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Pure Mathematics
A Rosetta Stone for Mathematics | Quanta Magazine In 1940 André Weil wrote a letter to his sister Simone outlining a grand vision for translating between three distinct mathematical worlds: number fields, function fields, and geometry. This Quanta piece explains how Weil’s analogy has animated decades of subsequent mathematics — culminating eventually in the Langlands program and Fermat’s Last Theorem — making it essential reading for anyone curious about the deep unity underlying seemingly unrelated mathematical structures. https://www.quantamagazine.org/a-rosetta-stone-for-mathematics-20240506/
Structure and Randomness — Celebration of Timothy Gowers’ Mathematics (Newton Institute) A celebration lecture series at the Newton Institute honoring Timothy Gowers, the Fields Medal–winning mathematician whose work on combinatorics and functional analysis straddles the boundary between structure and chaos. Gowers’ contributions to Ramsey theory and additive combinatorics are central here, and the title itself echoes Terry Tao’s influential book on the same themes.
The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved | Scientific American Number theorists have been trying to prove the Riemann Hypothesis — a conjecture about the distribution of prime numbers — for more than 160 years. This Scientific American piece covers a recent partial progress that has energized the field, explaining both the statement of the hypothesis and why a proof would have profound consequences for cryptography, number theory, and our understanding of randomness in primes. https://www.scientificamerican.com/article/the-riemann-hypothesis-the-biggest-problem-in-mathematics-is-a-step-closer/
Monumental Proof Settles Geometric Langlands Conjecture | Quanta Magazine In work 30 years in the making, a team of mathematicians proved a major piece of the geometric Langlands program — one of the deepest frameworks in modern mathematics connecting number theory, algebraic geometry, and representation theory. The Quanta article makes the achievement accessible, tracing the lineage from Weil’s vision through Frenkel’s popularizations to the final proof. https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/
Personal note on Langlands:
An interesting aside. The analogy to spectral analysis and waves may make it interesting to some of you. It’s a pretty comprehensive overview of the Langlands program — especially the video that summarizes the links between disparate fields that resulted in the proof of Fermat’s Last Theorem. It still leaves much unsaid (in layman terms) about certain areas — and makes you want more description on things like sheaves.
How the Square Root of 2 Became a Number | Quanta Magazine A beautifully paced history of how irrational numbers — specifically √2 — forced mathematicians to expand their conception of what a “number” actually is. The article traces the journey from the Pythagorean crisis (incommensurable lengths) through Dedekind cuts to the modern real number line, showing how useful concepts can linger for millennia before being rigorously defined. https://www.quantamagazine.org/how-the-square-root-of-2-became-a-number-20240621/
Ulam Spiral — Wikipedia The Ulam spiral is a visual arrangement of integers in a rectangular grid that reveals unexpected diagonal patterns when prime numbers are highlighted. Discovered by Stanisław Ulam while doodling during a boring conference, it inspired decades of research into prime distribution and remains one of the most visually striking results in elementary number theory. https://en.wikipedia.org/wiki/Ulam_spiral
Birthday Problem — Wikipedia The birthday problem asks: how many people must be in a room before there is a better-than-even chance that two share a birthday? The surprising answer — just 23 — is a classic result in probability theory illustrating how human intuition dramatically underestimates coincidences in combinatorial settings. It serves as a gateway to collision probability in cryptographic hash functions and data structures. https://en.wikipedia.org/wiki/Birthday_problem
Grad Students Find Inevitable Patterns in Big Sets of Numbers | Quanta Magazine Two graduate students made the first progress in decades on a Ramsey-theory problem about how arithmetic structure inevitably emerges from large sets of integers. Their proof advances our understanding of additive combinatorics — the study of when patterns are unavoidable — and was celebrated as a genuine breakthrough by the combinatorics community. https://www.quantamagazine.org/grad-students-find-inevitable-patterns-in-big-sets-of-numbers-20240805/
Elliptic Curve ‘Murmurations’ Found With AI Take Flight | Quanta Magazine Mathematicians discovered unexpected statistical patterns in families of elliptic curves — dubbed “murmurations” for their resemblance to starling flocks — using machine learning tools to spot regularities that human intuition missed. This piece documents the ongoing effort to explain these AI-discovered patterns rigorously, sitting at the intersection of number theory, representation theory, and AI-assisted mathematics. https://www.quantamagazine.org/elliptic-curve-murmurations-found-with-ai-take-flight-20240305/
How Base 3 Computing Beats Binary | Quanta Magazine Ternary (base-3) computing has been explored since the earliest days of computing but has rarely been built in hardware. This piece explains how balanced ternary — with digits −1, 0, +1 — offers natural advantages for certain cryptographic and signal-processing problems, and why renewed interest in non-binary architectures may finally find practical applications in post-quantum security. https://www.quantamagazine.org/how-base-3-computing-beats-binary-20240809/
Complexity Theory’s 50-Year Journey to the Limits of Knowledge | Quanta Magazine Meta-complexity theory asks not just whether problems are hard, but how hard it is to prove that problems are hard — a recursive self-referential frontier that has stumped researchers for half a century. This Quanta retrospective covers the intellectual history from Cook-Levin (1971) through the oracle barrier results to recent progress on circuit lower bounds, revealing why P vs NP remains so stubbornly resistant. https://www.quantamagazine.org/complexity-theorys-50-year-journey-to-the-limits-of-knowledge-20230817/
A Life Inspired by an Unexpected Genius | Quanta Magazine Ken Ono, a number theorist whose work deeply extends Ramanujan’s legacy, recounts how discovering Ramanujan’s story saved him during a painful struggle with parental pressure and perfectionism. The article weaves personal biography with mathematical content — Ramanujan’s mock theta functions, partition theory, and the collaboration between a self-taught genius and Hardy — making it as emotionally resonant as it is mathematically rich. https://www.quantamagazine.org/the-mathematician-ken-onos-life-inspired-by-ramanujan-20160519/
My search for Ramanujam: How I learned to count — Ken Ono and Amir D. Aczel (book reference)
Brahmagupta — Biography (MacTutor History of Mathematics) Brahmagupta (598–668 CE) was the foremost Indian mathematician of the early medieval period, making seminal advances in astronomy and number systems: algorithms for square roots, the solution of quadratic equations, and crucially the formal treatment of zero and negative numbers as algebraic quantities. His Brahmasphutasiddhanta influenced Islamic mathematics and — through translation — the development of algebra in Europe. https://mathshistory.st-andrews.ac.uk/Biographies/Brahmagupta/
The Theorist Who Sees Math in Art, Music and Writing | Quanta Magazine Sarah Hart, Gresham Professor of Geometry, turns a mathematical eye to literature — analyzing meter, structure, and combinatorics in poetry, fiction, and music composition. The interview illuminates how mathematical thinking reveals hidden architecture in artistic works, and why the perceived divide between the humanities and mathematics is more cultural than intellectual. https://www.quantamagazine.org/the-theorist-who-sees-math-in-art-music-and-writing-20240112/
A Pilot Project in Universal Algebra to Explore New Ways to Collaborate and Use Machine Assistance (Terry Tao blog, Sep 2024) Terry Tao describes an experiment in using AI tools — specifically large language models and formal proof assistants — to collaboratively prove results in universal algebra. The post is notable for Tao’s careful assessment of where AI assistance genuinely helps (search through large solution spaces) versus where it still falls short (conceptual insight), making it required reading for anyone thinking about the future of AI in mathematics. https://terrytao.wordpress.com/2024/09/25/a-pilot-project-in-universal-algebra-to-explore-new-ways-to-collaborate-and-use-machine-assistance/
Gabriel Peyré (@gabrielpeyre): Divergence, Gradient, and the Laplacian A compact tweet-thread by the applied mathematician Gabriel Peyré explaining that the divergence operator (both continuous and discrete) is the negative adjoint of the gradient — a manifestation of integration by parts. Together with the Laplacian, these operators are the workhorses of PDE-based image processing, optimal transport, and variational methods in signal processing. https://x.com/gabrielpeyre/status/1816699972789628992
Frank Nielsen (@FrnkNlsn): Introduction to Topological Data Analysis (book recommendation) Frank Nielsen recommends an introductory text on topological data analysis (TDA) — the field that uses homology and persistent diagrams to extract shape-level features from high-dimensional data. TDA has found applications in materials science, neuroscience (brain connectivity), and more recently in understanding the geometry of neural network loss landscapes. https://x.com/FrnkNlsn/status/1814972379187093563
Math Cafe (@Riazi_Cafe_en): “The Art of Mathematics” by Béla Bollobás A tweet highlighting Bollobás’s collection of elegant mathematical problems — a book that sits in the tradition of Yaglom and Pólya in demonstrating that the deepest mathematics often emerges from deceptively simple questions. Bollobás, a student of Paul Erdős and himself a titan of combinatorics, selects problems that reward creative thinking over mechanical computation. https://x.com/Riazi_Cafe_en/status/1819718342699286829
Mathematical Books
Challenging Mathematical Problems with Elementary Solutions Vol 1 — Yaglom & Yaglom A Soviet classic (1954) presenting non-elementary problems that can nonetheless be solved with only elementary techniques — a philosophy that trains mathematical ingenuity over symbol manipulation. Originally published by the Soviet Printing House for Technical-Theoretical Literature, it remains one of the best collections for developing combinatorial and geometric intuition. PDF: https://ia802800.us.archive.org/13/items/ChallengingMathematicalProblemsWithElementarySolutionsVol1DoverYaglomYaglom/
Applied Problems In Probability Theory — E. Wentzel Wentzel’s problem book takes an engineering-oriented approach to probability, grounding abstract theory in queuing, reliability, and statistical inference problems drawn from Soviet industrial applications. It is particularly strong on conditional probability, Markov chains, and the Poisson process — topics that remain central to data engineering and systems modeling. https://archive.org/details/wentzel-ovcharov-applied-problems-in-probability-theory
Exercises in Probability — Grimmett & Stirzaker (Oxford) The companion problem book to Grimmett and Stirzaker’s Probability and Random Processes — the standard graduate text in probability theory at Oxford. The exercises are demanding, covering branching processes, martingales, ergodic theory, and large deviations, and working through them is among the most rigorous ways to build probabilistic intuition for stochastic modeling. https://people.sabanciuniv.edu/atilgan/FE507_Fall2014/ProblemsProblems/Grimmett-Stirzaker_Oxford01_ExercisesInProbability.pdf
Probability Theory: The Logic of Science — Jaynes E.T. Jaynes’s magnum opus presents Bayesian probability as an extension of Aristotelian logic — not merely a computational tool but the correct normative framework for reasoning under uncertainty. It is polemical, rigorous, and deeply influential, providing the theoretical foundation for Bayesian machine learning, scientific inference, and maximum-entropy methods in statistical mechanics. http://www.med.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/JaynesProbabilityTheory.pdf
Winning Ways for Your Mathematical Plays — Conway Berlekamp, Conway, and Guy’s four-volume masterwork on combinatorial game theory — the rigorous mathematical analysis of two-player perfect-information games. Conway’s surreal numbers emerge naturally from the theory, and the book established the field of combinatorial game theory as a mature branch of mathematics, with applications from nim to go endgame analysis.
Conceptual Mathematics: A First Introduction to Categories — Lawvere & Schanuel Lawvere and Schanuel’s gentle introduction to category theory, intended for mathematicians at any level who want to understand the abstract language unifying algebra, topology, logic, and computer science. Written by Lawvere — one of the founders of categorical logic and topos theory — it is the most accessible path into a framework that has deeply influenced type theory and functional programming. https://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X
Category Theory Without Categories — John D. Cook A short blog post by John D. Cook exploring category-theoretic thinking — functors, natural transformations, universal properties — without requiring the reader to have studied formal category theory. Particularly valuable for software engineers who encounter categorical patterns in Haskell, Scala, or type theory and want conceptual grounding without the full formalism. https://www.johndcook.com/blog/2023/05/22/category-theory-without-categories/
Algorithms — Jeff Erickson A freely available comprehensive textbook on algorithms by Jeff Erickson of UIUC, covering graphs, dynamic programming, network flows, NP-hardness, and randomized algorithms with unusually clear exposition. The book is used in advanced undergraduate and graduate algorithms courses and is notable for its rigorous yet readable treatment of proof techniques alongside algorithmic design. https://jeffe.cs.illinois.edu/teaching/algorithms/