AI Maths

Introduction to Mathematics

Conventions in Mathematics

PART I : ALGEBRA

Foundations of Algebra

Theory

Sets, Functions and Relations

Logic and Proof

Order Theory and Lattices

Mereology and Collective Set Theory

Cardinality and the Axiom of Choice

Set-Theoretic Foundations

Formal Logic and Computability

Model Theory

Proof Theory and Type Theory

Universal Properties and Categories

Operator Theory

Operators on a Lattice

The Converse Relation as an Operator

Endomorphisms of a Relational Structure

Closure Operators and the Consequence Operator

The Order Automorphism Group

Operators on a Poset

* Theory

Orthocomplemented Lattices and the Involution

Orthomodular Lattices

Relation Algebras and the Converse

Involutive Categories and the Dagger Functor

Boolean Algebras with an Involution

Groupoids with an Involution

Involutive Set Theory and the Symmetric Difference

* Operator Theory

The Converse as an Adjoint

Involutions of the Operator Layer

The Converse Relation as an Adjoint

The Involution on the Closure Operators

Groups

Theory

Groups

Infinite Abelian Groups

Transformation Groups

Group Actions and Structure

Solvable and Nilpotent Groups

Generators, Presentations and Free Products

Combinatorial Group Theory

Infinite Groups

Coxeter Groups

Braid Groups

Group Cohomology

The Classification of Finite Simple Groups

Finite Simple Groups of Lie Type

Operator Theory

Inner Conjugation and the Class Operator

The Action of a Group on Itself

The Conjugation Representation

Left and Right Multiplication in a Group

The Cayley Action

The Signed Sandwich on a Group

Reflections as Signed Two-Sided Operators on a Group

The Signed Left Multiplication on a Group

The Graded Action on a Module over a Group

* Theory

Involutive Groups

Involutions and the Fixed-Point Subgroup

Involutive Group Actions

Free Involutions and the Quotient

The Commuting Algebra of an Involution

The Centre of an Involutive Group

Involutions and the Word Problem

* Operator Theory

Involutions on the Operator Layer

The Group Inversion as an Adjoint

The Adjoint of the Left Multiplication on a Group

The Signed Adjoint Sandwich on a Group

The Signed Adjoint of the Reflection on a Group

The Signed Adjoint of the Left Multiplication on a Group

The Graded Adjoint Action on a Module over a Group

Applications

Finite Groups and Symmetry

Rings and Fields

Theory

Rings

Commutative Rings

Reduced Rings and the Nilradical

Integral Domains

GCD Domains

Bézout Domains

Unique Factorisation Domains

Principal Ideal Domains

Euclidean Domains

Localization and the Fraction Field

Noetherian and Artinian Rings

Primary Decomposition

Integral Extensions and Krull Dimension

Dedekind Domains and Ideal Class Groups

Valuation Theory and Henselian Rings

Fields

Field Extensions

Splitting Fields and Algebraic Closure

Finite Fields

Galois Theory

Ring and Field Automorphisms

Cyclotomic Fields

Kummer Theory

Galois Cohomology

Class Field Theory

Algebraic Number Theory

Global Fields

Elliptic Curves

Algebraic Curves

The Riemann–Roch Theorem for Curves

Gröbner Bases and Elimination Theory

Invariant Theory

Symmetric Functions and Schur Functions

Representation Theory of Symmetric Groups

Schur–Weyl Duality

The Inverse Galois Problem

Ordered Fields

Real-Closed and Complete Ordered Fields

Algebraically Closed Fields

Semiprime Rings

Prime Rings

Non-Commutative Domains

Ore Domains and Division Rings of Fractions

Division Rings

Von Neumann Regular Rings

Real Algebraic Geometry

Operator Theory

Derivations of a Ring

Inner Automorphisms of a Ring

Left and Right Multiplication in a Ring

The Signed Sandwich on a Ring

Reflections as Signed Two-Sided Operators on a Ring

The Frobenius Operator on a Field of Characteristic p

The Signed Left Multiplication on a Ring

The Graded Action on a Module over a Ring

* Theory

Involutive Rings

Matrix Rings with an Involution

Rings with a Semilinear Involution

Involutions of a Group Ring

Involutive Local Rings

Involutions of a Polynomial Ring and the Symmetric Part

The Skew Field of a Ring with Involution

* Operator Theory

Involutions of the Endomorphism Ring

The Transpose as an Adjoint

Star-Derivations and the Skew Derivations

Matrix Rings and the Adjoint

The Adjoint of the Left Multiplication on a Ring

The Signed Adjoint Sandwich on a Ring

The Signed Adjoint of the Reflection on a Ring

The Signed Adjoint of the Left Multiplication on a Ring

The Graded Adjoint Action on a Module over a Ring

Applications

Polynomial Rings and Rational Functions

Examples of Rings and Fields

Linear Codes over Finite Fields

Linear Spaces

Theory

Modules

Direct Sums, Free Modules and Rank

Exact Sequences

Modules over a PID

Projective and Injective Modules

Vector Spaces

Linear Maps and Matrices

Eigenvalues and Diagonalisation

The General Linear Group

The Special Linear Group and the Determinant

Transvections and Elementary Transformations

Affine Spaces and Translations

Multilinear Spaces

The Balanced Product

The Regular Module and the Regular Bimodule

Flatness and Exactness

Extension of Scalars

Localization and Completion of Modules

Operator Theory

Endomorphisms of a Linear Space

The Transpose of a Linear Map

Algebras of Endomorphisms

The Regular Representation as an Algebra of Operators

The Signed Sandwich on a Linear Space

Reflections as Signed Two-Sided Operators on a Linear Space

The Signed Left Multiplication on a Linear Space

The Graded Action on a Module over a Linear Space

* Theory

Involutive Linear Spaces

Involutions of the Dual Space

Involutions of the Endomorphism Algebra

Involutions of a Graded Linear Space

Modules over an Involutive Ring

The Regular Bimodule over an Involutive Ring

Involutions of a Tensor Power

Involutive Subspaces and the Decomposition

* Operator Theory

The Adjoint of an Endomorphism

Unitary Endomorphisms

The Involution on the Dual Operator

The Adjoint of the Left Multiplication on a Linear Space

The Adjoints of the Regular Operators

The Signed Adjoint Sandwich on a Linear Space

The Signed Adjoint of the Reflection on a Linear Space

The Signed Adjoint of the Left Multiplication on a Linear Space

The Graded Adjoint Action on a Module over a Linear Space

Applications

Finitely Generated Abelian Groups

Modules over $k[x]$ and the Jordan Form

Vector Spaces over Finite Fields

Bilinear Algebras

Theory

Algebras

Associative Algebras

Unital Algebras

Ideals and Quotients of Algebras

Centre, Units, Zero Divisors and Division Algebras

Central Simple Algebras and the Brauer Group

Crossed Products

Separable Algebras

Frobenius Algebras

Automorphisms and Derivations of Algebras

The Operators on an Algebra

Tensor Powers and the Free Algebra

Quotients of the Tensor Algebra

Tensor Products of Algebras

Hopf Algebras

Quantum Groups

Deformation Quantization

Koszul Duality

Differential Graded Algebras

Differential Graded Categories

A-Infinity and L-Infinity Algebras

Calabi–Yau Algebras

Module Categories

Abelian and Grothendieck Categories

Homological Algebra

Derived Functors

Ext and Tor

Spectral Sequences

Derived Categories

K-Theory of Rings

Hochschild Homology

Cyclic Homology

Deformation Theory

Operads

Topoi

Sheaves on Sites

Descent Theory

Operator Theory

The Sandwich Operator on an Algebra

The Commutator Operator

The Hochschild Differential as an Operator

Left and Right Multiplication

The Signed Sandwich on an Algebra

Reflections as Signed Two-Sided Operators on an Algebra

The Signed Left Multiplication on an Algebra

The Graded Action on a Module over an Algebra

* Theory

Involutive Bilinear Algebras

Opposite Algebras and Anti-Isomorphisms

Involutions of the Tensor Algebra

Involutions of a Free Algebra

Involutive Graded Algebras

Unitary Elements of an Involutive Algebra

The Self-Adjoint Part of an Algebra

Involutions of a Path Algebra

Involutions of a Central Simple Algebra

* Operator Theory

Involutions of the Operator Algebra

The Adjoint in an Involutive Algebra

Unitary Operators of an Involutive Algebra

The Adjoint of the Left Multiplication on an Algebra

The Signed Adjoint Sandwich on an Algebra

The Signed Adjoint of the Reflection on an Algebra

The Signed Adjoint of the Left Multiplication on an Algebra

The Graded Adjoint Action on a Module over an Algebra

Applications

Examples of Algebras

Matrix Algebras

Polynomial Algebras

Group Algebras

Non-Associative Algebras and the Property Ladder

Division Algebras

Symmetric Bilinear Algebras

Theory

Symmetric Powers

The Symmetric Algebra

Divided Powers

Jordan Algebras

Special and Exceptional Jordan Algebras

The Lie–Jordan Decomposition of a Bilinear Product and the Jordan Triple System

Commutative Algebras

Formal Power Series and Completion

Hecke Algebras

Macdonald and Hall–Littlewood Polynomials

Operator Theory

Multiplication Operators on a Commutative Algebra

The Jordan Multiplication Operators

Operators on the Symmetric Algebra

The Derivations of a Commutative Algebra

The Polarisation Operator

The Left and Right Multiplication Operators on a Jordan Algebra

The Signed Sandwich on a Jordan Algebra

Reflections as Signed Two-Sided Operators on a Jordan Algebra

The Signed Left Multiplication on a Jordan Algebra

The Graded Action on a Module over a Jordan Algebra

* Theory

Jordan Algebras with an Involution

The Symmetric Algebra with an Involution

Commutative Algebras with an Involution

Real Forms and the Descent of an Algebra

Formally Real Algebras and the Sum of Squares

The Symmetric Powers of an Involutive Module

Jordan Triples with an Involution

Involution-Invariant Ideals of the Symmetric Algebra

* Operator Theory

Involutions of the Multiplication Operators

Adjoints in a Commutative Involutive Algebra

The Involution on the Multiplication Operators

The Adjoint in the Symmetric Algebra

The Adjoint of the Left Multiplication on a Jordan Algebra

The Signed Adjoint Sandwich on a Jordan Algebra

The Signed Adjoint of the Reflection on a Jordan Algebra

The Signed Adjoint of the Left Multiplication on a Jordan Algebra

The Graded Adjoint Action on a Module over a Jordan Algebra

Anti-symmetric Bilinear Algebras

Theory

Exterior Powers

The Exterior Algebra

The Determinant and Alternating Forms

Lie Algebras

Universal Enveloping Algebras

Structure of Lie Algebras

Root Systems and Classification

Representations of Lie Algebras

Lie Algebra Cohomology

Vertex Algebras

Superalgebras and Graded Structures

Graded Lie Algebras and Lie Superalgebras

Poisson and Gerstenhaber Algebras

Operator Theory

Derivations of a Lie Algebra

The Casimir Operator

The Cartan Operator

The Exterior Derivative on the Lie Algebra

The Killing Form Operator

The Adjoint Action of a Lie Algebra

The Signed Sandwich on a Graded Algebra

Reflections as Signed Two-Sided Operators on a Graded Algebra

The Signed Left Multiplication on a Graded Algebra

The Graded Action on a Module over a Graded Algebra

* Theory

Real Forms of a Complex Lie Algebra

Involutive Lie Superalgebras

Involutions of the Exterior Algebra

Involutions of the Universal Enveloping Algebra

Graded Lie Algebras with an Involution

Symmetric Pairs of a Lie Algebra

* Operator Theory

The Casimir Operator and the Involution

Involutions of the Enveloping Algebra

The Involution on the Exterior Derivative

The Adjoint Representation and the Involution

The Signed Adjoint Sandwich on a Graded Algebra

The Signed Adjoint of the Reflection on a Graded Algebra

The Signed Adjoint of the Left Multiplication on a Graded Algebra

The Graded Adjoint Action on a Module over a Graded Algebra

Applications

Grassmann Variables and Berezin Integration

Linear Spaces over Bilinear Algebras

Theory

Modules over an Algebra

Automorphisms of Modules over an Algebra

Simple and Semisimple Modules

Character Theory

Morita Equivalence

The Balanced Product over an Algebra

Change of Rings

Induced Representations

Projective Representations

Modular Representation Theory

Blocks and Defect Groups

Integral Representations

Quiver Representations and Representation Type

Auslander–Reiten Theory

Tilting Theory

Cluster Algebras

Operator Theory

Left and Right Multiplication of a Module

Module Endomorphisms

The Endomorphism Algebra of a Module

The Signed Sandwich on a Bimodule over an Algebra

Reflections as Signed Two-Sided Operators on a Bimodule over an Algebra

The Signed Left Multiplication on a Module over an Algebra

The Graded Action on a Module over a Bimodule over an Algebra

* Theory

Modules over an Involutive Algebra

The Adjoint of a Module Homomorphism

The Involution on the Endomorphism Ring of a Module

* Operator Theory

The Adjoint of the Sandwich on a Bimodule over an Algebra

Module Operators with an Involution

Involutions of the Module Endomorphism Ring

The Signed Adjoint Sandwich on a Bimodule over an Algebra

The Signed Adjoint of the Reflection on a Bimodule over an Algebra

The Signed Adjoint of the Left Multiplication on a Module over an Algebra

The Graded Adjoint Action on a Module over a Bimodule over an Algebra

Applications

Representations of Groups

Representations of Algebras

PART II : TOPOLOGY

Foundations of Topology

Theory

Metric, Uniform and Complete Spaces

Topological Spaces

Nets, Filters and Convergence

Metrisation and Separation Axioms

Paracompactness and Partitions of Unity

Baire Spaces and Category

Dimension Theory

Proximity Spaces

Bornology

Continuum Theory

Graph Theory

Operator Theory

Continuous Maps as Operators

The Orbit Map

Operators on a Fixed Set

Continuous Maps and the Orbit Map

Operators on a Boolean Algebra

The Boundary Operator of a Simplex

* Theory

Involutions on a Topological Space and the Fixed Set

Spaces with an Involution and the Orbit Space

The Orbit Space of a Free Involution

Equivariant Maps and Equivariant Homotopy

Two-Fold Coverings and the Borel Construction

Involutive Uniform Spaces

Involutive Proximity Spaces

Borsuk's Theorem and the Antipodal Involution

* Operator Theory

Equivariant Operators under a Continuous Involution

The Involution on the Cohomology Operators

The Antipodal Map and the Adjoint

Hermitian Pairings on a Topological Space

Topology on Groups

Theory

Topological Groups

Abelian Topological Groups

Pontryagin Duality

Profinite Groups and the Krull Topology

Representation Theory of Locally Compact Groups

Induced Representations of Locally Compact Groups

Mackey Theory

Linear Algebraic Groups

Property (T)

Geometric Group Theory

Hyperbolic Groups

Amenable Groups

Bass–Serre Theory

Buildings and Tits Systems

Bruhat–Tits Theory

Operator Theory

Operators on a Topological Group

Convolution on a Topological Group

The Signed Sandwich on a Topological Group

Reflections as Signed Two-Sided Operators on a Topological Group

The Signed Left Multiplication on a Topological Group

The Graded Action on a Module over a Topological Group

* Theory

Involutive Topological Groups

The Fixed-Point Subgroup of a Continuous Involution

Involutive Compact Groups

Involutive Profinite Groups

Involutions and Pontryagin Duality

Involutive Group Actions on a Space

* Operator Theory

The Adjoint of the Left Multiplication on a Topological Group

The Signed Adjoint Sandwich on a Topological Group

The Signed Adjoint of the Reflection on a Topological Group

The Signed Adjoint of the Left Multiplication on a Topological Group

The Graded Adjoint Action on a Module over a Topological Group

Topology on Rings and Fields

Theory

Topological Rings and Fields

Absolute Values, Valuations and Completions

Local Fields

Adeles and Ideles

Rigid Analytic Geometry

Berkovich Spaces

Formal Schemes

Adic Spaces

Perfectoid Spaces

Operator Theory

Operators on a Topological Ring

The Completion Operator

The Residue Operator of a Valued Field

The Frobenius Automorphism of a Finite Field

The Valuation Operator

Operators on a Non-Archimedean Field

The Left and Right Multiplication Operators on a Topological Ring

The Signed Sandwich on a Topological Ring

Reflections as Signed Two-Sided Operators on a Topological Ring

The Signed Left Multiplication on a Topological Ring

The Graded Action on a Module over a Topological Ring

* Theory

Involutive Topological Rings and Fields

The Involution and the Completion of a Ring

Involutive Valued Fields

Involutive Local Fields

Involutions of a Non-Archimedean Field

Involutive Topological Division Rings

* Operator Theory

The Involution on Bounded Operators of a Ring

Adjoints under the Residue Pairing

The Adjoint under a Hermitian Valuation

The Adjoint of the Left Multiplication on a Topological Ring

The Signed Adjoint Sandwich on a Topological Ring

The Signed Adjoint of the Reflection on a Topological Ring

The Signed Adjoint of the Left Multiplication on a Topological Ring

The Graded Adjoint Action on a Module over a Topological Ring

Applications

The $p$-adic Numbers

Topology on Linear Spaces

Theory

Topological Modules and Vector Spaces

Normed and Banach Spaces

Locally Convex Spaces

Fréchet Spaces

Duality Theory

Nuclear Spaces

Topological Tensor Products

Operator Theory

Bounded Operators on a Topological Vector Space

The Dual Operator

Operators on a Locally Convex Space

Bounded Operators and the Operator Norm

The Dual Operator and the Weak Topology

The Projection Operator on a Locally Convex Space

The Left and Right Multiplication Operators on a Topological Vector Space

The Signed Sandwich on a Topological Vector Space

Reflections as Signed Two-Sided Operators on a Topological Vector Space

The Signed Left Multiplication on a Topological Vector Space

The Graded Action on a Module over a Topological Vector Space

* Theory

Involutive Topological Linear Spaces

Locally Convex Spaces with an Involution

The Involution and the Dual Pairing

Involutive Normed Spaces

Involutive Banach Spaces

Involutive Fréchet Spaces

Involutive Nuclear Spaces

The Involution on a Topological Tensor Product

* Operator Theory

The Adjoint of a Bounded Operator

Involutions of the Bounded Operators

The Dual Pairing and the Adjoint

The Adjoint of the Left Multiplication on a Topological Vector Space

The Signed Adjoint Sandwich on a Topological Vector Space

The Signed Adjoint of the Reflection on a Topological Vector Space

The Signed Adjoint of the Left Multiplication on a Topological Vector Space

The Graded Adjoint Action on a Module over a Topological Vector Space

Topology on Bilinear Algebras

Theory

Topological Algebras and Banach Algebras

Locally Convex and Fréchet Algebras

Type I Groups

Topological K-Theory

Locally Compact Quantum Groups

Normed Division Algebras and the Hurwitz Theorem

Operator Theory

Operators on a Banach Algebra

Left and Right Multiplication in a Banach Algebra

The Operator Algebra of a Banach Space

The Gelfand Transform as an Operator

Operators on a C*-Algebra

Multipliers of a Banach Algebra

The Left and Right Multiplication Operators on a Banach Algebra

The Signed Sandwich on a Banach Algebra

Reflections as Signed Two-Sided Operators on a Banach Algebra

The Signed Left Multiplication on an Involutive Banach Algebra

The Graded Action on a Module over an Involutive Banach Algebra

* Theory

Involutive Topological Bilinear Algebras

Operator Algebras

Toeplitz Algebras

Graph C*-Algebras

Crossed Products of C*-Algebras

Groupoid C*-Algebras

K-Theory of Operator Algebras

KK-Theory

Involutive Banach Algebras and the Gelfand–Naimark Theorem

States and Positive Functionals on an Involutive Algebra

The Spectrum of a Self-Adjoint Element

Hermitian and Self-Adjoint Elements of a Banach Algebra

The Functional Calculus of a Self-Adjoint Element

Involutive Operator Algebras and the Commutant

The Involution and the Spectral Radius

Involutive Fréchet Algebras

* Operator Theory

Adjoints in a Banach Algebra

The Involution on the Operator Algebra

Self-Adjoint Operators of a Banach Algebra

The Adjoint of the Left Multiplication on an Involutive Banach Algebra

The Signed Adjoint Sandwich on a Banach Algebra

The Signed Adjoint of the Reflection on a Banach Algebra

The Signed Adjoint of the Left Multiplication on an Involutive Banach Algebra

The Graded Adjoint Action on a Module over an Involutive Banach Algebra

Topology on Bilinear Algebras with a degree-2 form

Theory (General)

Bilinear Forms

Quadratic Forms and Polarisation

Symplectic Forms and Poisson Brackets

Symplectic Reflection Algebras

Rational Cherednik Algebras

The Orthogonal Lie Algebra

The Cartan Involution and the Cartan Decomposition

Quadratic Forms over Algebras and Norms

Witt's Theorems

The Witt Group and the Grothendieck–Witt Ring

Clifford Algebras

Clifford Algebras in Finite Dimensions

Degenerate Clifford Algebras and the Radical

The Geometric Product and the Grade Decomposition

The Filtration and the Associated Graded Algebra

The Volume Element, Duality and the Hodge Star

The Hodge Star and the Real Structure of the Exterior Algebra

The Clifford Algebra as a Lie Algebra

The Low-Dimensional Classification

Bott Periodicity and the Classification

The Brauer–Wall Group and the Eightfold Way

Two-Sided Operators on a Clifford Algebra

One-Sided Operators on a Clifford Algebra

Theory, Two-Sided Operator with Inner Conjugation

The Sandwich on a Clifford Algebra

The Two-Sided Operators and the Spin Group

The Inner Conjugation on the Two-Sided Operators

The Spinor Norm and the Structure of the Orthogonal Group with Inner Conjugation

The Low-Dimensional Spin Groups and the Exceptional Isomorphisms with Inner Conjugation

The Clifford Invariant and the Spinor Genus with Inner Conjugation

Infinite-Dimensional Clifford Algebras and CAR with Inner Conjugation

Spin Factors and the Clifford Envelope with Inner Conjugation

Theory, One-Sided Operator with Inner Conjugation

Left Multiplication and the Clifford Module Structure

The One-Sided Action and the Spin Representation

Right Multiplication and the Opposite Algebra

Spin Representations and Clifford Modules with Inner Conjugation

Spinors as Minimal Left Ideals with Inner Conjugation

Spin Representations of the Orthogonal Lie Algebra with Inner Conjugation

Real Spinors and Reality Conditions with Inner Conjugation

Triality and Spin(8) with Inner Conjugation

Theory, Two-Sided Operator with Signed Inner Conjugation

The Graded Multiplication Operators

Two-Sided Operators with the Signed Product

The Sandwich with the Signed Product

Two-Sided Operators on a Clifford Algebra with Signed Inner Conjugation

The Grading of the Clifford Algebra with Signed Inner Conjugation

The Clifford, Pin and Spin Groups with Signed Inner Conjugation

The Two Pin Groups and the Double Covers of the Orthogonal Group with Inner Conjugation

Versors, Rotors and the Sandwich Action with Signed Inner Conjugation

Reflection Groups and Clifford Algebras with Signed Inner Conjugation

Theory, One-Sided Operator with Signed Inner Conjugation

The Signed Action on a Clifford Module

One-Sided Operators with the Signed Product

One-Sided Operators on a Clifford Algebra with Signed Inner Conjugation

Pin Representations and Clifford Modules with Signed Inner Conjugation

Spinors as Minimal Left Ideals with Signed Inner Conjugation

* Theory

Hilbert Algebras

Indefinite Inner Product Spaces

Krein Spaces

Pontryagin Spaces

The Fundamental Symmetry

Krein Algebras

The Indefinite GNS Construction

Krein–von Neumann Algebras

The Indefinite Modular Operator

Krein–Tomita–Takesaki Theory

The Completion of a Hilbert Algebra

The Left and the Right Regular Representation

The GNS Construction

Self-Adjoint Elements and the Positive Cone

The Modular Operator and Tomita-Takesaki Theory

The Modular Group and the KMS Condition

Von Neumann Algebras and the Hilbert Algebra Completeness

* Theory, Two-Sided Operator with Hermitian Adjoint

Hermitian Adjoints on a Hilbert Algebra

The Adjoint of the Left and the Right Multiplication

The Adjoint of the Sandwich on a Hilbert Algebra

The Two-Sided Operators and the Modular Conjugation

The Blade Form and the Hilbert Structure with Hermitian Adjoint

Hermitian Forms on a Hilbert Algebra with Hermitian Adjoint

Hermitian Forms over an Involution Ring and the Unitary Witt Group with Hermitian Adjoint

The Hermitian Sandwich on a Hilbert Algebra with Hermitian Adjoint

The Hermitian Sandwich in the Biquaternion Algebra with Hermitian Adjoint

The Unitary Slice and the Compact Real Form with Hermitian Adjoint

Positivity and the Hermitian Cone of a Hilbert Algebra with Hermitian Adjoint

Self-Adjoint and Skew Operators with Hermitian Adjoint

The Spectra of Self-Adjoint Operators with Hermitian Adjoint

Unitary Equivalence and Congruence of Operators with Hermitian Adjoint

The Hermitian Sylvester Equation with Hermitian Adjoint

Mixed Inner Conjugation and Hermitian Adjoint

Completely Positive Maps of a Hilbert Algebra with Hermitian Adjoint

J-Self-Adjoint and J-Unitary Operators

Spectral Theory on Krein Spaces

Definitizable Operators and the Krein–Naĭmark Theorem

The Hermitian Sandwich on a Krein Space

* Theory, One-Sided Operator with Hermitian Adjoint

The Adjoint of the Left Multiplication on a Hilbert Algebra

The Hilbert Adjoint on a Hilbert Module

Adjoints of the Intertwiners of a Hilbert Algebra

One-Sided Operators on a Hilbert Algebra with Hermitian Adjoint

Hermitian Modules over a Hilbert Algebra with Hermitian Adjoint

The Adjoint of the One-Sided Action with Hermitian Adjoint

Bilinear Operators on a Hermitian Module with Hermitian Adjoint

Invariant Operators and Intertwiners with Hermitian Adjoint

Dirac Operators with Hermitian Adjoint

Spinor Adjoints and the Dirac Adjoint with Hermitian Adjoint

Infinite-Dimensional Hermitian Operators and CAR with Hermitian Adjoint

* Theory, Two-Sided Operator with Signed Hermitian Adjoint

The Signed Adjoint of the Sandwich on a Hilbert Algebra

Adjoints of the Graded Operators of a Hilbert Algebra

Two-Sided Operators on a Hilbert Algebra with Signed Hermitian Adjoint

The Grading of a Hilbert Algebra with Signed Hermitian Adjoint

The Pin and Spin Groups with Signed Hermitian Adjoint

Versors, Rotors and the Sandwich Action with Signed Hermitian Adjoint

Reflection Groups and the Pin Lift with Signed Hermitian Adjoint

* Theory, One-Sided Operator with Signed Hermitian Adjoint

The Signed Adjoint on a Hilbert Module

Adjoints of the Signed Action of a Hilbert Algebra

One-Sided Operators on a Hilbert Algebra with Signed Hermitian Adjoint

Pin Representations and Hermitian Modules with Signed Hermitian Adjoint

Spinors as Minimal Left Ideals with Signed Hermitian Adjoint

Applications

The Number Systems as Clifford Algebras

Lattices and the Quaternion Lattice

Algebraic Topology

Theory

The Fundamental Group and Covering Spaces

CW Complexes and Cellular Approximation

Simplicial and Singular Homology

Cohomology and the Universal Coefficient Theorem

Cup and Cap Products

Poincaré Duality

Homotopy Groups and Fibrations

Classifying Spaces and Cohomology Operations

The Leray–Serre Spectral Sequence

Model Categories and Homotopy Theory

Higher Algebra and Higher Categories

Stable Homotopy Theory

Higher Algebraic K-Theory

Ramification Sequences and Bezoutian Forms

Degree Theory and the Brouwer Fixed Point Theorem

Operator Theory

The Transfer Map

Equivariant Maps as Operators

The Boundary Operator

The Cup Product as an Operator

The Suspension Operator

The Boundary Operator of a Simplicial Complex

* Theory

Equivariant Cohomology

Equivariant Homotopy Theory

Equivariant K-Theory

The Mod 2 Cohomology of an Involution

Smith Theory and the Fixed Sets of Periodic Maps

The Transfer and the Involution

The Gysin Sequence of a Two-Fold Covering

The Cohomology of an Orbit Space

Equivariant Obstruction Theory

* Operator Theory

Equivariant Operators and the Transfer

The Borel Construction on Operators

Sheaves and Cohomology

Theory

Presheaves and Sheaves

Sheaf Cohomology

Čech Cohomology

Derived Functors and Sheaf Cohomology

Sheaves in Algebraic Geometry

Operator Theory

Operators on a Sheaf

The Restriction and Extension Operators

The Coboundary Operator

The Sheaf of Operators

The Cup Product on Sheaf Cohomology

* Theory

Equivariant Sheaves and Descent

Sheaves with a Real Structure

The Involution on the Structure Sheaf

Cohomology with an Involution

Equivariant Derived Categories

Galois Descent for Sheaves

* Operator Theory

The Involution on the Coboundary

Hermitian Pairings of Sheaves

Descent and the Involution on the Cohomology

Algebraic Geometry

Theory

Algebraic Geometry

Schemes

Coherent Sheaves

Moduli Spaces

Stacks

Operator Theory

Operators on a Variety

The Galois Action as an Operator

The Frobenius Operator

The Pullback Operator of a Morphism

The Divisor Operator

The Sheaf of Differential Operators

* Theory

Real Structures on Varieties and Galois Descent

The Weil Restriction and the Trace Form

Real Algebraic Varieties

The Galois Action on the Cohomology

Involutions on a Scheme and the Quotient

Real Structures on a Curve

* Operator Theory

Real Structures on the Operator Layer

Galois Descent for the Operator Layer

Geometric Topology

Theory

Low-Dimensional Topology

Knot Theory

Mapping Class Groups

Lens Spaces

Wild and Exotic Manifolds

Cobordism and Surgery Theory

Thompson Groups and the Cantor Set

Symplectic and Contact Topology

Floer Homology

Operator Theory

The Involution as an Operator on the Homology

The Surgery Operator

The Involution on the Homology

The Dehn Twist as an Operator

The Mapping Class Group Action

The Handle Operator

* Theory

Involutions on Manifolds and Equivariant Surgery

The Classification of Involutions on Surfaces

Amphichiral Knots and the Orientation-Reversing Involution

Equivariant Knot Theory

Free Involutions and Lens Spaces

Periodic Maps and the Smith Theory

Involutions and the Cobordism of Group Actions

* Operator Theory

The Involution on the Homology Operators

Hermitian Pairings and the Signature

Hermitian Pairings and the Equivariant Signature

PART III : ANALYSIS

Foundations of Analysis

Theory

Measure Theory and Integration

Algorithmic Randomness and Chaitin's $\Omega$

Modes of Convergence

Descriptive Set Theory

Fourier Analysis on Euclidean Spaces

Convex Analysis

Nonsmooth and Variational Analysis

Potential Theory

Operator Theory

The Integral Operator

Convolution Operators

The Fourier Operator

The Translation Operator

The Difference Operator

* Theory

Positive Definite Functions and Hermitian Kernels

Reproducing Kernel Hilbert Spaces

Hermitian Measures and Complex Measures

The Fourier Transform and Conjugate Symmetry

Positive Definite Distributions

Hermitian Kernels and the Integral Operator

* Operator Theory

The Adjoint of an Integral Operator

Hermitian Integral Kernels

Involutions of the Convolution Operators

The Adjoint of the Fourier Operator

Analysis on Groups

Theory

Locally Compact Groups and Haar Measure

Harmonic Analysis on Groups

Analysis on Compact Groups

The Peter–Weyl Theorem

The Convolution Algebra $L^1(G)$

Noncommutative Harmonic Analysis

The Plancherel Theorem

Ergodic Theory of Group Actions

Homogeneous Dynamics

Ratner's Theorems

Equidistribution

Automorphic Forms

The Langlands Program

Operator Theory

Convolution on a Group

The Group Algebra as an Algebra of Operators

The Plancherel Operator

The Left and Right Regular Representation

The Signed Sandwich on the Group Algebra

Reflections as Signed Two-Sided Operators on the Group Algebra

The Signed Left Multiplication on the Group Algebra

The Graded Action on a Module over the Group Algebra

* Theory

The Group Algebra as an Involutive Algebra

Positive Definite Functions and the Gelfand–Raikov Theorem

Spherical Functions and the Zonal Spherical Function

Unitary Representations and the Plancherel Theorem

Hermitian Forms and the Group Algebra

The Involution on the Measure Algebra

* Operator Theory

Hermitian Operators on a Group Algebra

Unitary Representations and the Adjoint

The Signed Adjoint Sandwich on the Group Algebra

The Signed Adjoint of the Reflection on the Group Algebra

The Signed Adjoint of the Left Multiplication on the Group Algebra

The Graded Adjoint Action on a Module over the Group Algebra

The Adjoint of a Convolution Operator

Analysis on Rings and Fields

Theory

Analytic Functions and Power Series

Elliptic Functions and Integrals

Non-Archimedean Analysis

p-adic Analysis

p-adic Integration

p-adic Differential Equations

Rigid Analytic Functions

Non-Archimedean Functional Analysis

Adelic Analysis

Zeta Functions

L-Functions

Analytic Number Theory

The Prime Number Theorem

The Riemann Hypothesis

Modular Forms

Operator Theory

The Hecke Operator

Multiplication Operators on an L-Function

The Euler Product Operator

The Shift Operator on the Coefficients

The Shift Operator on a Dirichlet Series

The Signed Sandwich on the Algebra of Arithmetic Functions

Reflections as Signed Two-Sided Operators on the Algebra of Arithmetic Functions

The Signed Left Multiplication on the Algebra of Arithmetic Functions

The Graded Action on a Module over the Algebra of Arithmetic Functions

* Theory

The Functional Equation and the Conjugate Symmetry of an L-Function

Hermitian Forms over a Local Field

The Conjugate Symmetry and the Critical Line

Positivity and the Explicit Formula

Conjugate Orthogonality and the Hecke Characters

Hermitian Forms and the Zeta Function

Involutions of a Rigid Analytic Space

* Operator Theory

The Adjoint of the Hecke Operator

Hermitian Pairings of an L-Function

The Conjugate Symmetry of the Hecke Operator

The Adjoint of the Left Multiplication on the Algebra of Arithmetic Functions

The Signed Adjoint Sandwich on the Algebra of Arithmetic Functions

The Signed Adjoint of the Reflection on the Algebra of Arithmetic Functions

The Signed Adjoint of the Left Multiplication on the Algebra of Arithmetic Functions

The Graded Adjoint Action on a Module over the Algebra of Arithmetic Functions

Analysis on Linear Spaces

Theory

Banach and Hilbert Spaces

Interpolation Theory

Differential Calculus on Normed Spaces

Distributions and Fundamental Solutions

Distributions on Surfaces, Layers, and Jump Conditions

The Schwartz Kernel Theorem

Fredholm Theory

Unbounded Operators and Spectral Measures

Pseudodifferential Operators

Microlocal Analysis

Semiclassical Analysis

Nonlinear Functional Analysis

Fixed Point Theory and Degree Theory

Besov and Triebel–Lizorkin Spaces

Symmetric Tensors and Spherical Harmonics

Operator Theory

Bounded Operators on a Hilbert Space

Compact Operators

The Spectral Operator

Operators on a Banach Space

The Left and Right Multiplication Operators on a Hilbert Space

The Signed Sandwich on a Hilbert Space

Reflections as Signed Two-Sided Operators on a Hilbert Space

The Signed Left Multiplication on a Hilbert Space

The Graded Action on a Module over a Hilbert Space

* Theory

Hilbert Spaces

Sesquilinear Forms and the Lax–Milgram Theorem

Dirichlet Forms and the Hermitian Dirichlet Principle

Self-Adjoint Operators and the Spectral Theorem

Unitary Operators and the Spectral Measure

Hermitian Operators and the Numerical Range

The Adjoint of an Unbounded Operator

Positive Operators and the Square Root

The Friedrichs Extension of a Hermitian Form

The General Spectral Theorem on a Krein Space

The Resolvent of a J-Self-Adjoint Operator

Spectral Measures and the Krein–Naĭmark Dilation

* Operator Theory

Self-Adjoint Operators

Unitary Operators

The Adjoint of the Left Multiplication on a Hilbert Space

The Signed Adjoint Sandwich on a Hilbert Space

The Signed Adjoint of the Reflection on a Hilbert Space

The Signed Adjoint of the Left Multiplication on a Hilbert Space

The Graded Adjoint Action on a Module over a Hilbert Space

Analysis on Bilinear Algebras

Theory

Hypercomplex Analysis

Regularity and the Cauchy–Riemann Operator

Clifford Analysis

Geometric Calculus and the Vector Derivative

Clifford Modules and the Twisted Cauchy–Riemann Operator

Fueter Theory

Hypercomplex Integration

Riemann Boundary Value Problems and Singular Integral Equations

Harmonic Analysis over Hypercomplex Systems

Dirac Differential Operators

The Dirac–Kähler Equation

Holomorphic Functional Calculus

Cyclic Cohomology

Operator Theory

The Dirac Operator

The Cauchy Integral Operator

The Fischer Operator

Operators on a Clifford Module

* Theory

Hermitian Clifford Analysis and the Hermitian Monogenic Functions

The Hermitian Dirac Operator and the Fischer Decomposition

Hermitian Quaternionic Analysis and the Conjugate Cauchy–Riemann Operator

Hermitian Fueter Theory

The Hermitian Cauchy Integral and the Boundary Values

Hermitian Hilbert Modules over a Clifford Algebra

Hilbert and C*-Modules

Positive Definite Kernels in Clifford Analysis

* Operator Theory

The Hermitian Dirac Operator

The Hermitian Cauchy Kernel as an Adjoint

Adjoints on a Clifford Module

Convexity and Order

Theory

Jordan Algebras and the Positive Cone

Convex Sets and the Convex Hull

Convex Functions and the Legendre Transform

Ordered Vector Spaces and the Order Unit

Cones, Extremal Rays and the Choquet Theory

Helly's Theorem and the Approximation of Convex Sets

Operator Theory

Positive Operators on an Ordered Space

The Cone of Positive Operators

The Order Unit as an Operator

The Order Projection

Operators on a Convex Set

The Support Function Operator

The Left and Right Multiplication Operators on an Ordered Algebra

The Signed Sandwich on an Ordered Algebra

Reflections as Signed Two-Sided Operators on an Ordered Algebra

The Signed Left Multiplication on an Ordered Algebra

The Graded Action on a Module over an Ordered Algebra

* Theory

The Positive Cone of an Involutive Algebra

JB*-Algebras and the Gelfand–Naimark Theorem

Hermitian Elements and the Order Unit

The Cone of Positive Functionals

Positive Definite Forms and the Order

The Jordan Algebra of Self-Adjoint Elements

Ordered Involutive Algebras

The Hilbert Cone of an Involutive Algebra

Positive Definite Forms on an Ordered Space

* Operator Theory

The Adjoint of a Positive Operator

Self-Adjoint Elements and the Order

Positive Functionals and Self-Adjointness

The Involution on the Order Automorphisms

The Adjoint of the Left Multiplication on an Ordered Algebra

The Signed Adjoint Sandwich on an Ordered Algebra

The Signed Adjoint of the Reflection on an Ordered Algebra

The Signed Adjoint of the Left Multiplication on an Ordered Algebra

The Graded Adjoint Action on a Module over an Ordered Algebra

Smooth Manifolds and Differential Topology

Theory

Smooth Manifolds and Differential Geometry

Differential Forms and Stokes' Theorem

Sheaves and the de Rham Complex

Fibre Bundles, Connections and Curvature

Characteristic Classes

Differential Topology

Operator Theory

Differential Operators on a Manifold

The Covariant Derivative

The Lie Derivative

The Exterior Derivative

The Codifferential

* Theory

Hermitian Vector Bundles and the Chern Connection

Chern Classes of a Hermitian Bundle

Hermitian Metrics and the Levi-Civita Connection

Real Structures on a Smooth Manifold

Hermitian Manifolds and the Canonical Connection

Hermitian Structures and the Almost Complex Structure

* Operator Theory

The Formal Adjoint of a Differential Operator

Hermitian Connections and the Adjoint

The L2 Adjoint of a Differential Operator

Hermitian Metrics and the Codifferential

Differential Equations

Theory

Ordinary Differential Equations

Delay and Functional Differential Equations

Impulsive Differential Equations

Differential-Algebraic Equations

Partial Differential Equations

Fractional Differential Equations

Sobolev Spaces and Weak Solutions

The Calculus of Variations

Noether's Two Theorems

Optimal Control and the Pontryagin Maximum Principle

Minimal Surfaces

Teichmüller Theory

Integrable Systems

Soliton Theory

Lagrangian and Hamiltonian Systems

The Hamilton–Jacobi Equation and Separability

Multisymplectic and Covariant Hamiltonian Field Theory

Semigroups and Evolution Equations

Stochastic Differential Equations

Stochastic Partial Differential Equations

Operator Theory

Differential Operators

The Green Operator

The Sturm–Liouville Operator

The Resolvent Operator

The Fundamental Solution Operator

* Theory

Self-Adjoint Boundary Value Problems and the Sturm–Liouville Theory

Sesquilinear Forms and the Weak Formulation of an Elliptic Problem

The Dirichlet Principle and the Hermitian Functional

Variational Methods and the Hermitian Form

Self-Adjoint Elliptic Operators and the Spectral Theorem

The Adjoint Problem and the Green Function

The Lagrange Identity and the Self-Adjoint System

* Operator Theory

Self-Adjoint Sturm–Liouville Operators

The Adjoint Boundary Condition

Lie Groups

Theory

Lie Groups

The Lie Algebra and the Exponential Map

The Lie Correspondence and the Adjoint Representation

The Fundamental Group of a Lie Group

Homology of Classical Groups and Homogeneous Spaces

Hilbert's Fifth Problem and Infinite-Dimensional Lie Theory

Diffeomorphism Groups

Loop Groups

Kac–Moody Groups

p-adic Lie Groups

Lattices in Lie Groups

Arithmetic Groups

Operator Theory

Operators on a Lie Group

The Casimir Operator of a Lie Group

The Exponential Map as an Operator

The Regular Representation of a Lie Group

The Signed Sandwich on a Lie Algebra

Reflections as Signed Two-Sided Operators on a Lie Algebra

The Signed Left Multiplication on a Lie Algebra

The Graded Action on a Module over a Lie Algebra

* Theory

Real Forms of a Complex Lie Group and the Cartan Involution

Unitary Representations of a Lie Group

The Cartan Decomposition and the Cartan Involution

Hermitian Lie Groups and the Bounded Domain

The Involution on the Enveloping Algebra of a Lie Group

Unitary Representations and the Orbit Method

* Operator Theory

Hermitian Forms on a Lie Algebra

Unitary Representations and the Adjoint Operator

The Signed Adjoint Sandwich on a Lie Algebra

The Signed Adjoint of the Reflection on a Lie Algebra

The Signed Adjoint of the Left Multiplication on a Lie Algebra

The Graded Adjoint Action on a Module over a Lie Algebra

Probability and Ergodic Theory

Theory

Measure-Theoretic Probability

Independence and Conditional Expectation

Laws of Large Numbers and the Central Limit Theorem

Martingales

Markov Chains and Processes

Brownian Motion and Stochastic Calculus

Functional Integration and the Rigorous Path Integral

Ergodic Theory

The Probabilistic Method

Probabilistic Number Theory

Random Walks on Groups

Operator Theory

The Markov Operator

The Transition Operator

The Ergodic Operator

The Conditional Expectation Operator

The Shift Operator of a Process

The Signed Sandwich on the Algebra of Random Variables

Reflections as Signed Two-Sided Operators on the Algebra of Random Variables

The Signed Left Multiplication on the Algebra of Random Variables

The Graded Action on a Module over the Algebra of Random Variables

* Theory

Reversible Markov Chains and Time Reversal

Non-Commutative Probability and the Involutive Algebra of Random Variables

The Reversibility of a Stationary Process

The Involution on the Algebra of Random Variables

The Covariance Function and Hermitian Positivity

The Characteristic Function and Conjugate Symmetry

* Operator Theory

The Adjoint of the Markov Operator

The Involution on the Operator Algebra of a Process

The Adjoint of the Transition Operator

The Adjoint of the Left Multiplication on the Algebra of Random Variables

Reversible Operators and Self-Adjointness

The Signed Adjoint Sandwich on the Algebra of Random Variables

The Signed Adjoint of the Reflection on the Algebra of Random Variables

The Signed Adjoint of the Left Multiplication on the Algebra of Random Variables

The Graded Adjoint Action on a Module over the Algebra of Random Variables

Dynamical Systems

Theory

Topological Dynamics

Smooth Dynamical Systems

Hyperbolic Dynamics and Anosov Systems

Random Dynamical Systems

Bifurcation Theory

Chaos and Strange Attractors

Symbolic Dynamics

Dynamics and Number Theory

Operator Theory

The Evolution Operator

The Koopman Operator

The Transfer Operator

The Flow Operator

The Poincaré Map

* Theory

Reversible Dynamical Systems and Time-Reversal Symmetry

Equivariant Dynamics under an Involution

Symmetric Periodic Orbits and the Involution

Reversible Systems and the KAM Theorem

Equivariant Bifurcation Theory

Invariant Tori under an Involution

* Operator Theory

The Adjoint of the Koopman Operator

Reversible Operators and the Involution

Time Reversal and the Transfer Operator

The Involution on the Flow Operator

PART IV : GEOMETRY

Foundations of Geometry

Theory

Curvature and Geodesics

Riemannian Geometry

Metric Geometry

Pseudo-Riemannian and Lorentzian Geometry

Fractal Geometry

Gromov–Hausdorff Convergence

Operator Theory

Isometries as Operators

The Geodesic Flow Operator

The Curvature Operator

The Shape Operator

The Parallel Transport Operator

* Theory

The Geodesic Symmetry and Locally Symmetric Spaces

Riemannian Symmetric Spaces and the Involution

Isometric Involutions and the Two-Fold Quotient of a Riemannian Manifold

Isometric Involutions and the Fixed-Point Set

Real Structures on a Riemannian Manifold

Hermitian Manifolds and the Geodesic Involution

* Operator Theory

The Adjoint of the Geodesic Operator

Isometric Involutions on the Operator Layer

The Involution on the Curvature Operator

Hermitian Manifolds and the Adjoint

Geometry on Groups

Theory

The Three Two-Dimensional Algebras and the Three Kinds of Rotation

Isometries and Orthogonal Transformations

The Rotation Group and Orientation

The Unitary and Symplectic Groups

Matrix Groups and Classical Groups

Symmetry, Point and Crystallographic Groups

Operator Theory

Operators on a Symmetry Group

The Symmetry Operators of a Space

The Convolution Operator on a Symmetric Space

The Translation Operator on a Symmetry Group

The Signed Sandwich on a Symmetry Group

Reflections as Signed Two-Sided Operators on a Symmetry Group

The Signed Left Multiplication on a Symmetry Group

The Graded Action on a Module over a Symmetry Group

* Theory

The Unitary Group and the Hermitian Symmetric Space

Unitary Groups and Their Involution

The Orthogonal Group and the Involutive Automorphism

Hermitian Symmetric Spaces and the Group Involution

The Involution on the Symmetry Group of a Space

* Operator Theory

The Adjoint of a Symmetry Operator

Unitary Operators of a Symmetry Group

The Involution on the Translation Operators

The Signed Adjoint Sandwich on a Symmetry Group

The Signed Adjoint of the Reflection on a Symmetry Group

The Signed Adjoint of the Left Multiplication on a Symmetry Group

The Graded Adjoint Action on a Module over a Symmetry Group

Geometry on Rings and Fields

Theory

Euclidean Geometry

Spherical Geometry

Hyperbolic Geometry

Non-Euclidean Geometry

Conformal Geometry

Möbius and Lie Sphere Geometry

Operator Theory

Two-Sided Operator

Operators on a Projective Space

The Möbius Transformation as an Operator

One-Sided Operator

The Projection Operator

Signed Two-Sided Operator

The Conformal Operator

Geodesic Reflection as an Operator

Signed One-Sided Operator

Operators on Hyperbolic Space

* Theory

Unitary Geometry over a Field with Involution

Hermitian Forms and Unitary Geometry

Real Structures on a Projective Space

Orthogonal Geometry and the Involution

Hermitian Spaces over a Local Field

* Operator Theory

Two-Sided Operator with Adjoint

The Adjoint under a Hermitian Pairing

Hermitian Structures and the Projection Operator

The Involution on the Möbius Operators

One-Sided Operator with Adjoint

Involutions of the Projection Operators

Signed Two-Sided Operator with Adjoint

The Signed Adjoint of the Conformal Operator

Signed One-Sided Operator with Adjoint

The Signed Adjoint of the Geodesic Reflection

Geometry on Linear Spaces

Theory

Grassmannians and Stiefel Manifolds

Homogeneous Spaces

Transformation Groups and the Erlangen Program

Symmetric Spaces

Flag Manifolds

Symplectic Geometry

Contact Geometry

Poisson Geometry

Lie–Poisson Reduction and the Euler–Poincaré Equation

Kähler Geometry

Hermitian Geometry and Almost Complex Structures

Calabi–Yau Manifolds

Operator Theory

Operators on a Complex Manifold

The Bergman Operator

The Almost Complex Operator

The Kähler Form Operator

The Curvature Operator of a Complex Manifold

The Left and Right Multiplication Operators on a Complex Vector Space

The Signed Sandwich on a Complex Vector Space

Reflections as Signed Two-Sided Operators on a Complex Vector Space

The Signed Left Multiplication on a Complex Vector Space

The Graded Action on a Module over a Complex Vector Space

* Theory

Hermitian Symmetric Spaces and the Bergman Metric

Real Structures on a Complex Manifold

Complex Manifolds with an Antiholomorphic Involution

Kähler Manifolds and the Hermitian Form

The Involution on a Complex Vector Space

Hermitian Geometry and the Unitary Group

* Operator Theory

The Adjoint of a Hermitian Operator

Involutions of the Bergman Operator

The Involution on the Kähler Operator

The Adjoint of the Left Multiplication on a Complex Vector Space

The Signed Adjoint Sandwich on a Complex Vector Space

The Signed Adjoint of the Reflection on a Complex Vector Space

The Signed Adjoint of the Left Multiplication on a Complex Vector Space

The Graded Adjoint Action on a Module over a Complex Vector Space

Geometry on Bilinear Algebras

Theory

Quaternionic Geometry

Hyperkähler Geometry

G2 and Spin(7) Manifolds

Spin Geometry

Supergeometry

The Conformal Model of Euclidean Space

Projective Geometry

Projective Geometric Algebra and Dual Quaternions

Curves and Surfaces in the Biquaternion Moving Frame

Operator Theory

The Spinor Operator

The Twistor Operator

The Clifford Multiplication Operator

The Codifferential on a Spinor Bundle

The Penrose Operator

The Left and Right Multiplication Operators on a Clifford Algebra

The Signed Sandwich on a Clifford Algebra

Reflections as Signed Two-Sided Operators on a Clifford Algebra

The Signed Left Multiplication on a Clifford Algebra

The Graded Action on a Module over a Clifford Algebra

* Theory

Twistor Spaces and the Hermitian Structure of a Conformal Manifold

Hermitian Spin Geometry and the Twistor Correspondence

Quaternionic Geometry and the Conjugate Structure

Hyperkähler Manifolds and the Twistor Space

Hermitian Clifford Structures

* Operator Theory

The Adjoint of the Clifford Multiplication

Involutions of the Twistor Operator

The Adjoint of the Twistor Operator

The Adjoint of the Left Multiplication on a Clifford Algebra

The Signed Adjoint Sandwich on a Clifford Algebra

The Signed Adjoint of the Reflection on a Clifford Algebra

The Signed Adjoint of the Left Multiplication on a Clifford Algebra

The Graded Adjoint Action on a Module over a Clifford Algebra

Synthesis of Geometry and Analysis

Theory

Geometric Measure Theory

Harmonic Maps

The Geodesic Flow

Ricci Flow

Mean Curvature Flow

The Atiyah–Singer Index Theorem and K-Theory

Spectral Triples and Noncommutative Geometry

Operator Theory

The Signature Operator

The Hodge Laplacian

The Dirac Operator on a Manifold

* Theory

Polarised Hodge Structures and the Hodge–Riemann Relations

The Hermitian Index Theorem and the Signature Operator

Hermitian Metrics and the Hodge Theory

The Signature Operator and the Involution

The Involution on the Space of Connections

* Operator Theory

The Adjoint of the Signature Operator

The Dirac Operator and Its Adjoint

Hermitian Pairings and the Index

PART V : CATALOGUES

Catalogue of Algebra

List of Algebraic Structures

List of Universal Properties

List of Categories

List of Groups

List of Finite Simple Groups

List of Abelian Groups

List of Group Constructions

List of Group Actions

List of Automorphism Groups

List of Representations

List of Rings

List of Structures from Rings to Fields

List of Structures from Rings to Division Rings

List of Maximal Structures Before Failure

List of Commutative Rings

List of Reduced, Local and Product Rings

List of Local Rings and Valuations

List of Commutative Integral Domains

List of Non-Commutative Domains

List of Unique Factorisation Domains

List of Principal Ideal and Euclidean Domains

List of Commutative Fields

List of Finite Fields

List of Non-Commutative Division Rings

List of Non-Commutative Rings

List of Zero Divisors and Nilpotents

List of Zero Divisors and Prime Ideals

List of Rings by Their Idempotents

List of Artinian and Noetherian Rings

List of Homological Algebra Constructions

List of Modules

List of Algebras

List of Associative Algebras

List of Unital Algebras

List of Central Simple Algebras

List of Matrix Rings

List of Algebras by Dimension

List of Number Systems by Property

List of Division Algebras

List of Non-Examples and Counter-Examples in Algebra

Catalogue of Topology

List of Topological Spaces

List of Topological Spaces by Separation and Compactness

List of Metric and Uniform Spaces

List of Topological Groups

List of Topological Homology and Cohomology

List of Homotopy Theories

List of Geometric Forms

List of Clifford Algebras and Spin Groups

List of Geometric Lattices and Integral Forms

List of Geometric Schemes and Varieties

List of Non-Examples in Topology

Catalogue of Analysis

List of Function Spaces

List of Norms and Seminorms

List of Measures

List of Modes of Convergence

List of Inequalities

List of Integral Transforms

List of Special Functions

List of Orthogonal Polynomials

List of Constants

List of Differential Operators and Equations

List of Distributions

List of Probability Distributions and Processes

List of Dynamical Systems

List of Lie Groups

List of Geometric Bundles and Invariants

List of Non-Examples in Analysis

Catalogue of Geometry

List of Transformation Groups

List of Linear Geometric Groups

List of Classical Geometric Groups

List of Projective Geometric Groups

List of Affine and Euclidean Groups

List of Isometry and Symmetry Groups

List of Discrete Geometric Groups

List of Möbius and Conformal Groups

List of Diffeomorphism and Homeomorphism Groups

List of Klein Geometries

List of Erlangen Program Geometries

List of Geometric Manifolds

List of Riemannian Geometries

List of Symplectic Geometries

List of Geometric Surfaces and Curves

List of Non-Examples in Geometry

PART VI : SYNTHETIC STUDIES

Booleans

Algebra

Boolean Algebras and Lattices

Heyting Algebras and Intuitionistic Logic

MV-Algebras and Many-Valued Logic

Effect Algebras and Orthomodular Lattices

Quantales and Frames

Topology

Boolean Rings and Stone Duality

Natural Numbers

Algebra

The Natural Numbers ($\mathbb{N}$)

Peano Arithmetic and Model Theory

Computability Theory

Special Functions

Combinatorial Functions and Generating Functions

Integers

Algebra

The Integers ($\mathbb{Z}$)

Applications

Modular Arithmetic and the Ring of Residues

Rational Numbers

Algebra

The Rational Numbers ($\mathbb{Q}$)

Analysis

Diophantine Approximation and Continued Fractions

Real Numbers

Algebra

The Real Numbers ($\mathbb{R}$)

Real Algebra

Worked Examples in the Real Algebra

Topology

Real Norm and Invertibility

Real Topology

Analysis

Real Analysis

O-Minimality

Real Integration

Real Special Functions

Real Harmonic Analysis

Real Exponential and Lie Group Structure

Focus on Element Representations

Real Element Representations

Real Polar Element Representation

Real Regular Element Representation

Geometry

Real Line Geometry and Isometries

Real Automorphisms and Derivations

Complex Numbers

Algebra

The Complex Numbers ($\mathbb{C}$)

Complex Algebra

Galois Theory of ℂ/ℝ

Worked Examples in the Complex Algebra

Topology

Complex Norm and Invertibility

Complex Topology

Focus on Subspaces

Complex Subspaces

Analysis

Complex Analysis

Several Complex Variables

Complex Integration

Complex Special Functions

Complex Harmonic Analysis

Complex Exponential and Lie Group Structure

Focus on Element Representations

Complex Element Representations

Complex Polar Element Representation

Complex Regular Element Representation

Complex Two-Component Element Representation

Geometry

Rotations and Reflections in the Complex Plane

Complex Automorphisms and Derivations

Split-Complex Numbers

Algebra

Split-Complex Algebra

Split-Complex Idempotents and Projections

Split-Complex Ideals and Peirce Decomposition

Split-Complex Zero Divisors

Worked Examples in the Split-Complex Algebra

Topology

Split-Complex Norm and Invertibility

Split-Complex Topology

Focus on Subspaces

Split-Complex Subspaces

Analysis

Split Complex Analysis

Split-Complex Integration

Split-Complex Special Functions

Split-Complex Harmonic Analysis

Split-Complex Exponential and Lie Group Structure

Focus on Element Representations

Split-Complex Element Representations

Split-Complex Polar Element Representation

Split-Complex Regular Element Representation

Split-Complex Two-Component Element Representation

Geometry

Split-Complex Null Quadric and Projective Geometry

Hyperbolic Rotations

Split-Complex Automorphisms and Derivations

Dual Numbers

Algebra

Dual-Numbers Algebra

Dual-Numbers Ideals and the Maximal Ideal

Dual-Numbers Zero Divisors

Worked Examples in the Dual-Number Algebra

Topology

Dual-Numbers Norm and Invertibility

Dual-Numbers Topology

Focus on Subspaces

Dual-Number Subspaces

Analysis

Dual-Numbers Analysis

Dual-Numbers Integration

Dual-Numbers Special Functions

Dual-Numbers Harmonic Analysis

Dual-Numbers Exponential and Lie Group Structure

Focus on Element Representations

Dual-Numbers Element Representations

Dual-Numbers Polar Element Representation

Dual-Numbers Matrix Element Representation

Geometry

Dual-Numbers Null Quadric and Projective Geometry

Shears and Parabolic Rotations

Dual-Numbers Automorphisms and Derivations

Quaternions

Algebra

Quaternion Algebra

Quaternion Ideals and Simplicity

Quaternion Roots of Minus One

Worked Examples in the Quaternion Algebra

Topology

Quaternion Norm and Invertibility

Quaternion Metrics

Quaternion Topology

Focus on Subspaces

The Scalar and Vector Subspaces of $\mathbb{H}$

Analysis

Quaternion Analysis

Quaternion Regular Functions

Bloch's Theorem and the Bloch Constant for Monogenic Functions

Bohr's Phenomenon for Monogenic Functions

Quaternion Spectral Theory

Fueter Theory for Quaternions

Quaternion Integration

Quaternion Exponential and Lie Group Structure

Quaternion Special Functions

Quaternion Harmonic Analysis

The Generalised Hamilton–Rodrigues Calculus

Quaternion Augmented Statistics

Focus on Element Representations

Quaternion Element Representations

Quaternion Polar Element Representation

Quaternion Four-Vector Element Representation

Quaternion 4x4 Regular Matrix Element Representation

Quaternion 2x2 Matrix Element Representation

Quaternion Other Algebraic Element Representations

Geometry

Quaternion Rotations and Reflections

Four-Dimensional Rotations

Quaternion Involutions and Projections

Quaternion Automorphisms and Derivations

Quaternion Geometry

Riemannian Quaternions

Split-Quaternions

Algebra

Split-Quaternion Algebra

Split-Quaternion Idempotents and Projections

Split-Quaternion Ideals and Peirce Decomposition

Split-Quaternion Roots of Minus One

Split-Quaternion Zero Divisors

Worked Examples in the Split-Quaternion Algebra

Topology

Split-Quaternion Norm and Invertibility

Split-Quaternion Topology

Focus on Subspaces

Split-Quaternion Scalar and Vector Subspaces

Split-Quaternion Split-Complex Subspaces

Split-Quaternion Subspaces and the Involutions

Split-Quaternion Relations Between Subspaces

Split-Quaternion Involution Lattice

Analysis

Split-Quaternion Analysis

Split-Quaternion Regular Functions

Split-Quaternion Spectral Theory

Fueter Theory for Split-Quaternions

Split-Quaternion Integration

Split-Quaternion Exponential and Lie Group Structure

Split-Quaternion Elementary Functions

Split-Quaternion Higher Special Functions

Split-Quaternion Harmonic Analysis

Focus on Element Representations

Split-Quaternion Element Representations

Split-Quaternion Polar Element Representation

Split-Quaternion Matrix Element Representations

Split-Quaternion Four-Vector Element Representation

Split-Quaternion Other Algebraic Element Representations

Geometry

Split-Quaternion Null Quadric and Projective Geometry

Split-Quaternion Rotations and the Lorentz Group

Split-Quaternion Automorphisms and Derivations

Split-Quaternions and Hyperbolic Geometry

Split-Quaternion Geometry

Biquaternions

Algebra

Introduction

Biquaternion Algebra

Introduction to the Six Subspaces

Comparison of the Six Subspaces

Different Ways to Consider Biquaternions

Multiplication and Degree-2 Forms

Biquaternion Multiplication

Biquaternion Jordan Algebra

Biquaternion Lie Algebra

Structure and Modules

Biquaternion Idempotents and Projections

Biquaternion Ideals and Peirce Decomposition

Modules over the Biquaternion Algebra

The Regular Module and the Regular Bimodule over the Biquaternion Algebra

The Enveloping Algebra of the Biquaternion Algebra and the Bi-module Structure

Elements and Operators

Biquaternion Square Roots of Minus One, Zero and Plus One

Biquaternion Square Roots of a General Element

Biquaternion Zero Divisors

Biquaternion Involution Lattice

Biquaternion Relations Between Subspaces

Worked Examples in the Biquaternion Algebra

The Six Subspaces

The Six Subspaces and the Products

The Six Subspaces and the Jordan Algebra

The Six Subspaces and the Lie Algebra

The Six Subspaces and the Units

The Six Subspaces and the Zero Divisors

The Six Subspaces and the Idempotents and Projections

The Six Subspaces and the Ideals

The Six Subspaces and the Roots of Minus One

The Six Subspaces and the Two Matrix Representations

The Six Subspaces and the Forms

The Six Subspaces and the Analysis

Basic Representations

Biquaternion 2×2 Matrix Element Representation

Biquaternion 4×4 Regular Matrix Element Representation

Topology

Introduction to Topology on the Biquaternions

Introduction to Topology on the Biquaternions

The Three Topologies of the Biquaternion Algebra

Topology Induced by the Bilinear Form

Form

The Bilinear Form on the Biquaternion Algebra

Biquaternion Norm and Invertibility

The Clifford Structure of the Biquaternion Algebra

Biquaternion Topology

The Biquaternion Unit Group as a Topological Group

Biquaternion Orders and Finite Groups of Units

Biquaternion Four-Vector Element Representation

Biquaternion Polar Element Representation

Biquaternion Partial Polar Element Representations

Biquaternion Other Algebraic Element Representations

The Fierz–Kofink Identities and the Classification of Spinors

Operators

Bilinear Operators on the Biquaternion Algebra with Hermitian Adjoint

Association and the Transpose on the Biquaternion Algebra

Biquaternion Four-Vector Operator Representation

Biquaternion Twisted Spinor Operator Representation

Topology Induced by the Hermitian Form

Form

The Hermitian Form on the Biquaternion Algebra

The Canonical Hermitian Form on the Regular Module of the Biquaternion Algebra

The Euclidean Topology of the Biquaternion Algebra

The Unitary Group of the Biquaternion Algebra

Hermitian Modules over the Biquaternion Algebra with Hermitian Adjoint

Hermitian Forms over the Biquaternion Algebra and the Unitary Witt Group with Hermitian Adjoint

Real Spinors and Reality Conditions on the Biquaternion Algebra with Hermitian Adjoint

Operators

Two-Sided Operators on the Biquaternion Algebra with Hermitian Adjoint

One-Sided Operators on the Biquaternion Algebra with Hermitian Adjoint

Mixed Inner Conjugation on the Biquaternion Algebra with Hermitian Adjoint

The Spectra of the Operators on the Biquaternion Algebra with Hermitian Adjoint

Positivity and the Hermitian Cone of the Biquaternion Algebra with Hermitian Adjoint

Completely Positive Maps of the Biquaternion Algebra with Hermitian Adjoint

Biquaternion 2×2 Matrix Operator Representation

Biquaternion 4×4 Regular Matrix Operator Representation

Topology Induced by the Krein Form

Form

The Biquaternion Krein Form and Its Signature

The Fundamental Symmetry of the Biquaternion Algebra

The Three Pairings of the Biquaternion Algebra

The Krein Gram Matrix and the Restrictions of the Form

Krein Orthogonality and the Fundamental Decomposition

The Isotropic Structure of the Krein Form

The Krein Level Sets and the Hyperbolic Structure

Operators

J-Self-Adjoint and J-Unitary Operators on the Biquaternion Algebra

Indefinite Positivity and the Krein Cone of the Biquaternion Algebra

The Indefinite Spectra of the Operators on the Biquaternion Algebra

J-Normal Operators and the Indefinite Spectral Theorem

The Krein Isometry Group and Its J-Contractions

The Indefinite Hermitian Sandwich on the Biquaternion Algebra

The Krein Cartan Decomposition of the Operator Algebra

The Biquaternion Algebra and the Krein–von Neumann Property

The Tomita Operator and the J-Modular Pair for the Biquaternion Algebra

Synthesis and relations between the 3 topologies

Form

The Forms in the Matrix Representation of the Biquaternion Algebra

The Minkowski Form and the Modulus-Squared Map in the Matrix Representation

Topology

The Unit Group and the Frobenius Norm in the Matrix Representation

The Six Subspaces

The Centre Subspace under the Three Topologies

The Vector Subspace under the Three Topologies

The Quaternion Subspace under the Three Topologies

The Anti-Quaternion Subspace under the Three Topologies

The Hermitian Subspace under the Three Topologies

The Anti-Hermitian Subspace under the Three Topologies

Basic Representations

The 2×2 Matrix Representation under the Three Topologies

The 4×4 Regular Matrix Representation under the Three Topologies

Analysis

Biquaternion Analysis

Biquaternion Regular Functions

Biquaternion Spectral Theory

The Hermitian Sylvester Equation

Fueter Theory for Biquaternions

Biquaternion Integration

Biquaternion Lie Group and Exponential Structure

Biquaternion Elementary Functions

Biquaternion Higher Special Functions

Biquaternion Discrete Harmonic Analysis

Biquaternion Continuous Harmonic Analysis

Geometry

Biquaternion Rotations and Lorentz Transformations

Biquaternion Automorphisms and Derivations

Biquaternion Lorentzian and Conformal Geometry

Biquaternion Spin Geometry

Biquaternion Versors and the Orthogonal Group

Biquaternion Objects and Their Matrix Correspondences

Split-Biquaternions

Algebra

Split-Biquaternion Algebra

Complex Split Biquaternions and the Clifford Algebra Cl(3)

Split-Biquaternion Idempotents and Projections

Split-Biquaternion Ideals and Peirce Decomposition

Split-Biquaternion Roots of Minus One

Split-Biquaternion Zero Divisors

Worked Examples in the Split-Biquaternion Algebra

Topology

Split-Biquaternion Norm and Invertibility

Split-Biquaternion Topology

Focus on Subspaces

Split-Biquaternion Split-Complex Subspace

Split-Biquaternion Quaternion Subspace

Split-Biquaternion Hermitian Subspace

Split-Biquaternion Anti-Hermitian Subspace

Split-Biquaternion Relations Between Subspaces

Split-Biquaternion Involution Lattice

Analysis

Split-Biquaternion Analysis

Split-Biquaternion Regular Functions

Split-Biquaternion Spectral Theory

Fueter Theory for Split-Biquaternions

Split-Biquaternion Integration

Split-Biquaternion Exponential and Lie Group Structure

Split-Biquaternion Elementary Functions

Split-Biquaternion Higher Special Functions

Split-Biquaternion Discrete Harmonic Analysis

Split-Biquaternion Continuous Harmonic Analysis

Focus on Element Representations

Split-Biquaternion Element Representation Theory

Split-Biquaternion Four-Vector Element Representation

Split-Biquaternion Polar Element Representation

Split-Biquaternion Algebraic Element Representations

Geometry

Split-Biquaternion Null Quadric and Projective Geometry

Split-Biquaternion Rotations and the Lorentz Group

Split-Biquaternion Automorphisms and Derivations

Split-Biquaternions and Hyperbolic Geometry

Split-Biquaternion Geometry

Structural Comparison around Biquaternions

Overview

The Eight Algebras Compared

Algebra

Comparison of the Algebras

Topology

Comparison of Norms and Invertibility

Focus on Subspaces

Comparison of Subspaces and Involutions

Analysis

Comparison of Analysis

Comparison of Integration

Comparison of Spectral Theory

Comparison of the Exponential and Lie Group Structure

Comparison of Special Functions

Comparison of Harmonic Analysis

Focus on Element Representations

Comparison of Element Representations and Matrix Models

Comparison of the Polar Element Representation

Geometry

Comparison of Geometry

Comparison of Automorphisms and Derivations

Octonions

Algebra

Octonion Algebra

Octonion Norm and Invertibility

Complex Octonions and the Clifford Algebra Cl(6)

Split Bioctonions and the Clifford Algebra Cl(7)

Maximal Totally Isotropic Subspaces and Their Unitary Symmetries

Focus on Element Representations

Octonion Element Representations

Octonions and the Exceptional Lie Groups

Geometry

Octonion Geometry

Octonions and Exceptional Geometry

Analysis

Octonion Analysis

Integration

Octonion Integration

Special Functions

Octonion Special Functions

Harmonic Analysis

Octonion Harmonic Analysis