Earlier proofs and lesson routes

Each lesson names the earlier proofs it uses and gives their precise receiving calculations. These routes collect the programme links present in the current readings. A link can also identify a later comparison; an availability notice supplies no proof.

• Metric and topological foundations
  • Localizing symbols with moving metrics
  • Polynomial and contour interfaces for stable boundary models
  • Banach estimates and measure foundations
  • Polynomial and contour interfaces for stable boundary models
  • Fourier transforms, finite spectra and convex separation
  • Stable modes and the algebra of boundary data
• Banach estimates, quotient spaces and compact parameter arguments
  • Metric and topological foundations
  • From Weyl symbols to operators and changes of coordinates
  • Local inverses and distance-weighted elliptic estimates
  • Arctangent, circular parameters and the original pi
  • Fourier transforms, finite spectra and convex separation
• Fourier transforms, finite spectra and convex separation
  • Quantitative estimates for quadratic Fourier multipliers
  • Metric and topological foundations
  • Metric and topological foundations
  • Metric and topological foundations
  • Polynomial and contour interfaces for stable boundary models
  • Fourier transforms, finite spectra and convex separation
  • Banach estimates, quotient spaces and compact parameter arguments
  • Banach estimates, quotient spaces and compact parameter arguments
• Polynomial and contour interfaces for stable boundary models
  • Stable modes and the algebra of boundary data
  • Metric and topological foundations
  • Metric and topological foundations
  • Fourier transforms, finite spectra and convex separation
  • Metric and topological foundations
  • Banach and Hilbert foundations
• Spectral measures with the original operator domain retained
  • Dirichlet realizations, spectral projectors, and local extensions
  • Banach foundations
  • Traces that survive passage to cohomology
  • When a moving symbol scale controls an operator
  • From local energy to global divergence equations
• Localizing symbols with moving metrics
  • Metric and topological foundations
• Quadratic Fourier multipliers at a moving scale
  • Localizing symbols with moving metrics
  • Fourier transforms, finite spectra and convex separation
  • Polynomial and contour tools
  • Metric and topological foundations, Section 6
  • Banach estimates, quotient spaces and compact parameter arguments
  • Approximation on compact sets
• Two measuring scales, one Weyl product
  • Localizing symbols with moving metrics
  • Quadratic Fourier multipliers at a moving scale
  • Banach estimates, quotient spaces and compact parameter arguments
  • the Fourier prerequisite
  • complete real/complex receiving maps, Section17.5
  • Approximation on compact sets
  • Quadratic Fourier multipliers at a moving scale
  • An anisotropic model
  • Theorem 8.1
• From Weyl symbols to operators and changes of coordinates
  • Localizing symbols with moving metrics
  • Quadratic Fourier multipliers at a moving scale
  • Fourier transforms, finite spectra and convex separation
  • Two measuring scales, one Weyl product
  • Section6 of the Banach foundations
  • Section14.3
• Symbols, operators and Sobolev scales
  • Fourier transforms, finite spectra and convex separation
  • Quadratic Fourier multipliers at a moving scale
  • Two measuring scales, one Weyl product
  • Sections4.1–4.5 of the Weyl-product lesson
  • Sections8.1–8.4 of the Fourier lesson
  • The Banach and measure lesson
  • the metric and calculus lesson
  • original Gauss finite-bound theorem G24–G26
  • the complete Schur proof, Section1.1
  • Stable prerequisites
  • Fourier prerequisites
  • Metric foundations
  • Banach foundations
  • Conormal transmission
• Positivity through a moving family of scalar probes
  • Fourier transforms, finite spectra and convex separation
  • Banach estimates, quotient spaces and compact parameter arguments
  • Symbols, operators and Sobolev scales
  • Quadratic Fourier multipliers at a moving scale
  • the full original-metric calculation
  • Section 7
  • Section 8
  • the full conversion proof
  • Section 8
  • Approximation on compact sets
  • the Euclidean lesson
  • the Sobolev chapter
• When a moving symbol scale controls an operator
  • Two measuring scales, one Weyl product
  • Localizing symbols with moving metrics
  • Quadratic Fourier multipliers at a moving scale
  • From Weyl symbols to operators and changes of coordinates
  • Positivity through a moving family of scalar probes
  • Banach estimates, quotient spaces and compact parameter arguments
  • Spectral measures with the original operator domain retained
• When a nonnegative scalar symbol acquires a negative part
  • Two measuring scales, one Weyl product
  • From Weyl symbols to operators and changes of coordinates
  • Localizing symbols with moving metrics
  • When a moving symbol scale controls an operator
  • The metric and calculus lesson, Sections12–14
  • The Fourier lesson, Sections1–3 and7–8
  • The geometric lesson, Sections16.4–16.5
  • The Banach and measure lesson, Sections16–17
  • Positivity through a moving family of scalar probes
• Singularities along a submanifold and smooth boundary passage
  • Quadratic Fourier multipliers at a moving scale
  • Localizing symbols with moving metrics
  • Two measuring scales, one Weyl product
  • Metric foundations
  • Banach estimates and measure foundations
• Detecting regularity without choosing coordinates
  • From symbol estimates to operators on every Sobolev scale
  • Quadratic Fourier multipliers at a moving scale
  • Banach estimates, quotient spaces and compact parameter arguments
  • Two measuring scales, one Weyl product
  • Local inverses and distance-weighted elliptic estimates
  • the original Fourier inverses
  • the nonlinear completed measure theorem
  • the wave kernel chapter
• Local inverses and distance-weighted elliptic estimates
  • Fourier transforms, finite spectra and convex separation
  • Detecting regularity without choosing coordinates
  • Banach estimates, quotient spaces and compact parameter arguments
  • Finite defects under perturbation
  • The full calculus of a lower-bounded self-adjoint operator
  • the Fourier chapter, Section7.2
• Curved weights and the directions in which support can end
  • Local inverses and distance-weighted elliptic estimates
  • Fourier transforms, finite spectra and convex separation
• Detecting a solution from infinite-order silence at one point
  • Local inverses and distance-weighted elliptic estimates
  • Curved weights and the directions in which support can end
  • Metric and topological foundations
• From local energy to global divergence equations
  • Local inverses and distance-weighted elliptic estimates
  • Curved weights and the directions in which support can end
  • Banach estimates, quotient spaces and compact parameter arguments
  • Spectral measures with the original operator domain retained
  • Singularities along a submanifold, Section 16.9
  • Detecting regularity without choosing coordinates
• Positive energy and vanishing at the far end of space
  • Fourier transforms, finite spectra and convex separation
  • Local inverses and distance-weighted elliptic estimates
  • Curved weights and the directions in which support can end
  • Detecting a solution from infinite-order silence at one point
• Inverting mixed symbols without commuting matrix factors
  • From symbol estimates to operators on every Sobolev scale
• Polynomial inverse expansion with ordered matrix coefficients
  • Inverting mixed symbols without commuting matrix factors
• Composition of mixed symbols with two different remainder estimates
  • Quadratic Fourier multipliers at a moving scale
  • Two measuring scales, one Weyl product
  • From symbol estimates to operators on every Sobolev scale
  • Inverting mixed symbols without commuting matrix factors
• Mixed symbols on every real two-parameter Sobolev scale
  • From symbol estimates to operators on every Sobolev scale
  • Composition of mixed symbols with two different remainder estimates
  • Banach estimates, quotient spaces and compact parameter arguments
  • Fourier transforms, finite spectra and convex separation
  • Inverting mixed symbols without commuting matrix factors
• Totally characteristic operators on the half space
  • From symbol estimates to operators on every Sobolev scale
  • Singularities along a submanifold and smooth boundary passage
  • Detecting regularity without choosing coordinates
  • the Fourier lesson, Sections1–2 and7–8
  • the Weyl-product lesson, Sections4.1–4.5
  • the measure chapter, Sections16.1–16.5
  • the calculus chapter, Section13.6
  • Section14.7 of the Banach foundations
  • Section14.8 of the Banach foundations
  • the Banach chapter, Section6
• Global boundary operators, compressed wave fronts, and normal extension
  • Singularities along a submanifold and smooth boundary passage
  • Totally characteristic operators on the half space
  • From symbol estimates to operators on every Sobolev scale
  • Detecting regularity without choosing coordinates
• Changing an interior frame to extend an invertible matrix
  • finite matrix foundation
  • metric and calculus foundation
  • Bott and Euclidean index reduction
  • closed-manifold symbol and index lesson
• Finite defects under perturbation
  • Banach estimates, quotient spaces and compact parameter arguments
• Traces that survive passage to cohomology
  • Banach estimates, quotient spaces and compact parameter arguments
  • Finite defects under perturbation
  • Spectral measures with the original operator domain retained
  • Lower-bounded selfadjoint operators and their spectral calculus, Section 6
• Weyl kernels, operator traces, and a finite trace-class test
  • Symbols, operators and Sobolev scales
  • Traces that survive passage to cohomology
  • Banach estimates, quotient spaces and compact parameter arguments
• Scaled Weyl parametrices and the surviving differential degree
  • Weyl products for a varying metric
  • Metric operator bounds
  • Symbols, operators and Sobolev scales
  • Traces that survive passage to cohomology
  • Weyl kernels, operator traces, and a finite trace-class test
• Radial compression and index transport for product-metric symbols
  • Weyl products for a varying metric
  • Quadratic Fourier multipliers at a moving scale
  • Metric operator bounds
  • Weyl kernels, operator traces, and a finite trace-class test
  • Traces that survive passage to cohomology
  • scaled Weyl coefficient proof
  • matrix Weyl boundary lesson
  • Fourier prerequisites
  • Quadratic Fourier multipliers
  • Weyl products
  • Finite defects
  • scaled Weyl coefficient proof
• Relative Weyl projectors and the Chern cutoff form
  • Two measuring scales, one Weyl product
  • When a moving symbol scale controls an operator
  • Finite defects under perturbation
  • Weyl kernels, operator traces, and a finite trace-class test
  • Scaled Weyl parametrices and the surviving differential degree
  • Traces that survive passage to cohomology
  • Radial symbols and index transport
  • the Fourier prerequisite
• From ordered Weyl errors to the exterior index form
  • Scaled Weyl parametrices and the surviving differential degree
  • When a moving symbol scale controls an operator
  • Relative Weyl projectors and the Chern cutoff form
  • Fredholm stability
  • Weyl traces
  • traces and complexes
• The matrix Weyl index in the original phase coordinates
  • Relative Weyl projectors and the Chern cutoff form
  • Radial compression and index transport for product-metric symbols
  • Scaled Weyl parametrices and the surviving differential degree
  • Weyl kernels, operator traces, and a finite trace-class test
  • From ordered Weyl errors to the exterior index form
• Alternation and the index under linear phase changes
  • The matrix Weyl index in the original phase coordinates
  • Scaled Weyl parametrices and the surviving differential degree
  • Metric operator bounds
  • Relative Weyl projectors and the Chern cutoff form
  • From ordered Weyl errors to the exterior index form
  • Finite defects under perturbation
• Symbols, finite defects, and the index on a closed manifold
  • Symbols, operators and Sobolev scales
  • Detecting regularity without choosing coordinates
  • Finite defects under perturbation
  • Traces that survive passage to cohomology
  • Fourier transforms, finite spectra and convex separation
• When finite defects force one-sided ellipticity
  • Finite defects under perturbation
  • From symbol estimates to operators on every Sobolev scale
  • Symbols, finite defects, and the index on a closed manifold
  • Linear distribution tests and Hermitian adjoints
  • the compact-cylinder Fourier foundations
• Nonelliptic Fredholm operators on their exact adapted spaces
  • Two-parameter Sobolev weights and conjugated operators
  • Finite defects under perturbation
  • When finite defects force one-sided ellipticity
  • the metric foundation, Sections 13.1–13.8
  • the Banach foundation, (LP1)–(LP5)
  • the metric foundation, (OC33)–(OC37)
  • the Banach foundation, Section 11
  • the Banach foundation, Section 15.3
  • the Fredholm lesson, (F1)
  • the proved bounded inverse theorem, (B8)–(B9)
  • the metric foundation, Section 13.10, (OC38)–(OC41)
  • the Banach foundation, Section 10
  • in the trace lesson, (T5)
• Elliptic complexes, diagonal traces, and fixed points
  • Detecting regularity without choosing coordinates
  • Symbols, finite defects, and the index on a closed manifold
  • Traces that survive passage to cohomology
  • the earlier Banach foundation
• The Bott operator, suspension, and reduction of the index to Euclidean space
  • Symbols, finite defects, and the index on a closed manifold
  • Finite defects under perturbation
  • From symbol estimates to operators on every Sobolev scale
  • Localizing symbols when the measuring scale moves
  • Two measuring scales, one Weyl product
  • When a moving symbol scale controls an operator
  • Detecting regularity without choosing coordinates
  • Quadratic Fourier multipliers at a moving scale
  • From Weyl symbols to operators and changes of coordinates
  • Metric and topological foundations
  • Lower-bounded spectral calculus
  • Banach and Hilbert foundations
  • Traces that survive passage to cohomology
  • Changing an interior frame to extend an invertible matrix
  • the matrix-extension lesson
• Characteristic classes, Bott normalization, fixed points, and external index theories
  • Symbols, finite defects, and the index on a closed manifold
  • Relative Weyl projectors and the Chern cutoff form
  • The Bott operator, suspension, and reduction of the index to Euclidean space
  • Elliptic complexes, diagonal traces, and fixed points
  • Nonelliptic Fredholm operators on their exact adapted spaces
  • From ordered Weyl errors to the exterior index form
  • Detecting regularity without choosing coordinates
  • Lower bounds and spectral calculus
  • Traces that survive passage to cohomology
• Building a local inverse from radial singularities
  • Fourier transforms, finite spectra and convex separation
  • Symbols, operators and Sobolev scales
  • Detecting regularity without choosing coordinates
  • Metric and topological foundations
  • the earlier divergence lesson
• Causal kernels, initial data, and short-time geometry
  • Fourier transforms, finite spectra and convex separation
  • Singularities along a submanifold and smooth boundary passage
  • Detecting regularity without choosing coordinates
  • the metric and calculus foundation
• A wave kernel that cancels at a curved boundary
  • Building a local inverse from radial singularities
  • Causal kernels, initial data, and short-time geometry
  • Causal kernels, initial data, and short-time geometry
  • the Fourier prerequisite bridge
  • the metric foundation bridge
  • the geometric calculus lesson
  • the divergence lesson
  • the conormal lesson
• Stable modes and the algebra of boundary data
  • Metric and topological foundations
  • Metric and topological foundations
  • Polynomial and contour interfaces for stable boundary models
  • Cauchy data from jumps and residues
  • Solving an elliptic system from compatible boundary measurements
  • Cauchy data from jumps and residues, Section9
• Boundary energy, local inverses, and harmonic data
  • Local inverses and distance-weighted elliptic estimates
  • Two measuring scales, one Weyl product
  • Fourier transforms, finite spectra and convex separation
  • Symbols, operators and Sobolev scales
  • Detecting regularity without choosing coordinates
  • Finite defects under perturbation
  • Polynomial and contour tools
  • Metric foundations
  • Geometric microlocal calculus
  • Stable modes and the algebra of boundary data
  • Symbols, operators and Sobolev scales
  • Singularities along a submanifold and smooth boundary passage
  • Cauchy data and the Calderón projector
  • Reducing arbitrary boundary data to a boundary system
• Cauchy data from jumps and residues
  • Stable modes and the algebra of boundary data
  • Singularities along a submanifold and smooth boundary passage
  • Detecting regularity without choosing coordinates
• Solving an elliptic system from compatible boundary measurements
  • Euclidean symbol calculus
  • coordinate-free microlocal calculus
  • Singularities along a submanifold and smooth boundary passage
  • Cauchy data from jumps and residues
  • Fredholm stability
  • Stable modes and the algebra of boundary data
  • Cauchy data from jumps and residues, Section9
• Fredholm boundary problems with first-order Calderón defects
  • Inverting mixed symbols without changing their scales
  • Composing symbols with two independent orders
  • Sobolev mapping with normal and tangential weights
  • Finite inverse expansions for normal polynomials
  • Cauchy data from jumps and residues
  • Solving an elliptic system from compatible boundary measurements
  • Finite defects under perturbation
  • Geometric symbol and kernel calculus
  • Mixed symbols on every real two-parameter Sobolev scale
• Doubling a boundary problem and computing its index
  • Stable modes and the algebra of boundary data
  • Fredholm boundary problems with first-order Calderón defects
  • Finite defects under perturbation
  • Symbols, finite defects, and the index on a closed manifold
  • Symbols, operators and Sobolev scales
  • Detecting regularity without choosing coordinates
  • Mixed symbols on every real two-parameter Sobolev scale
• Boundary wave fronts for elliptic systems
  • Global boundary calculus on the compressed cotangent bundle
  • Sobolev mapping with normal and tangential weights
  • Fredholm boundary problems with first-order Calderón defects
  • Doubling a boundary problem and computing its index
• Reducing first-order boundary data to a split trace
  • Stable modes and the algebra of boundary data
  • Cauchy data from jumps and residues
  • Fredholm boundary problems with first-order Calderón defects
  • Finite defects under perturbation
  • Symbols, finite defects, and the index on a closed manifold
  • The Bott operator, suspension, and reduction of the index to Euclidean space
  • Doubling a boundary problem and computing its index
  • GI1–GI32 and AL1–AL18
  • complete composition proof
  • half-space receiving maps
• Reducing arbitrary-order boundary problems to first order
  • Stable modes and the algebra of boundary data
  • Composition in the mixed symbol calculus
  • Fredholm boundary problems with first-order Calderón defects
  • Finite defects under perturbation
  • Reducing first-order boundary data to a split trace
  • Doubling a boundary problem and computing its index
  • Symbols, finite defects, and the index on a closed manifold
• Reducing arbitrary boundary data to a boundary system
  • Solving an elliptic system from compatible boundary measurements
  • Fredholm boundary problems with first-order Calderón defects
  • Cauchy data from jumps and residues
  • The calculus of pseudodifferential operators on a manifold
  • Fourier transforms, finite spectra and convex separation
  • Banach estimates, quotient spaces and compact parameter arguments
  • Mixed symbols on every real two-parameter Sobolev scale
  • Doubling a boundary problem and computing its index
  • Polynomial and contour interfaces for stable boundary models
• Dirichlet realizations, spectral projectors, and local extensions
  • Boundary energy, local inverses, and harmonic data
  • Local inverses and distance-weighted elliptic estimates
  • Spectral measures with the original operator domain retained
  • Fourier transforms, finite spectra and convex separation
  • Traces that survive passage to cohomology
  • Metric and topological foundations
  • Banach estimates, quotient spaces and compact parameter arguments