The Mathematician newsletter
📚 Academic Research
Topologically 1-based T-minimal Structures (arxiv:math) . Topological 1-basedness, t-minimal theories with independent neighborhood property, group existence, type-definable abelian group, open subset, topological group, Hrushovski-Pillay 1-based classification analog
Algebraization of infinite summation (arxiv:math) . Algebraization of infinite summation via an algebraic framework, infinitizing group/ring axioms, focusing on associativity, Eilenberg-Mazer swindle generalizations, partial/unconditional summation, topology interactions, and new algebraic quotient structures
The Galois Theory (a ring theoretic approach) (arxiv:math) . Ring-theoretic approach to Galois theory via central simple algebras, End_K(L), automorphisms, and Double Centralizer Theorem for G-extensions
Decomposing the Diagonals of Invariant Fields (arxiv:math) . Decomposing diagonals of invariant fields via Chow groups, Sylow properties, unramified cohomology, and Galois extensions
Analogue of the Galois Theory for normal fields and B-extensions (characteristic free approach) (arxiv:math) . Analogue of Galois Theory for normal and B-extensions; characteristic-free approach via central simple algebras, double centralizer theorem, differential operators
Twisted conjugacy classes in Lie groups (arxiv:math) . Twisted conjugacy classes, Reidemeister numbers, topological R_infty-property for connected solvable, nilpotent, and SL(2,R), GL(2,R) Lie groups
On Graham's rearrangement conjecture over $\mathbb{F}_2^n$ (arxiv:math) . Graham's rearrangement conjecture over F_2^n; valid orderings; Freiman–Ruzsa; absorption method; Cayley graphs; additive combinatorics
Large implies henselian (arxiv:math) . Large implies henselian: equivalence via elementary extensions; étale-open topology; finite-closed topology; separably closed, bounded fields; étale morphisms; Lampe question
On the Parallel Complexity of Identifying Groups and Quasigroups via Decompositions (arxiv:cs) . Parallel complexity of identifying groups and quasigroups via decompositions; uses WL/Counting WL, AC^3 labeling, Kayal and Nezhmetdinov's decomposition, and NC central-quasigroup isomorphism results for efficiency
Some bi-quadratic Pólya fields and large Pólya groups of compositum of simplest cubic and quintic fields (arxiv:math) . Bi-quadratic Pólya fields K = Q(√p,√qrs), Q(√2p,√qrs), Q(√2p,√2qrs) with quadratic-residue relations; five ramified primes; Po(K) in Shank's cubic and Lehmer quintic compositums; infinite index-1 cases
Power maps on General Linear groups over finite principal ideal local rings of length two (arxiv:math) . Power maps on GL_n over finite length-two local rings via Hensel lifting and canonical matrix forms
Affine étale group schemes over Tambara fields (arxiv:math) . Classification of finite étale extensions and finite affine étale group schemes over G-Tambara functors, Galois descent from Tambara algebraic closure, and new families extending Lindenstrauss–Richter–Zou
Examples and Nonexamples of Distal Metric Structures (arxiv:math) . Examines distal properties in metric valued fields, topological dynamics, and models like Dual Linear Continua, analyzing indiscernible sequences and automorphism groups for stability and distality
First-order theory of torsion-free Tarski monsters (arxiv:math) . First-order theory of torsion-free direct limits of hyperbolic groups; simple torsion-free Tarski monsters with same positive theory as free group
On integral extensions between the abelianization functor and its symmetric powers (arxiv:math) . Ext-groups between abelianization and its dth symmetric power for finitely generated free groups; Arone’s complex; explicit projective resolution; comparison with Franjou–Pirashvili; Ext^1, Ext^2, Ext^(d-1), Ext^(d-2) computations
Stabilizers and NIP arithmetic regularity (arxiv:math) . New proof of NIP arithmetic regularity for finite groups via stabilizers, Alon–Fox–Zhao lemma, Sisask techniques, tripling, and Polynomial Bogolyubov–Ruzsa conjecture
Algorithmic Construction of Real Hyperfields from Minimal Axioms (arxiv:math) . Algorithmic construction of finite real hyperfields with cyclic positive cones, C-characteristic, Krasner quotients, and classification up to order 17
Duality on group algebras over finite chain rings: applications to additive group codes (arxiv:math) . Analyzes S|R chain rings, viewing S[G] as R-bimodule; characterizes additive group codes via left ideals; introduces trace-Euclidean inner product; ACPs connect orthogonality, anti-automorphisms, and representations
Non-orientable regular hypermaps of arbitrary hyperbolic type (arxiv:math) . Non-orientable regular hypermaps of hyperbolic type: infinite primes, PSL(2,p) automorphism groups, non-orientable surface carriers, and hyperbolic triples (ℓ,m,n) in compact surfaces
On Banach subalgebras of $\mathscr{H}^\infty$ consisting of lacunary Dirichlet series (arxiv:math) . Banach subalgebras of H^∞ consisting of lacunary Dirichlet series, multiplicative subsemigroups S, unitization, H^2_S Hilbert spaces, multiplier algebras, and Wiener 1/f analogue for H^∞_S
Unital Specker $\ell$-groups and boolean multispaces (arxiv:math) . Boolean multispaces, unital Specker ℓ-groups, dual equivalences, MV-algebras, Γ functor, Priestley duals, finite (co)limits, divisibility order
Greenberg's conjecture and Inverse Galois problem: metacyclic-nonmodular group of type 1 whose abelianization is $\mathbb{Z}/2 \mathbb{Z}\times\mathbb{Z}/2^m \mathbb{Z}$, $m\geq 2$ (arxiv:math) . Metacyclic-nonmodular Galois groups of type 1 with abelianization Z/2Z × Z/2^mZ (m≥2) over maximal unramified/pro-2 extensions, Greenberg conjecture techniques, real quadratic/biquadratic/Frohlich multiquadratic fields
Sites and Grothendieck toposes: an introduction (arxiv:math) . Intro to topos theory: Grothendieck topologies, sites, Giraud’s theorem, morphisms, points, subtoposes, localizations, geometric logic, classifying toposes
Cohomological nonvanishing for algebraic fundamental groups of ball quotients (arxiv:math) . Cohomological nonvanishing for profinite completions of cocompact arithmetic ball quotients in PU(n,1)
The algebraic $K$-theory of Green functors (arxiv:math) . Computes higher algebraic K-theory of Green functors for cyclic p-groups via a spectral sequence; exact K-theory of constant F2- and Z Green functors; introduces Green meadows and K0-freeness results
Splittings in a complete local ring and decomposition its group of units (arxiv:math) . Explores splittings in complete local rings, units group decompositions, separability criteria for field extensions, and conditions for formal étaleness and smoothness over perfect fields
Rational Witt vectors and associated sheaves (arxiv:math) . Study of sheafification of Witt vectors and maps in Grothendieck topologies; includes results on rational functions and correspondence for Dedekind rings
On the Homogeneous Space of a Pair of Associative Composition Algebras (arxiv:math) . Equivariant cohomology of Hamiltonian quaternion homogeneous spaces; equivariant K-theory; Grothendieck ring of locally free modules on singular matrices over associative composition algebras; rank notions and determinantal varieties
Finiteness properties of Subgroups of Houghton Groups of full Hirsch length (arxiv:math) . Investigates finiteness properties of subgroups of Houghton groups H_n, proving type F_n-1 but not FP_n for subgroups mapping onto Z^n-1, via Jordan–Wielandt and wreath-product analysis
Generation of Grothendieck topologies, provability and operations on subtoposes (arxiv:math) . Grothendieck topologies, subtoposes, dualities, provability translation, inner/outer operations, and pushforward/pullback preservation
On the second partial Global Euler-Poincare characteristics for Galois cohomology (arxiv:math) . Second partial Euler-Poincaré characteristic for Galois cohomology, explicit formula by expanding ramification, G(K,S) presentation, deformation ring counterexamples
Polynomial extension of Van der Waerden's Theorem near zero (arxiv:math) . Polynomial Van der Waerden's theorem near zero for dense subrings of real numbers, combinatorial partitions, polynomial sequences, and sum conditions in a near-zero interval
The action of an inverse semigroup on its Stone-Čech compactification (arxiv:math) . Stone-Čech transformation groupoid of an inverse semigroup, amenability, Hausdorffness, and tight groupoid connections
Cholesky decomposition for symmetric matrices over finite fields (arxiv:math) . Development of Cholesky factorization for positive matrices over finite fields, extending LPM cone, and exploring Frobenius compatibility and group operations
Symmetric spaces for groups over involutive algebras and applications to Higgs bundles (arxiv:math) . Symplectic and indefinite orthogonal groups over involutive algebras, Hermitian cases, complexified tangent spaces, Higgs bundles, polystability, harmonicity, and representations
Extending fusion rules with finite subgroups: For a general understanding of quotient or gauging (arxiv:math) . Method for simple current extensions in conformal field theories, deriving $Z_N$ symmetry fusion rings, modular partition functions, and applications to topological orders, domain walls, and RG flows
Transversality Methods for Homotopy Groups of Stable Loci in Affine GIT Quotients (arxiv:stat) . Extends dimensional transversality methods to affine GIT quotients, constructing a G equivariant bundle over parameter subgroup orbit, showing homotopies avoid unstable strata by dimension criteria
Number Fields With Large Pólya Groups (arxiv:math) . Finiteness theorems for Pólya groups in number fields: Galois, solvable CM-fields, real quadratic extended R-D type; classifications of imaginary bi-/tri-quadratic with Pólya index one; GRH-based lists and genus-number relations
Trace Minimization and Roots in ${\rm PSL}(2,\mathbb{R})$ (arxiv:math) . Investigates discreteness and freeness in PSL(2,R) generated by A,B; analyzes roots R^m=A, S^n=B, commutator trace tau, and Trace Minimization Algorithm, with powers, traces, and cases
🤝 Community, Mentoring & Teaching — Girls' Angle, MOP, courses and academic life
Girls’ Angle Bulletin, Volume 18, Number 6 (girlsangle.wordpress.com) . Interview with Greta Panova; creativity in math, math competitions, Caitlin Cunjak’s Fibonacci matrices, graph constructions from codes, Summer Fun problem solutions, OEIS sequences
Mathematical family (cameroncounts.wordpress.com) . Reflections on mathematical mentorship, collaboration, and the concept of mathematical families involving notable mathematicians like G. H. Hardy, Paul Erdős, and more
How to make the most out of MOP (blog.evanchen.cc) . MOP studying strategies: in-person group-work, moments of surprise, learning from peers, expert-led class structure, and signaling wisdom
Thoughts on teaching multivariable calculus (blog.evanchen.cc) . MIT 18.02 recitations emphasize writing polished notes, concrete examples: determinants, linear independence, normal vectors; plus sound-bites, repetition, flexible planning, and office-hours tweaks for student learning
Introduction to Hilbert Spaces: An Adventure In Infinite Dimensions (boffosocko.com) . UCLA Extension MATH 900: Introduction to Hilbert Spaces, orthogonality, orthonormal bases, Bessel’s inequality, linear operators, Legendre polynomials, Fourier series, Sobolev spaces, l2 and L2 spaces, course notes and bibliography by Chris Aldrich
Retirement party (cameroncounts.wordpress.com) . Retirement party at Cameron Counts blog features talks by Mia Tackney on design of experiments and Rosemary methods in clinical trials, Scott Harper on advanced combinatorics and Cauchy numbers, plus reflections on finite group theory progress and a Zizzi dinner
📣 Conferences, Talks & Research Projects — ICECA, SLMath, surreal arithmetic, Erdos→OEIS
A crowdsourced project to link up erdosproblems.com to the OEIS (terrytao.wordpress.com) . Crowdsourced linkage of Erdos problems to OEIS via a GitHub YAML workflow and discussion forums
International Conference on Enumerative Combinatorics and Applications ICECA 2025 (August 25-27, 2025) (gilkalai.wordpress.com) . ICECA 2025 online conference August 25–27 features Richard Stanley lecture on Symmetric functions from a Ramanujan theta function; Toufic Mansour organizes, with links to abstracts and videos from prior editions and interviews with combinatorialists
The elementary theory of surreal arithmetic is bi-interpretable with set theory, Kobe, Japan, September 2025 (jdh.hamkins.org) . Talk on the elementary theory of surreal arithmetic; bi-interpretability with set theory (V,∈); surreal field birthday order; ZFC bi-interpretation; collaboration with Junhong Chen and Ruizhi Yang; Kobe 2025
SLMath announces new research programs (terrytao.wordpress.com) . SLMath announces three new research initiatives with open applications starting Sep 1, 2025, announced by Terence Tao
🧵 Math Culture, Crafts & Reviews — embroidery, Dudeney puzzles, book review, anecdotes
Embroidering an aperiodic tiling (samjshah.com) . Sam Shah embroidery project uses aperiodic tiling with the spectre variant of the hat tile; threads, backstitch and satin stitch; water-soluble transfer paper; Threaded Theorems instructions; Scientific American explanation of aperiodic tilings
Review: The Mathematician’s Library, by Thomas K. Briggs (aperiodical.com) . Briggs’s Mathematician’s Library surveys global mathematical texts from India, China, and beyond, via six chronological sections and visually rich selections
A few more notes about Tom L (blog.computationalcomplexity.org) . Anecdotes about Tom Lehrer, Tom L., obituaries, novelty songs, Jane Morgan’s death, updates to Tom L.'s Elements, complexity classes, and related musical and mathematical references
Piecework (futilitycloset.com) . Henry Dudeney's Canterbury Puzzles (1907) asks how to cut an equilateral triangle into four pieces forming a square; 'strip of squares over triangles' offers proof
📜 Mathematical History & Biographies — Ada Lovelace, Smale, von Neumann and more
How a poet’s long-forgotten daughter created the first algorithm (adamgrant.micro.blog) . Ada Lovelace, daughter of Lord Byron, pioneered early algorithms, envisioned computers manipulating symbols, and contributed to Babbage's Mechanical Analytical Engine with her visionary insights
Finding a Horseshoe on the Beaches of Rio (thatsmaths.com) . Steve Smale's 1960s Rio beach work on chaos theory and the horseshoe map; IMPA in Rio; Johnson administration fallout; Norman Levinson letter; chaos theory origins
the MANIAC, or when machines strike deep inside enemy lines (am17an.bearblog.dev) . Explores the lives of Paul Ehrenfest and John von Neumann, the rise of Artificial General Intelligence, and the philosophical implications of machine superiority
From τὰ φυσικά (ta physika) to physics – XLIX (thonyc.wordpress.com) . Examines the evolution of mathematical disciplines like algebra, geometry, trigonometry, logarithms, calculus, and their role in seventeenth-century natural philosophy and physics development
The Martians of Science (markbernstein.org) . Budapest-born physicists von Neumann, Szilard, Teller, Bohr? No, five Martians at Los Alamos; group biography detailing their paths from Budapest to Germany to the US, their roles in the atomic bomb project, and the political shifts that followed
André ou Jean Ville (1910-1989) (xianblog.wordpress.com) . Overview of Jean/André Ville's life, contributions to martingale inequalities, counterexamples, early computer use, collaboration with mathematicians like Gödel, Popper, and Wald, and his diverse career
God created the real numbers (ethanheilman.com) . Kronecker on integers, Cantor infinity, Cantorian theology, Descartes, Hilbert, and the hierarchy of creation in mathematics and nature
⚙️ Computation, Algorithms & Complexity — ZK/proofs, Busy Beaver, Wolfram toolkit, contest algos
The (sometimes rocky) road to a paper (lmao.bearblog.dev) . Ligerito: a linear-algebraic view of succinct proofs, ZODA, tensorizable codes, sumcheck integration, and WHIRgerito optimizations
Scoopert! (radar.spacebar.org) . Tom 7 Radar discusses a convex polyhedron with Rupert's property, a non-Rupert result on arXiv, DOOM: The Dark Ages, Shantae and the Seven Sirens, and high-performance rational interval arithmetic
Learning Decision Process Theory with a Wolfram Language Toolkit (blog.wolfram.com) . Free Wolfram Language toolkit and 38-lesson course explore decision process theory, game theory basics, time-dependent field theory, DPT toolkit, Prisoner’s Dilemma, and Wolfram U Course Assistant
Linkage (11011110.github.io) . 3d and layered QR codes, developable surfaces from flat strips, AI slop in knowledge, matroid parity, cubical spheres, topological book embeddings, LATIN call, Wikipedia search critique
A thesis week (blog.mitrichev.ch) . Mitrichev outlines randomized solving for problem H, uses edge-based backtracking, notes DP over DFS subtrees with at most backward edges, duplicating vertices for nonbipartite matching
Busy Beaver Hunters Reach Numbers That Overwhelm Ordinary Math (quantamagazine.org) . Busy Beaver BB(6) lower bounds explode beyond decimal notation; 6-rule Turing machines run for astronomically many steps, via shift overflow counters, tackled by Busy Beaver Challenge members like Doucette, Ligocki, Kropitz, and mxdys
🧠 Formalization & Proof Assistants — Lean, Isabelle, Z3, Autoformalization
Tic-tac-toe meets Lean 4 (ochagavia.nl) . Lean 4 tic-tac-toe formalization with proofs, well-formedness, Vector 9, Fin 9, GameStatus, initialGameState, and Curry–Howard connections
Definite integrals, I: easy cases over finite intervals (lawrencecpaulson.github.io) . Definite integrals, FTC variants, arcsin derivative, endpoint divergence, continuous antiderivatives, Isar proofs, derivative_intros, continuous_on, fundamental_theorem_of_calculus_interior, WolframAlpha, Maple, Isabelle proof scripting
Beyond Booleans (overreacted.io) . Lean's propositions-as-types, proofs as values, Not and by decide, and how 2+2=4 becomes a proof object; contrast with TypeScript booleans and the Prop type
Knuckledragger Analysis Etudes (philipzucker.com) . Explores balls, open sets, and contraction via Z3-backed tactics, unfolding lemmas, axioms, and bagged intermediate goals to prove real analysis properties in a knuckledragger style
Autoformalization and Verifiable Superintelligence with Christian Szegedy - #745 (twimlai.com) . Autoformalization, formal reasoning, AI safety, Morph Labs, Inception architecture, adversarial examples, LLMs, autoformalization with large language models
The same colored grass (buttondown.com/j2kun) . Dan Abramov's lean+Mathlib journey on real numbers, Spivak's calculus critique, Emily Riehl's homotopy type theory talk, and the joy of cross-pollinating programming types with mathematical rigor
🔬 Pure Math Deep Dives — Conformal maps, Hadamard, multilinear, spectral/number theory
Science Notes / Pick Two Points (gregegan.net) . Geometric probability of random point pairs in a square; 4D hypercube mapping; parallel faces, Jacobian, and volume integrals; generalisation to d-dimensions and convex polytopes
The Craft of Conformal Maps (galileo-unbound.blog) . Conformal maps, Riemann sphere, Schwarz–Christoffel equation, Joukowski map, Laplace equation, field lines, stereographic projection, Mercator, Lambert, CFT
Hadamard Factorization Theorem (kuniga.me) . Hadamard Factorization, genus and order of entire functions; Weierstrass, Jensen’s formula, Poisson kernel; Hadamard's theorem: h ≤ λ ≤ h+1; lemmas on order, genus, and canonical products
Multilinear polynomials: survival kit (blog.lambdaclass.com) . Multilinear polynomials, hypercube interpolation, Lagrange basis, coordinates via evaluations, tensor product structure, and variable-dependence tests for products p_k(X)
What is mathematics? A classification based on universals (ebellani.github.io) . Classification of mathematics by universals: realism vs nominalism (including Kantian conceptualism), Franklin 2014, De Wulf 1911; taxonomy of mathematical definitions; figures and references; author contact
‘Ten Martini’ Proof Uses Number Theory to Explain Quantum Fractals (quantamagazine.org) . Ten Martini problem links Hofstadter butterfly fractals to Cantor set, almost-periodic functions, Avila’s global theory, and graphene experiments
The biggest math symbol (johndcook.com) . Riemann P-symbol (Papperitz) for solutions to Riemann’s differential equation with three regular singular points a, b, c and Möbius-transformation behavior
Pascal's Determinant (sriku.org) . Explores Pascal matrix determinants via Pascal steps, row/column convolution invariance, and recursive/inductive proofs using Julia/ determinant concepts