Study Guide
A Self-Study Guide for Advanced Mathematics
A comprehensive guide to teaching yourself advanced mathematics, with recommended learning paths, book selections, study strategies, and advice for self-directed learners.
Writing
Blog
94 posts · Page 2 of 7
Mathematics
Andrey Kolmogorov: The Father of Modern Probability
The extraordinary life and vast mathematical legacy of Andrey Kolmogorov, who laid the axiomatic foundations of probability theory and made transformative contributions to topology, turbulence, algorithmic complexity, and a dozen other fields.
Resources
The Best Books for Each Major Mathematics Subject
A carefully curated guide to the best textbooks for every major area of university mathematics, from calculus and linear algebra through graduate-level subjects, with honest assessments and recommendations for different learning styles.
Mathematics
The Dominated Convergence Theorem: When Can We Swap Limits and Integrals?
We prove the Dominated Convergence Theorem — the central result of Lebesgue integration theory that gives sharp conditions for interchanging limits and integrals — and compare it to weaker results like the Monotone Convergence Theorem.
Study Guide
How to Choose Your Mathematics Specialization
A guide to choosing your area of specialization in mathematics, with an overview of major fields, what each one is about, and practical advice for making this important decision.
Career
Why Writing Well Matters in Mathematics
An argument for why clear mathematical writing is not a luxury but a necessity — with practical advice on how to improve your mathematical prose.
Mathematics
Augustin-Louis Cauchy: The Man Who Made Analysis Rigorous
The story of Augustin-Louis Cauchy, the prolific French mathematician who single-handedly transformed calculus from a collection of powerful but loosely justified techniques into the rigorous discipline of mathematical analysis.
Resources
GeoGebra and Desmos: Visualizing Mathematics
A hands-on guide to GeoGebra and Desmos, two powerful free tools for visualizing mathematical concepts, from function graphing and geometric constructions to 3D surfaces and interactive demonstrations.
Mathematics
Evaluating a Stunning Double Integral: Exponential, Logarithm, and Inverse Hyperbolic Cosine
We evaluate a double integral involving an exponential quadratic form and a logarithmic kernel in closed form, arriving at a result featuring inverse hyperbolic cosine. The proof uses polar reduction, partial fractions, integration by parts, and a Möbius substitution.
Study Guide
The Power of Working Through Exercises: Why Passive Reading Fails
Why solving exercises is the most important part of learning mathematics, how to approach exercises effectively, and how to use them to build deep understanding.
Mathematics
The Stone-Weierstrass Theorem: Approximating Continuous Functions
We state and prove the Stone-Weierstrass theorem — the powerful generalization of Weierstrass's classical approximation theorem that characterizes when subalgebras of continuous functions are dense — and explore its wide-ranging applications.
Mathematics
Grigori Perelman: The Genius Who Refused a Million Dollars
The remarkable story of Grigori Perelman, the reclusive Russian mathematician who proved the Poincaré conjecture — one of the great unsolved problems in mathematics — and then refused the Fields Medal and the million-dollar Millennium Prize.
Career
How Mathematical Research Actually Works
A behind-the-scenes look at the process of mathematical research — how problems are chosen, how proofs are discovered, how papers get published, and what the daily reality looks like.
Mathematics
The Uniform Boundedness Principle: A Cornerstone of Functional Analysis
We prove the Uniform Boundedness Principle (Banach-Steinhaus theorem) — that a pointwise bounded family of bounded linear operators on a Banach space is uniformly bounded — and explore its far-reaching consequences in functional analysis.
Resources
Essential Tools for Writing Mathematics Papers
A practical guide to the essential tools for writing mathematics papers: LaTeX editors, reference managers, diagram tools, collaboration platforms, and writing workflows used by working mathematicians.