Part I: Algebraic Structures & Topological Foundations
Groups and Group Theory
- Group: Abstract set with a binary operation satisfying associativity, identity, and invertibility. Captures symmetry and structure.
- Subgroup: A subset that is itself a group under the same operation.
- Normal Subgroup: Invariant under conjugation; key to forming quotient groups.
- Quotient Group: The result of identifying elements differing by a normal subgroup; often used in symmetry reduction.
- Group Homomorphism & Isomorphism: Structure-preserving maps between groups; isomorphisms show structural equivalence.
- Group Action: A group acting on a set reveals symmetry in geometry and algebra.
Rings and Fields
- Ring: A set with two operations: addition (forming an abelian group) and multiplication (associative, not necessarily invertible).
- Ideal: Subset of a ring absorbing multiplication, critical in quotient ring constructions.
- Quotient Ring: Formed by collapsing an ideal to zero; key in algebraic geometry and number theory.
- Field: A ring where every nonzero element is invertible under multiplication.
- Field Extensions: Understanding solutions of polynomials by expanding base fields.
Metric and Topological Spaces
- Metric Space: Set with a distance function satisfying symmetry, triangle inequality, and positivity.
- Topological Space: Abstracts nearness via open sets, enabling general continuity and convergence.
- Open/Closed Sets: Defined via the topology; crucial in understanding continuity and limits.
- Basis and Subbasis: Foundations for generating topologies.
- Continuous Maps: Functions preserving the topological structure.
- Homeomorphisms: Topological equivalence between spaces.
- Compactness: Generalizes finite coverings; key in functional analysis and topology.
- Connectedness & Path-connectedness: Concepts describing whether spaces can be split or traversed via paths.
- Separation Axioms (T0-T4): Classify topologies based on distinguishability of points and closed sets.
Part II: Linear and Multilinear Algebra
Vector Spaces & Linear Maps
- Vector Space: Algebraic structure closed under vector addition and scalar multiplication.
- Subspace, Span, Basis, Dimension: Concepts to quantify and describe vector spaces.
- Linear Map: Preserves linear structure; fundamental to analysis and geometry.
- Isomorphisms and Automorphisms: Describe when vector spaces or transformations preserve full structure.
Spectral Theory
- Eigenvectors and Eigenvalues: Describe intrinsic directions and scalings in transformations.
- Diagonalization and Jordan Form: Simplify understanding of linear operators.
- Invariant Subspaces: Subspaces preserved under linear operators.
Inner Product Spaces
- Inner Product, Norm, and Orthogonality: Generalize geometric ideas to abstract spaces.
- Orthogonal Projections and Orthonormal Bases: Central in Fourier analysis and optimization.
Duality and Tensor Products
- Dual Space: Space of linear functionals; fundamental in functional analysis.
- Tensor Products: Construct multilinear objects; bridge between linear algebra and geometry.
Part III: Abstract and Homological Algebra
Modules
- Module over a Ring: Generalization of vector spaces over arbitrary rings.
- Submodules, Quotients, Exact Sequences: Central tools in homological algebra.
Homological Tools
- Chain Complexes and Homology: Algebraic method of encoding topological information.
- Derived Functors and Ext/Tor: Measure obstructions to exactness.
Part IV: Geometry and Topology
Manifolds and Differential Topology
- Manifolds: Locally Euclidean spaces that may globally curve or twist.
- Charts, Atlases, and Smooth Structures: Tools for handling differentiability on curved spaces.
- Tangent and Cotangent Spaces: Linear approximations of manifolds at a point.
Differential Forms and Integration
- Exterior Algebra and Wedge Product: Basis for multivariable integration.
- Stokes' Theorem: Unifies several classical theorems (e.g., Green, Gauss).
Algebraic Topology
- Fundamental Group (\pi_1): Captures loops and holes.
- Covering Spaces: Reveal the structure of \pi_1 through unwrapping.
- Homology and Cohomology Groups: Classify spaces via cycles and boundaries.
- Exact Sequences and Mayer-Vietoris: Tools for computation and analysis.
Part V: Functional and Real Analysis
Measure and Integration
- Measure Space and Sigma-Algebra: Framework for assigning sizes to sets.
- Measurable Functions and Lebesgue Integration: Robust tools for handling limits and convergence.
Function Spaces
- L^p Spaces: Normed spaces essential in PDEs and harmonic analysis.
- Convergence Theorems: Dominated, Monotone, Fatou's Lemma.
Topology of Function Spaces
- Pointwise vs. Uniform Convergence: Critical in analysis and topology.
- Compactness Criteria (Arzelà–Ascoli): Classify precompact sets in function spaces.
Part VI: Complex Analysis and Riemann Surfaces
Complex Functions
- Holomorphic and Meromorphic Functions: Central to complex geometry.
- Cauchy-Riemann Equations: Characterize complex differentiability.
- Cauchy's Integral Theorem and Formula: Cornerstones of complex integration.
Residue Calculus and Applications
- Residue Theorem: Efficient method for evaluating integrals.
- Laurent Series and Poles: Tools to understand function behavior near singularities.
Riemann Surfaces
- Complex Manifolds of Dimension One: Model sheets on which multivalued functions become single-valued.
- Branch Cuts and Covering Maps: Resolve multi-valuedness in complex functions.
Part VII: Algebraic Geometry and Number Theory
Varieties and Schemes
- Affine and Projective Varieties: Defined by polynomial equations.
- Schemes: Generalization using spectrum of rings, bridging geometry and algebra.
- Sheaves and Sheaf Cohomology: Local-to-global tools to track data.
Number Fields and Arithmetic Geometry
- Number Fields and Ring of Integers: Extensions of \mathbb{Q} with rich arithmetic.
- Dedekind Domains and Factorization of Ideals: Structure in rings beyond \mathbb{Z}.
- Galois Theory: Deep interplay between field extensions and symmetry groups.
- Frobenius Element and Artin Symbol: Key players in unramified extensions.
Part VIII: Category and Homotopy Theory
Category Theory
- Categories, Functors, and Natural Transformations: Framework for comparing structures.
- Limits, Colimits, Adjunctions: Abstract constructions with concrete implications.
Homotopy and Higher Structures
- Homotopy Theory: Studies spaces up to continuous deformation.
- Higher Categories and n-Categories: Extend the categorical framework to stratified morphisms.
Part IX: Foundations and Set Theory
ZFC and Beyond
- Axiomatic Set Theory (ZFC): Foundation for most of mathematics.
- Ordinal and Cardinal Numbers: Measure order type and size.
- Large Cardinals and Consistency Strengths: Extend foundational horizons.
- Forcing and Model Theory: Build new set-theoretic universes.