From Zero to
Real Mathematics

• Algebraic operations and factoring
• Functions and their graphs
• Exponential, logarithmic, and trigonometric functions
• Equations and inequalities
• Basic analytic geometry

Minimum to Move On:the topics above, using Iezzi as a guide for specific gaps

• Propositional logic and quantifiers
• Implication, negation, contraposition
• Proof by induction
• Proof by contradiction
• Sets, functions, relations

Minimum to Move On:the topics above mastered: you can write complete proofs

• Divisibility, the Euclidean algorithm, and greatest common divisor
• Prime numbers and unique factorization
• Congruences and modular arithmetic
• Chinese Remainder Theorem in simple examples
• Fermat's little theorem and Euler's theorem
• Elementary arithmetic functions at an introductory level
• First simple Diophantine problems

Minimum to Move On:mastery of divisibility, congruences, the Euclidean algorithm, unique factorization, and the first arithmetic theorems, solving simple exercises without relying on ready-made solutions

• Definition of sequence convergence
• Definition of limit and continuity
• Derivatives and differentiation rules
• Mean Value Theorem and applications
• Riemann integral and Fundamental Theorem of Calculus
• Sequences and series: convergence and convergence tests

Minimum to Move On:the topics above, using Caminha as the main text

• Linear systems and row reduction
• Vector spaces, subspaces, basis, and dimension
• Linear transformations and matrices
• Determinants
• Eigenvalues, eigenvectors, and diagonalization
• Inner product and orthogonality

Minimum to Move On:the topics above, using Plácido Andrade as the main text

• Partial derivatives, gradient, and Jacobian
• Chain rule in several variables
• Local extrema and Lagrange multipliers
• Multiple integrals and change of variables
• Theorems of Green, Stokes, and Gauss
• Vector fields and conservative fields

Minimum to Move On:the topics above, with conceptual mastery of total differentiability vs. partial derivatives, interpretation of the Jacobian as a linear map, and understanding of the hypotheses of Green's, Gauss's, and Stokes's theorems

• Construction of the real numbers and completeness
• Topology of the line: open sets, closed sets, compact sets
• ε-δ definition of limit and continuity
• Intermediate Value Theorem and Heine-Cantor theorem
• Differentiability and Mean Value Theorem
• Riemann integral: definition and properties
• Pointwise and uniform convergence of sequences of functions

Minimum to Move On:the topics above with genuine rigor: computing is not enough — you must be able to prove

• σ-algebras and measure spaces
• Lebesgue measure on ℝⁿ
• Lebesgue integral and its properties
• Convergence theorems: Monotone, Dominated, Fatou
• Lᵖ spaces and Hölder's and Minkowski's inequalities
• Product measures and the Fubini-Tonelli theorem

Minimum to Move On:the topics above, using Bartle or Cohn as the main text

• Probability spaces and events
• Random variables as measurable functions
• Distributions and laws of random variables
• Expectation as an integral
• Independence
• Modes of convergence of random variables
• Borel-Cantelli lemma
• Law of Large Numbers
• Central Limit Theorem in basic formulation

Minimum to Move On:understanding random variables as measurable functions, expectation as an integral, independence, the main modes of convergence, and the central ideas behind the Law of Large Numbers and the Central Limit Theorem

• Normed and Banach spaces
• Hilbert spaces: orthogonality, Schauder bases
• Bounded linear operators and the dual space
• Hahn-Banach theorem and its consequences
• Open mapping theorem and closed graph theorem
• Uniform boundedness principle
• Spectrum of linear operators

Minimum to Move On:understanding normed, Banach, and Hilbert spaces; continuous linear operators; basic duality; Hahn-Banach, Banach-Steinhaus, open mapping, and closed graph at an operational level; and fundamental examples such as ℓᵖ, Lᵖ, C(K), and separable Hilbert spaces

• C*-algebras: definition, examples, and morphisms
• Gelfand-Naimark theorem
• Spectral theorem for normal operators
• Compact operators and spectral decomposition
• Fredholm index and Fredholm operators

Minimum to Move On:the topics above via Murphy. Simon and Abramovich & Aliprantis are enrichment and reference

• Normal convergence and compactness in spaces of analytic functions
• Montel's theorem and normal families
• Riemann mapping theorem
• Weierstrass product for entire functions
• Meromorphic functions and Mittag-Leffler theorem
• Introduction to Riemann surfaces

Minimum to Move On:the topics above via Conway.

• Formalism of states, observables, and time evolution in Hilbert spaces
• Free Schrödinger equation and first one-dimensional examples
• Symmetric, self-adjoint, and essentially self-adjoint operators in fundamental examples
• Distinction between bounded and unbounded operators
• Spectral theorem for self-adjoint operators at a conceptual and operational level
• Simple Hamiltonians: free particle, potential well, and harmonic oscillator
• Uncertainty principle and canonical commutation relations
• Idea of Stone's theorem and unitary evolution
• Role of symmetries, Lie groups, and representations in basic examples

Minimum to Move On:understanding the Hilbert space formalism, the Schrödinger equation in simple examples, unbounded self-adjoint operators, spectrum, unitary evolution, and the role of canonical commutation relations — without needing to master WKB, path integrals, or geometric quantization

• Existence and uniqueness theorems in a more mature formulation
• Continuous dependence on initial conditions
• Flows associated with vector fields
• Autonomous systems
• Equilibrium points
• Stability and asymptotic stability
• Linearization
• Phase portraits
• Orbits and invariant sets
• Early bifurcations

Minimum to Move On:understanding flows of ODEs, stability of equilibria, linearization, phase portraits, and the idea of studying the qualitative behavior of solutions without relying on explicit formulas

• Classical and weak formulations of PDEs
• Distributions and weak derivatives
• Fourier transform in PDE
• Lp spaces as the foundation for function spaces
• First contact with Sobolev spaces and energy norms
• Energy methods
• Boundary-value problems
• Duality and solvability at an introductory level
• Variational formulations
• Basic elliptic problems
• Basic parabolic problems
• Basic hyperbolic problems
• Initial notions of regularity

Minimum to Move On:understanding the difference between classical and weak solutions, knowing the role of distributions and weak derivatives, recognizing Sobolev spaces at an introductory level, understanding energy methods and variational formulations, and distinguishing elliptic, parabolic, and hyperbolic equations

• Categorical language as a way of organizing algebra, without losing contact with concrete examples
• Universal properties in basic constructions: products, coproducts, quotients, free objects, and initial/terminal objects
• Groups revisited structurally: homomorphisms, quotients, actions, products, free groups, Sylow, and decompositions
• Rings and modules as natural extensions of linear algebra, including ideals, quotients, free modules, and presentations
• Factorization in domains: UFDs, PIDs, Euclidean domains, polynomials, and arithmetic examples
• Linear algebra in algebraic language: free modules, determinants, canonical forms, and polynomial ring actions
• Fields and extensions: algebraic extensions, finite fields, classical constructions, and a return to Galois theory
• Functors, equivalences, limits, colimits, and adjunctions at an operational level
• Tensor products, change of base, Hom, duality, and multilinear algebra
• First contact with complexes, homology, Tor, Ext, abelian categories, and derived categories at a panoramic level

Minimum to Move On:understanding how categories and universal properties organize recurring algebraic constructions; being able to move between groups, rings, modules, fields, and linear algebra in structural language; and recognizing tensors, Hom, exact sequences, and homological notions as tools that reappear in algebra, topology, and geometry

• Transition from curves and surfaces to manifolds
• Euclidean submanifolds and geometric examples
• Vector fields and differential forms in a geometric context
• Tangent and cotangent bundles in concrete examples
• Connections and covariant derivative
• Curvature of a connection
• First ideas on characteristic classes
• Geometric intuition for fundamental examples

Minimum to Move On:understanding the transition from classical curve-and-surface geometry to manifolds, interpreting fields, forms, connections, and curvature in fundamental examples, and recognizing the geometric role of connections without needing to master characteristic classes in depth

• Differentiable manifolds: atlas, charts, smooth structure
• Tangent and cotangent bundles, vector fields, and flows
• Submanifolds and the implicit function theorem on manifolds
• Differential forms and the exterior derivative
• Orientation and integration on manifolds
• Stokes' theorem on manifolds

Minimum to Move On:the topics above via Tu

• Riemannian metrics
• Levi-Civita connection
• Covariant derivative
• Geodesics and parallel transport
• Sectional curvature
• Ricci curvature and scalar curvature at an introductory level
• Fundamental examples: Euclidean spaces, sphere, torus, and low-dimensional manifolds
• First ideas of global geometry, where appropriate

Minimum to Move On:understanding Riemannian metrics, the Levi-Civita connection, geodesics, parallel transport, and sectional, Ricci, and scalar curvatures in fundamental examples — without needing to master comparison geometry, global theorems, or advanced geometric analysis

• Manifolds as global geometric spaces
• Presheaves, sheaves, and concrete examples
• Local sections, global sections, and the gluing condition
• Morphisms of sheaves and short exact sequences
• Sheaves of functions, differential forms, constant sheaves, and geometric examples
• The idea of cohomology as a measure of global obstructions
• First contact with Cech cohomology or sheaf cohomology, without requiring complete technical mastery

Minimum to Move On:understanding what a sheaf is, how local sections glue together, why cohomology measures global information not visible locally, and how this language prepares for algebraic geometry and algebraic topology

• Fundamental group π₁ and computation in simple spaces
• Covering space theory and path lifting
• Singular homology: definition and computation
• Long exact sequence of a pair
• Categories, functors, and natural transformations
• Adjunctions and the Yoneda lemma

Minimum to Move On:the topics above; one algebraic topology book and one category theory book are sufficient

• Noetherian and Artinian rings
• Modules: localization and completion
• Primary ideals and primary decomposition
• Krull dimension
• Nakayama's lemma
• Hilbert's Nullstellensatz

Minimum to Move On:the topics above via AM or Reid

• Affine and projective algebraic varieties at an introductory level
• Ideals, coordinate rings, and the algebra-geometry correspondence
• Zariski topology
• Regular morphisms and rational maps in simple examples
• Initial notions of projective varieties
• First contact with sheaves or schemes, only if the chosen text reaches that far

Minimum to Move On:understanding the correspondence between ideals and algebraic sets, coordinate rings, the Zariski topology, and basic examples of affine and projective varieties

• Affine schemes and the spectrum of a ring
• Zariski topology on Spec(A)
• Structure sheaves and locally ringed spaces
• Morphisms of schemes
• Gluing of schemes
• Local and global properties in fundamental examples
• First ideas on sheaf cohomology, if sufficient maturity is present

Minimum to Move On:understanding affine schemes, structure sheaves, morphisms, and gluing of schemes in basic examples; Vakil should be treated as a long-term horizon, not as mandatory linear reading