elliptic operators on manifolds.
method Using Calderón projectors and mixed order Sobolev spaces, the paper describes the space of boundary values and characterizes Fredholm and regular realisations.
result Characterization of boundary conditions for elliptic operators leading to Fredholm and regular realisations.
The study proves curvature rigidity for convex polytopes.
problem Proving curvature rigidity for convex polytopes.
method Using Fredholm theory for Dirac operators and a theorem of Fefferman and Phong.
result Scalar curvature rigidity theorem for convex polytopes proved.
Characterizes Fredholm conditions for group-invariant pseudodifferential operators.
problem Understanding Fredholm conditions for operators on quotient spaces.
method Study of symbol C∗-algebra and topology of primitive ideal spectrum. result Explicit characterization of Fredholm operators in terms of principal symbol and group action.
Extends Atiyah-Singer Dirac operator study to non-compact spacetimes.
problem Analyzing Dirac operator on non-compact spacetimes with non-compact Cauchy hypersurface.
method Building on previous works, extends Fredholm result to non-compact Lorentzian spaces, using von Neumann algebras and Galois coverings.
result Γ-Fredholmness of the Dirac operator under APS boundary conditions.
The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. The proof is based on an …
This article provides a version of scale calculus geared towards a notion of (nonlinear) Fredholm maps between certain types of Frechet spaces, retaining as many as possible of the properties Fredholm maps between Banach spaces enjoy, and the existence of a constant rank theorem for such maps. It does so by extending t…
We provide a thorough construction of a system of compatible determinant line bundles over spaces of Fredholm operators, fully verify that this system satisfies a number of important properties, and include explicit formulas for all relevant isomorphisms between these line bundles. We also completely describe all possi…
We show that the existence of a Fredholm element of the zero calculus of pseudodifferential operators on a compact manifold with boundary with a given elliptic symbol is determined, up to stability, by the vanishing of the Atiyah-Bott obstruction. It follows that, up to small deformations and stability, the same symbol…
Develops calculus for QFB metrics, proving Fredholm properties and decay of harmonic forms.
problem Analyzing differential operators on quasi-fibered boundary metrics.
method Introduces pseudodifferential calculus, principal symbols, and Fredholm theory.
result Hodge-deRham operator is Fredholm on QFB Sobolev spaces and L2 harmonic forms decay. 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 …
We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds with boundary. We emphasize var…
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. Derives numerical formulas for elliptic differential operators on specific groupoids.
problem Index problem of elliptic differential operators on boundary groupoids.
method Similar to Moroianu and Nistor's renormalized trace approach, focusing on eta and Atiyah-Singer terms.
result For q≥3, K-theoretic and Fredholm indices are given by the Atiyah-Singer term. Introduces Floer functions and Floerfolds for intrinsic properties.
problem Complex transformation of Hessian under chart transition.
method Introduces Floer functions and Floerfolds to address intrinsic properties.
result Floer functions and Floerfolds provide intrinsic conditions for Hessian.
We explain the topology of the space, so called, Fredholm-Lagrangian-Grassmannain and the quantity ``Maslov index'' for paths in this space based on the standard theory of Functional Analysis. Our standing point is to define the Maslov index for arbitrary paths in terms of the fundamental spectral property of the Fredh…
We consider two calculi of pseudodifferential operators on manifolds with fibered boundary: Mazzeo's edge calculus, which has as local model the operators associated to products of closed manifolds with asymptotically hyperbolic spaces, and the phi calculus of Mazzeo and the second author, which is similarly modeled on…
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…
Study hypoelliptic operators on Carnot manifolds, extending index theory results.
problem Index theory of hypoelliptic operators on Carnot manifolds.
method Operator K-theory and geometric K-homology.
result Compute Fredholm index of hypoelliptic operators on Carnot manifolds.
For every connected manifold with corners we use a homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space),…
Develops tools for studying intersections of elliptic operators, focusing on J-holomorphic maps.
problem Intersection questions for families of elliptic operators.
method Equivariant Brill-Noether theory applied to Fredholm operators.
result Wendl's super-rigidity conjecture is proven.
A guide for solving first-order elliptic boundary value problems.
problem Solving first-order elliptic boundary value problems on manifolds.
method Operator methods and general elliptic boundary conditions.
result Characterization of a new subclass of elliptic boundary conditions.
Formula calculates index for CR operators on surfaces with boundary punctures.
problem Computing the index for Cauchy-Riemann operators on surfaces with boundary punctures.
method Large antilinear deformations method, generalized to punctured surfaces.
result Involves a non-standard weighted count of boundary zeros in the Euler characteristic term.
Study of eta invariant for non-compact manifolds via Dirac-type operators.
problem Defining and studying the relative eta invariant for non-compact manifolds.
method Defined the relative eta function and studied its variation and gluing law.
result Shows the relative eta invariant coincides with a previously defined version.
Absolute index theorem for warped product manifolds.
problem Equivariant index computation for manifolds with warped product structures.
method Warped product structure, Fredholm operator, Atiyah-Segal-Singer index theorem.
result Equivariant relative index theorem for manifolds with warped product structures.
Study of elliptic boundary value problems on non-compact manifolds.
problem Analyzing elliptic differential operators on manifolds with non-compact boundaries.
method Regularity theory and trace theorems for sections in the maximal domain under various assumptions.
result Systematic study of local and nonlocal boundary conditions, including the Atiyah-Patodi-Singer condition.
The aim of this paper is to present the construction, out of the Kohn-Rossi complex, of a new hypoelliptic operator QL on almost CR manifolds equipped with a real structure. The operator acts on all (p,q)-forms, but when restricted to (p,0)-forms and (p,n)-forms it is a sum of squares up to sign factor and lower o…
This paper provides a K-theoretic obstruction for higher kernel dimension for Dirac operators. For this we use a fibre-wise Dirac operator that gives rise to a family of Fredholm operators representing a class in topological K-theory. Then Chern classes of this K-class contain some information about the kernel of…
In an earlier paper, we showed that the moduli space of deformations of a smooth, compact, orientable special Lagrangian submanifold L in a symplectic manifold X with a non-integrable almost complex structure is a smooth manifold of dimension H^1(L), the space of harmonic 1-forms on L. We proved this first by showing t…
The Fredholm integral equation of the first kind improves solutions for ill-posed supervised learning problems with limited data.
problem Ill-posed supervised learning problems with insufficient data.
method Using the Fredholm integral equation of the first kind (FIFK) with semi-supervised assumptions and MSDF methods.
result Improved accuracy and stability in solutions for ill-posed problems.
Let $\Y$ be a smooth connected manifold, $Σ\subset\C$ an open set and $(σ,y)\to\scrP_y(σ)$ a family of unbounded Fredholm operators D⊂H1→H2 of index 0 depending smoothly on $(y,σ)\in \Y\times Σ$ and holomorphically on σ. We show how to associate to $\scrP$, under mild hypotheses, a smooth vector bundle …
We find G2-manifolds with specific asymptotic properties.
problem Existence and structure of G2-manifolds with ALC asymptotics.
method Robust Fredholm theory for ALC spaces, proving existence and rigidity results.
result Existence of a G2-analogue of the Atiyah-Hitchin metric and good moduli theory for ALC G2-holonomy metrics.
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…