Schubert positivity has a positive rule under two conjectures
If GRH and a derandomization assumption hold, every positive Schubert coefficient gets a poly-time verifiable certificate.
math.HO
History and Overview
Biographies, philosophy of mathematics, mathematics education, recreational mathematics, communication of mathematics, ethics in mathematics
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Sangaku tablets get a searchable
Bilingual, deeply zoomable records bring Edo-period temple geometry to scholars, teachers, and online visitors.
Mathematicians must build AI
The essay argues math should embrace formalization, symbolic tools, and discovery, not fear neural theorem provers.
Non-Euclidean geometry was a byproduct
He was rebuilding geometry on measurement and bodily contact; the split from physics came decades later.
Euclid's algorithm began as a musical procedure
Hippasus's 6:8:9:12 chord turns octave-and-fifth ratios into the 'reverse subtraction' behind Euclid's Book VII.
Survey maps which bent functions escape the Maiorana-McFarland class
A single subspace condition separates new bent functions from rearranged Maiorana-McFarland ones.
· “Bent Functions and the Completed Maiorana-McFarland Class”
78% of school students write correct cycles after one puzzle lesson
In a 45-minute lesson, most Grades 5–11 students built the cycle representation and used it to judge solvability.
· “Group Theory in School Mathematics? Teaching Permutation Cycles Through the 15-Puzzle”
Math's true goal is digesting proofs
A five-stage pipeline shows why an AI proof boom will make human understanding the scarcest resource.
LLMs recombine math but can't invent new concepts
A new taxonomy says mathematical capability doesn't transfer across modes, so aggregate AI math scores mislead.
Giorgi's 1934 rule made Heaviside's method rigorous
His rule is essentially the Laplace-transform method — and it led Giorgi into fractional calculus.
Wiles's proof is deep because five virtues reinforce each other
The proof's depth comes from the interplay of difficulty, originality, fruitfulness, unity, and explanatory power.
Freshman's dream error is worst possible except at the origin in 2D
Error volume decays like C·ε almost everywhere, and like C·ε log(1/ε) at one exceptional point.
· “Estimation of calculation errors and resolution of singularities”
Two, three, or one: a tire's side count is a modeling choice
The same rim can serve as a shared edge or as a non-separating loop; the convention decides.
A 12th-century magic square hides a tesseract's 384 symmetries
Every valid rearrangement of the 16 numbers matches one of the 384 symmetries of the four-dimensional hypercube.
In 1876, Lucas used a chessboard to certify a 39-digit prime
The same residue test, run by computer, now certifies Mersenne primes with tens of millions of digits.
From chaos targeting to cow synchronization: a memorial to Erik Bollt
A tribute from family and colleagues gathers his contributions to chaos control, data-driven modeling, and mentorship.
AI-generated proofs strip mathematics of human value
For the author, math's worth lies in the human act of writing proofs; machine theorems cannot carry it.
Why (-2)x(-3)=6 isn't just a rule to memorize
A classroom journey from tally marks to place value, zero, and negatives ends at one directional idea.
· “From Counting Objects to Negative Numbers, The Journey to Understanding Why (-)x(-)=+ ?”
Discrete curvature converges to the Gauss-Bonnet-Chern integrand
Index expectation on fine triangulations gives the continuum curvature, so finite geometry is not just an analogy.
Question-guided app lifts math mastery at every Bloom level
In a study of 92 Hungarian secondary students, almost all mastered apply-level boundedness while a matched control reached only 7 percent.
· “Question-Guided Learning for Mathematics Education: An Autonomous Online Inquiry-Based Approach”
Sylvester's identity shrinks m×m matrix spectra to n×n
When a matrix factors as A=TSW, its spectrum and linear dynamics reduce exactly to n×n problems.
Banaszczyk's theorem yields low-discrepancy colorings
One convex-geometric result, made algorithmic, organizes the best bounds on balancing, rounding, and online coloring.
· “Discrepancy Theory: An Algorithmic and Geometric Perspective”
AI makes the human core of math more valuable
As machines automate routine proofs, judgment, problem framing, and insight become the scarce skills.
HOMFLYPT Yang-Baxter operators get explicit homology
Face maps from the third Reidemeister move yield closed formulas for low-dimensional groups
Every homotopy class between spheres becomes polynomial
On complex quadrics, every sphere homotopy class is algebraic; on even-cellular varieties, bundles match exactly.
Five steps to work through any math text on your own
Overviews, definitions, proofs, problems, and a final big-picture pass turn reading into active learning.
One melody, any tempo: canons with infinitely many solutions
Prime-factor or logarithmic maps fix the vertical interval for every tempo ratio, so voices never drift apart.
· “Infinite Canons: Maximally Self-Similar Melodic Lines and Canons with Infinite Solutions”
Polyform fence maxima: 128, 35, and 116 tiles
Exact maxima for pentominoes, tetrahexes, and hexiamonds, with extremal configurations counted.
Galois' unfinished proof is a coset factorization
Left cosets land in conjugate fields; right cosets in the fixed field, bridging substitutions to automorphisms.
· “The Significance of Proposition II in Galois' M\'emoire, The Origin of Galois Automorphisms”
Navier-Stokes mild solutions exist and are unique in critical spaces
Local solutions for every divergence-free initial data in H^{n/2-1}, global for small data, with uniqueness in the critical class.
One integral family proves e and π irrational
Closed-form formulas for e's continued-fraction convergents turn two classic irrationality proofs into one short argument.
Triangular deck size forces Bulgarian solitaire to a 1,2,...,k finish
A single card move repeated forever either hits the staircase configuration or falls into a small, fully described cycle.
· “A short survey the game Bulgarian solitaire and related games”
Two circles, not a double helix, govern the Graz twin stair
A simple two-circle model shows why the flights meet, share treads, and part again—and what reconciliation requires.
· “The Stairs of Reconciliation: A Mathematical Tourist in Graz”
STACK calculus tests need fewer fixes
Symbolic step validation beats fixed Moodle patterns on reliability and regrading load
Games and flipped class keep summer ODEs on track
Courtroom trials, Jeopardy review, and Laplace vaults helped students stay active in a six-week course
· “Fast-Track ODE: A Flipped, Game-Based Summer Differential Equations Course”
Pre/post cycles lift Calculus scores and student readiness
Low-stakes previews and reflections in honors Calc I raised means from 70% to 89% with full participation.
· “Pre/Post-Assessment Cycles in Calculus: Supporting Student Preparation, Reflection, and Learning”
Hankel kept permanence by letting arithmetic laws be revised
Deliberative conservatism, not absolute preservation, explains his anti-Kantian program
AI tools reshape what counts as proof and who counts as a mathematician
Interactive theorem provers and language models are redistributing mathematical labor between humans and machines, changing trust, rigor,and
Over 200 statistics PhDs have joined Greek universities since 1982
A historical account traces the field from the 1828 census through six founding professors and the post-1982 expansion.
Treat math training as strategic infrastructure
As AI systems begin producing research-level mathematics, the US is defunding the human pipeline needed to verify, interpret, and govern it.
Pirahã lack Frege's #, not one-to-one matching
Stable token practice realizes the operator that turns matching into arithmetic.
· “The Pirah\~a and the cognitive gap in Frege's theorem: Hume's principle without the #”
Math students virtually unwrap scrolls in one seminar
Theory, genetic algorithms, and a low-cost optical lab produced readable text for every group
· “ArcheoLab: A hands-on course to engage students in applied mathematics”
Gondishapur seeded Baghdad’s House of Wisdom
Migrating physicians and hospital models from Sasanian Iran enabled Arabic synthesis of Greek, Persian and Indian knowledge.
62826 encodes cube roots of unity on the τ-circle
The palindrome formed by 6/28/26 corresponds to a reciprocal polynomial with roots at angles τ/3 and a symmetric pair.
· “Palindromes on the τ-circle: A note for Palindrome Tau Day, 6/28/26”
AI challenge: prove these formulas for π
Numerically checkable identities whose proofs may demand non-obvious mathematics
Matrices compose to recover finite field extensions
The maps for chained extensions recover the direct map up to permutations, letting one 6x6 matrix over F_2 show both F_64/F_8/F_2 and F_64/F
Plato and Augustine make mathematics a path to knowing God
Number trains reason to recognize order, completing its work only when the mind reaches the divine.
· “The Purpose of Mathematics according to Plato and Augustine”
New database gathers mathematical models into one knowledge graph
Researchers can now locate formulas, quantities and assumptions without scanning separate publications.
Proof assistant improves explicitness in student paper proofs
Quasi-experiment shows carryover from Waterproof to Dutch pen-and-paper work despite English interface.
Memorial note records the life and death of algebraist Winfried Bruns
A former student's tribute documents his textbooks, the Normaliz software, and his passing days before a planned birthday conference.
Ring C[D] turns ODE solution rules into algebraic facts
Kernels and cosets replace heuristics once differential operators are polynomials acting on smooth functions.
Graph links 11.7M arXiv theorems to Lean formalizations
Slogan embeddings produce 47,952 verified cross-matches and support retrieval near existing reranked baselines without an LM reranker.
n coprime to φ(n) has exactly one group of order n
The condition identifies all orders where the only group is the cyclic one.
Five classes classify all groups of order p^3
Every group whose order is a cube of a prime falls into one of three abelian or two non-abelian isomorphism types, with explicit maps suppli
AI solve times grow slower than human times on math problems
Power-law fits on fifty Project Euler problems give exponents below one for most models, showing no support for machines scaling worse with
· “Human vs Machine Mathematical Difficulty on Project Euler: An Experimental Analysis”
Simon comments on Askey's paper about Szego
The note discusses the account written for Szego's 60th anniversary and the latter's work.
Convex solids are lonely exactly when they have constant width
Non-constant-width solids all share a thin cuboid friend and connect in chains of length at most three
One motive splits to yield Pascal
A lecture-note proof that one geometric decomposition of Grassmannians explains classical combinatorial recursions in many guises.
Notes arrange number theory topics in five chapters for CS
Coverage runs from divisibility and modular equations to Möbius inversion for use in discrete math or crypto courses.
Dice bingo boards form nontransitive win cycles
A slower solo board can still win more often, producing rock-paper-scissors triples and positions that require mixed strategies.
· “Optimal Play, Nontransitivity, and Nash Equilibria in Dice Bingo”
Asymmetric formula rivals Euler's Gradus for consonance
f(p/q) = p + Ω^(q) matches data while equating Gradus to weighted harmonic coincidence counts.
· “An Asymmetric Formula for Interval Consonance and its Relation to Harmonic Coincidence”
Single identity recovers classical polylog and zeta formulas
Specializations of the dyadic Cauchy-kernel identity yield duplication formulas, Hasse-Sondow series, and eta zeros
· “The Dyadic Cauchy-Kernel Identity: Several Roads Back to Classical Objects”
Eisenstein integers parametrize all sphere triangulations
The method produces flat metrics with twelve cone points of deficit pi/3 and parametrizes their space of shapes.
Sphere to plane shows Inonu contraction
The geometric transition introduces the physicist's key idea and its place in modern theory on the centennial of his birth.
LLMs solve 98 of 100 research math questions
New collection of 100 questions with known answers shows top models succeeding after repeated evaluation stages.
Laurent series turned isolated singularities into sources of precise data
Extending Cauchy's theorem to annular domains in 1843 replaced avoidance strategies with expansions whose negative powers encode singularity
npm package reuses Waterproof features for proof teaching
Rich formatting and clear input areas abstracted for integration into varied educational tools for proof assistants and languages.
· “Waterproof Editor: an educational environment for proof assistants and programming languages”
Marx reversed on infinitesimals after Boucharlat textbook
Engels kept endorsing them; any Robinson link runs through Fermat adequality instead
· “Marx versus Engels on infinitesimals: Chimera or triumph?”
Grothendieck groups turn monoids into invariants in four fields
The universal group completion of a commutative monoid produces Euler characteristics, K0 of modules, and representation rings as the start
Al-Kashi wrote the first general square-and-multiply rule in 1427
Traces the algorithm from Pingala's metrical counting to al-Kashi's 1427 rule and modern cryptography.
Gibbons conjecture term spread informally
Literature trace finds no explicit conjecture in 1995 Carbou paper, showing how names attach through repeated use.
Notes trace the exact point where Riemann integration stops working
By building measurable sets first, the notes show how the Lebesgue integral restores the ability to pass limits inside the integral.
Riemann supplied explicit log derivatives of zeta at s=1/2
Formulas give first ratio with pi and gamma, second with Catalan's constant plus sum over zero squares
Lotus encodes Euclid subtractions as blowup sequence
The anthyphairetic process on coprime pairs (a,b) corresponds to successive blowups resolving y^a - x^b = 0, with all generated numbers plac
12-TET tone networks split into chiral mirror pairs
Affine classification of decomposed cyclic graphs identifies an orientation-reversing family only in the 3-by-4 case, supplying a foundation
Semantic text format must anchor digital math exams
A human-editable format with meaning, not images or handwriting, becomes the record of student work and shapes future mathematical practice.
Binomial distribution converges to Gaussian in tempered distributions
De Moivre's 1733 indicator calculations are recovered as the special case of a general convergence in the space of distributions.
· “From Coefficients to Distributions: De~Moivre and the Operational View of Probability”
Gamma(p/q) reduces to Beta values that are Kontsevich-Zagier periods
Explicit formulas turn Gamma evaluations at fractions into period computations via the Beta function at rational points.
Three regimes fix perimeter and area limits for self-similar fractals
From N pieces and scale factor r one reads off whether length grows without bound while area stays finite or reaches zero.
· “A Scaling-Parameter Framework for Perimeter and Area in Self-Similar Planar Fractals”
Memories of a mathematics professor span forty years across countries
Recollections from Russia to Arizona mix personal stories with historical changes in the world.
· “My Warm, Randomly Recorded, Recollections of Professor Richard Askey”
