Core · Algebra
Finish abstract linear algebra
Chapters 6–9: inner product spaces, spectral theorem, operators, determinants.
Think of the spectral theorem as the point where algebra and geometry become a single language.
A lifelong mathematical education
My mathematics journey.
A self-directed roadmap from rigorous undergraduate mathematics to the foundations of graduate research, designed around clarity, insight, and the beauty of ideas.
This is a garden, not a race. The order of the core subjects matters; the pace is my own.
Where I begin
My starting point is already proof-based: real analysis through Abbott, abstract linear algebra through Axler, and multivariable analysis through Shurman.
Completed
Understanding Analysis
2nd edition
In progress
Linear Algebra Done Right
4th edition
In progress
Calculus and Analysis in Euclidean Space
Starting snapshot: Axler chapters 1–5 completed; approximately one third of Shurman completed. The checkboxes below mark future reading goals and are saved only in this browser; they do not update public GitHub content.
0 reading goals marked completed
Optional personal reading tracker · saved locally on this device
Stage 0 · Complete the foundations
Roughly 6–12 months
Core · Analysis
Finish multivariable analysis
Differentiation, inverse/implicit function theorems, integration, differential forms.
The inverse function theorem and generalized Stokes’ theorem will pay dividends throughout geometry and analysis.
Stage 1 · The undergraduate core
Roughly 1–2 yearsCore · Structure and symmetry
Abstract algebra
Groups, actions, rings, fields, quotients, Galois theory.
A Book of Abstract Algebra
Alternative gentle start.
Use Artin for its geometric viewpoint.
Core · Spaces and continuity
Point-set topology
Compactness, connectedness, continuity, product spaces, covering spaces.
Topology
Alternative: a comprehensive, systematic treatment.
Core · Analysis and geometry
Complex analysis
Geometric intuition for holomorphic functions, integration and conformal maps.
Rigorous companion for major theorems and exercises.
The remarkable rigidity of complex differentiability makes this a particularly rewarding subject.
Core · Change over time
Differential equations
Existence and uniqueness, phase portraits, stability and nonlinear dynamics.
Think of an ODE as defining a flow, not just something that must be solved.
Companion · Arithmetic
Number theory
Primes, congruences and Diophantine equations.
Excellent alongside abstract algebra and often full of pleasant surprises.
Companion · Randomness
Probability
Counting, conditional probability, random variables and distributions.
Harvard Stat 110 has lectures and extensive self-study resources.
Stage 2 · The graduate bridge
Roughly 1–2 yearsAnalysis
Measure and integration
Measures, Lebesgue integration, convergence, product measures, Lᵖ spaces.
Return to Axler’s lucid style, this time with graduate-level analysis.
Geometry
Manifolds and forms
Curvature, surfaces and forms, explained geometrically.
Smooth structures, tangent spaces, vector fields and Stokes’ theorem.
Analysis
Functional analysis
Banach and Hilbert spaces, duality, operator theory and spectral methods.
Finite-dimensional linear algebra grows into infinite-dimensional geometry.
Topology
Algebraic topology
Fundamental groups, covering spaces, homology and cohomology.
Use algebraic invariants to distinguish spaces.
Analysis and mathematical physics
Fourier analysis and PDEs
Fourier series, transforms and oscillatory methods.
Heat, wave and Laplace equations, introductory PDE methods.
Partial Differential Equations
Later reference for deeper PDE theory.
Stage 3 · The graduate core
Roughly 1.5–3 yearsAt this level, textbooks overlap and the subject boundaries start to dissolve. Use one main book per area; keep the denser texts for problem sets and reference.
Abstract algebra
Graduate algebra
Abstract Algebra
A deep reference.
Introduction to Commutative Algebra
After modules and ideals.
Groups, fields, modules, representations and commutative algebra.
Real analysis
Graduate analysis
Real Analysis
Following Axler.
Continue into advanced integration, Hilbert spaces and functional analysis.
Complex analysis
Advanced function theory
Complex Analysis
Build fluency in analytic continuation, normal families, conformal mapping and residues.
Differential geometry
Modern manifolds
Introduction to Smooth Manifolds
Read first.
Introduction to Riemannian Manifolds
Continue here after smooth manifolds.
Topology
Advanced algebraic topology
Algebraic Topology
Continue through homology, cohomology, duality and CW complexes.
Algebraic geometry
Polynomial geometry
Ideals, Varieties, and Algorithms
Read first.
Algebraic Geometry: A First Course
Continue here.
The mathematics garden
These side paths are not distractions. They keep the larger education playful, broad and full of connections. Begin them when the necessary foundations are in place.
| Subject | Beautiful introduction | When |
|---|---|---|
|
Combinatorics |
generatingfunctionology |
Anytime |
|
Set theory and logic |
Naive Set Theory 
A Mathematical Introduction to Logic |
Anytime |
|
Numerical analysis |
Numerical Linear Algebra |
After Axler |
|
Graph theory |
Graph Theory |
After proof foundations |
|
Algebraic number theory |
Number Fields |
After Galois theory |
|
Optimization |
Convex Optimization |
After multivariable calculus |
|
Category theory |
Basic Category Theory |
After algebra and topology |
|
Mathematical physics |
Mathematical Methods of Classical Mechanics |
After ODEs and geometry |
Stage 4 · Qualifying-exam readiness
Final 6–12 months, overlapping Stage 3There is no single universal American mathematics PhD entrance exam. Preliminary and qualifying exam syllabi differ by university. Build depth first; use old examinations later to diagnose gaps and practice constructing proofs under constraints.
First benchmark
Undergraduate preliminary exams
Use Berkeley’s mathematics preliminary exam resources to test the analysis and algebra foundations.
Advanced benchmark
Graduate qualifying exams
Use Stanford’s qualifying exam information and Harvard’s graduate qualifying exams as broad targets. Check each department’s current syllabus and links before planning exam-style drills.
Sequence: Try selected problems untimed, identify missing theory, return to textbook exercises, and only then attempt full timed papers. The aim is understanding—not just speed.
A sustainable study rhythm
At a hobbyist pace of approximately 8–10 hours per week, this roadmap is a multi-year project. A reasonable planning horizon is five to eight years or longer; the exact pace is less important than steady, joyful work.
| Activity | Weekly time | Why |
|---|---|---|
|
Read and reconstruct proofs |
3 hours |
Develop insight and mathematical taste. |
|
Solve exercises |
4 hours |
Turn passive understanding into real skill. |
|
Review earlier subjects |
1 hour |
Strengthen long-term recall and connections. |
|
Explore a side subject |
1 hour |
Keep the journey broad and enjoyable. |
For each important theorem, ask what its assumptions mean, find an example and a counterexample, and try to recover the proof several days later.
The most rewarding mathematical moments are often the ones where two ideas, once thought unrelated, suddenly turn out to be the same idea in different languages.