The Mathematician 28-10-2025
Unraveling AI & formal math, historic mathematical notation, intriguing puzzles, visual demos, research practices, and deep math dives!
🤖 AI & Formal Mathematics
Formal or not formal? That is the question in AI for theorem proving. (xenaproject​.wordpress​.com). Contrasts informal vs formal AI theorem proving, Lean, mathlib, IMO experiments, and concerns about LLMs in formalized mathematics
Can AI solve quantum-many body problems? (condensedconcepts​.blogspot​.com). CMT-Benchmark: a 50-problem dataset evaluating AI as a research assistant for condensed matter theory methods like HF, ED, QMC, DMRG, and Monte Carlo
Stability of natural latents in information theoretic terms (lesswrong​.com). Information-theoretic proof of mediator-derived redundancy in latents, via mediation and redundancy bounds and mutual information
ICan'tBelieveICanProveItCanSort! (unnamed​.website). Lean 4 proof of a sorting algorithm using mvcgen and grind, with permutation and sortedness proofs and test comparisons
đź“ś History & Biographies
Chapter 6: The Problem Child (analog-antiquarian​.net). Kepler's early life, education, and move from Graz to Bohemia as he engages with heliocentrism and calendar reform debates
The Evolution of Mathematical Notation: Part 1 (appetrosyan​.github​.io). Tracing the evolution from tally marks and Roman numerals to decimal positional systems, algebraic notation, and their impact on teaching and computing
C. N. Yang, Dead at 103 (4gravitons​.com). Chen Ning Yang and Yang–Mills theory: symmetry, confinement, and impact on fundamental forces
The Evolution of Mathematical Notation: Part 2 (appetrosyan​.github​.io). Comparing mathematical notation to programming, exploring closed-form formulas, primes, Fibonacci, Gauss, Wilson's theorem, and implementation implications
The Origin Story of Eurisko, the Most Advanced Math/CS Track in the USA (justinmath​.com). Eu risko: high school math/CS track implementing neural networks, backpropagation, game trees, and evolutionary algorithms from scratch
Perwich Letter Deciphered. (languagehat​.com). Decrypting a 17th-century William Perwich cipher: columnar transposition, nulls, and historical-context codes
Examining the First Mechanical Calculator (hackaday​.com). Explores Blaise Pascal's Pascaline and a modern transparent analogue, carry mechanisms, surveying units, and historical context
🧩 Puzzles & Recreational Math
Why Busy Beaver Hunters Fear the Antihydra (benbrubaker​.com). Explains Antihydra, Collatz conjecture, BB(5) and BB(6) connection, and why Antihydra halting is hard to prove
The hyperbolic spiral as a trisectrix (11011110​.github​.io). Hyperbolic spiral as a trisectrix: using a projected spiral, secant-like construction, and equal-part subdivision on a simplified setup
Card Dealing Math (blog​.tanyakhovanova​.com). Explore under-down dealing, Josephus problem analogies, and new sequences added to OEIS through single-suit and multi-deck card patterns
First Shape Found That Can’t Pass Through Itself (quantamagazine​.org). Rupert’s puzzle: which convex polyhedra can or cannot pass a tunnel through themselves; the Noperthedron disproves universal Rupert property
987654321 / 123456789 (johndcook​.com). Explores the ratio 987654321/123456789 across bases, showing it nears b-2 with floating point limits noted
🧰 Technical Visualizations & Demos
ASP level generation examples from our PCG textbook (kmjn​.org). ASP level-generation demos for mazes and dungeon-crawler layouts using clingo (ASP 3–5) with width params, connectivity, style biases, and playability simulations
how to draw a tetrapod (dotat​.at). Explains a three.js tetrapod animation and a CAD-based construction method using a cube, tetrahedron, truncated cones, and taper geometry
Draw high dimensional tensors as a matrix of matrices (blog​.ezyang​.com). Show how to visualize high-dimensional tensors as a matrix of matrices across 0D–5D, using PyTorch's view and split to reveal axes
From fields to surface-specific points, geomorphons, and networks (spatialists​.ch). Transformations from fields to surface-specific points, geomorphons, and surface networks using R for DEM analysis
Spacing the circles on the Smith chart (johndcook​.com). Spacing Smith chart circles via inverse mapping from w to z using f(z)=(z-1)/(z+1) and g(w)=(1+w)/(1-w)
🎓 Research Practice & Intuition
Aarhus Summer School in Analysis and Probability (fabricebaudoin​.blog). Aarhus Summer School in Analysis and Probability: five-hour lectures, funding for early-career researchers, and courses by Murugan, Rizzi, Shanmugalingam, and Tindel
Bill's Bad Advice (blog​.computationalcomplexity​.org). Bill Gasarch advocates ignoring busywork constraints to pursue research joy, with examples from Muffins and Ramsey Projects
Intuition Through Repetition (justinmath​.com). Intuition through concrete numerical examples in ML concepts, teaching via running numbers before abstraction
Empirical Partial Derivatives (brianschrader​.com). Explains partial derivatives via randomized trials as empirical derivatives and links calculus to real-world experimentation
Note (hsu​.cy). Thought without words vs. compression: wordless high-dimensional thought, rigor of notation, and expert intuition in mathematics
đź” Math & Physics Deep Dives
Science Notes / Ellipsoid Gravity (gregegan​.net). Ellipsoid interior gravity potential, solid ellipsoids, shells, and related integrals; U0, Ua, Ub, Uc, and elliptic integrals in 3D gravity models
The Riemann Zeta Function (kuniga​.me). Overview of the Riemann zeta function: Dirichlet series, Euler product, Mellin transform, functional equation, and zeros (trivial and nontrivial)
Some Math/Physics Items (math​.columbia​.edu). Costello's approach via self-dual theory, twistor space, Grothendieck-Riemann-Roch; Scholze's Langlands work; Gestalten; beta function interpretations
The Standard Model (Part 1) (johncarlosbaez​.wordpress​.com). Overview of the Standard Model introduction, particle cast, interactions, and upcoming corrections in Baez–Huerta notes
Positive Geometry Could Save Physics — or Destroy It (backreaction​.blogspot​.com). Positive geometry methods examined for physics, focusing on amplituhedrons, Grassmannians, and potential impacts on quantum gravity and particle physics
đź“š Academic Research
A proof of the Kim-Vu sandwich conjecture (arxiv:math). Proves the Kim–Vu sandwich conjecture, constructing a high-probability coupling between random d-regular graphs and Erdős–Rényi graphs for d = ω(log n). This resolves a central open problem in random-graph theory, enabling precise comparisons between d-regular and Erdős–Rényi models and advancing probabilistic combinatorics and practical robustness analyses
Grauert's Approximation Theorem in any Characteristic and Applications (arxiv:math). Extends Grauert’s division and approximation theorems from complex-analytic convergent power series to arbitrary real-valued fields and any characteristic. Provides semi-universal deformations and splitting lemmas for singularities in broad arithmetic settings, profoundly expanding deformation theory and singularity analysis beyond classical complex-analytic frameworks and applications
String graphs are quasi-isometric to planar graphs (arxiv:math). Shows every countable string graph admits a planar graph with comparable distances, so string graphs are quasi-isometric to planar graphs. Yields metric-dimension bounds, accessibility results, and algorithmic constructions, impacting geometric group theory, graph metrics, and practical conversions between string and planar graph representations with algorithms
Irreducibility and Galois groups of random reciprocal polynomials of large degree (arxiv:math). Establishes high-probability irreducibility and (near-)full hyperoctahedral Galois groups for broad families of random reciprocal polynomials via Fourier-analytic and number-theoretic methods. Combines p-adic Fourier analysis, character theory, and discriminant/cohomological methods to settle probabilistic Galois behavior, informing random polynomial theory and arithmetic statistics and computational number theory
Module structure of Weyl algebras (arxiv:math). Survey and synthesis tracing Stafford’s foundational results on Weyl-algebra modules, geometric constructions and later developments, highlighting open problems in quantized symplectic singularities. Clarifies origins, geometric constructions, and modern parametrization results, presenting open questions that reconnect algebraic, geometric, and representation-theoretic threads in D-module theory with significant research directions