https://cluster-site.onrender.com/tags/mathematics/ Recent content in Mathematics on Tenu Tech Brief Hugo -- 0.146.0 en-us Wed, 25 Feb 2026 07:59:05 +0000 https://cluster-site.onrender.com/posts/early-evidence-of-vibe-proving-with-consumer-llms-a-case-study-on-spectral-region-characterization-with-chatgpt-5.2-thinking/ Tue, 24 Feb 2026 05:00:00 +0000 https://cluster-site.onrender.com/posts/early-evidence-of-vibe-proving-with-consumer-llms-a-case-study-on-spectral-region-characterization-with-chatgpt-5.2-thinking/ • LLMs increasingly used as scientific copilots, but research-level math evidence limited. • Case study uses ChatGPT-5.2 (Thinking) to resolve Conjecture 20 on spectral region of 4 https://cluster-site.onrender.com/posts/chinese-scientists-push-limits-of-300-year-old-math-problem-in-sphere-packing/ Thu, 19 Feb 2026 09:56:58 +0000 https://cluster-site.onrender.com/posts/chinese-scientists-push-limits-of-300-year-old-math-problem-in-sphere-packing/ • Chinese researchers at Fudan, Peking, and SAIS used AI to solve long‑standing kissing number problem. • AI system Packing Star computed optimal sphere packings in dimensions beyo https://cluster-site.onrender.com/posts/six-math-essentials/ Mon, 16 Feb 2026 16:59:26 +0000 https://cluster-site.onrender.com/posts/six-math-essentials/ • Just a brief announcement that I have been working with Quanta Books to publish a short book in popular mathematics entitled ‘Six Math Essentials’, which will cover six of the fu https://cluster-site.onrender.com/posts/the-univalence-principle/ Wed, 11 Feb 2026 02:26:50 +0000 https://cluster-site.onrender.com/posts/the-univalence-principle/ • The Univalence Principle Posted by Mike Shulman (guest post by Dimitris Tsementzis, about joint work with Benedikt Ahrens, Paige North, and Mike Shulman) The Univalence Principle https://cluster-site.onrender.com/posts/ipam-industrial-short-course-in-generative-ai-algorithms-deadline-for-applications-closing-soon/ Mon, 02 Feb 2026 22:32:45 +0000 https://cluster-site.onrender.com/posts/ipam-industrial-short-course-in-generative-ai-algorithms-deadline-for-applications-closing-soon/ • Updates on my research and expository papers, discussion of open problems, and other maths-related topics. • By Terence Tao Home About Career advice On writing Books Mastodon+ Ap https://cluster-site.onrender.com/posts/categorifying-riemanns-functional-equation/ Mon, 26 Jan 2026 00:19:49 +0000 https://cluster-site.onrender.com/posts/categorifying-riemanns-functional-equation/ • John Baez extends his categorification of the Riemann zeta function to its functional equation. • The completed zeta ξ(s) includes a real prime factor, key to symmetry. • Categor https://cluster-site.onrender.com/posts/a-crowdsourced-repository-for-optimization-constants/ Fri, 23 Jan 2026 03:24:40 +0000 https://cluster-site.onrender.com/posts/a-crowdsourced-repository-for-optimization-constants/ • Thomas Bloom’s Erdös problem site now thrives with AI‑assisted solutions. • Human, AI, and hybrid contributions are discussed and approved as updates. • A curated problem databas https://cluster-site.onrender.com/posts/rogers-theorem-on-sieving/ Mon, 19 Jan 2026 17:45:39 +0000 https://cluster-site.onrender.com/posts/rogers-theorem-on-sieving/ • Rogers theorem addresses density of integers remaining after removing finite congruence classes in sieve theory. • The theorem appears only in Halberstam & Roth’s 1966 text, unpu https://cluster-site.onrender.com/posts/the-integrated-explicit-analytic-number-theory-network/ Fri, 16 Jan 2026 07:24:59 +0000 https://cluster-site.onrender.com/posts/the-integrated-explicit-analytic-number-theory-network/ • Like many other areas of modern analysis, analytic number theory often relies on the convenient device of asymptotic notation to express its results. • It is common to use notati https://cluster-site.onrender.com/posts/polynomial-towers-and-inverse-gowers-theory-for-bounded-exponent-groups/ Tue, 06 Jan 2026 17:43:12 +0000 https://cluster-site.onrender.com/posts/polynomial-towers-and-inverse-gowers-theory-for-bounded-exponent-groups/ • Updates on my research and expository papers, discussion of open problems, and other maths-related topics. • By Terence Tao Home About Career advice On writing Books Mastodon+ Ap https://cluster-site.onrender.com/posts/coxeter-and-dynkin-diagrams/ Tue, 06 Jan 2026 15:44:13 +0000 https://cluster-site.onrender.com/posts/coxeter-and-dynkin-diagrams/ • The n-Category Café Skip to the Main Content Note:These pages make extensive use of the latest XHTML and CSSStandards. • They ought to look great in any standards-compliant moder https://cluster-site.onrender.com/posts/the-maximal-length-of-the-erd%C5%91s-herzog-piranian-lemniscate-in-high-degree/ Tue, 16 Dec 2025 06:18:34 +0000 https://cluster-site.onrender.com/posts/the-maximal-length-of-the-erd%C5%91s-herzog-piranian-lemniscate-in-high-degree/ • I’ve just uploaded to the arXiv my preprint The maximal length of the Erdős-Herzog-Piranian lemniscate in high degree. • This paper resolves (in the asymptotic regime of sufficie https://cluster-site.onrender.com/posts/the-equational-theories-project-advancing-collaborative-mathematical-research-at-scale/ Tue, 09 Dec 2025 17:13:55 +0000 https://cluster-site.onrender.com/posts/the-equational-theories-project-advancing-collaborative-mathematical-research-at-scale/ • Matthew Bolan, Joachim Breitner, Jose Brox, Nicholas Carlini, Mario Carneiro, Floris van Doorn, Martin Dvorak, Andrés Goens, Aaron Hill, Harald Husum, Hernán Ibarra Mejia, Zoltan https://cluster-site.onrender.com/posts/the-story-of-erd%C5%91s-problem-%231026/ Tue, 09 Dec 2025 03:11:49 +0000 https://cluster-site.onrender.com/posts/the-story-of-erd%C5%91s-problem-%231026/ • Problem 1026 on the Erdős problem web site recently got solved through an interesting combination of existing literature, online collaboration, and AI tools. • The purpose of thi https://cluster-site.onrender.com/posts/octonions-and-the-standard-model-part-13/ Sun, 07 Dec 2025 16:04:43 +0000 https://cluster-site.onrender.com/posts/octonions-and-the-standard-model-part-13/ • Octonions and the Standard Model (Part 13) Posted by John Baez When Lee and Yang suggested that the laws of physics might not be invariant under spatial reflection - that there’s https://cluster-site.onrender.com/posts/logx--c-revisited/ Wed, 03 Dec 2025 02:23:50 +0000 https://cluster-site.onrender.com/posts/logx--c-revisited/ • The n-Category Café Skip to the Main Content Note:These pages make extensive use of the latest XHTML and CSSStandards. • They ought to look great in any standards-compliant moder https://cluster-site.onrender.com/posts/quantitative-correlations-and-some-problems-on-prime-factors-of-consecutive-integers/ Tue, 02 Dec 2025 04:31:12 +0000 https://cluster-site.onrender.com/posts/quantitative-correlations-and-some-problems-on-prime-factors-of-consecutive-integers/ • Updates on my research and expository papers, discussion of open problems, and other maths-related topics. • By Terence Tao Home About Career advice On writing Books Mastodon+ Ap https://cluster-site.onrender.com/posts/octonions-and-the-standard-model-part-12/ Sun, 30 Nov 2025 17:27:18 +0000 https://cluster-site.onrender.com/posts/octonions-and-the-standard-model-part-12/ • Octonions and the Standard Model (Part 12) Posted by John Baez Having spent a lot of time pondering the octonionic projective plane and its possible role in the Standard Model of https://cluster-site.onrender.com/posts/beyond-the-geometry-of-music/ Sat, 22 Nov 2025 16:05:28 +0000 https://cluster-site.onrender.com/posts/beyond-the-geometry-of-music/ • The n-Category Café Skip to the Main Content Note:These pages make extensive use of the latest XHTML and CSSStandards. • They ought to look great in any standards-compliant moder https://cluster-site.onrender.com/posts/the-inverse-cube-force-law/ Wed, 05 Nov 2025 17:08:14 +0000 https://cluster-site.onrender.com/posts/the-inverse-cube-force-law/ • The n-Category Café Skip to the Main Content Note:These pages make extensive use of the latest XHTML and CSSStandards. • They ought to look great in any standards-compliant moder https://cluster-site.onrender.com/posts/second-quantization-and-the-kepler-problem/ Mon, 03 Nov 2025 08:51:55 +0000 https://cluster-site.onrender.com/posts/second-quantization-and-the-kepler-problem/ • The n-Category Café Skip to the Main Content Note:These pages make extensive use of the latest XHTML and CSSStandards. • They ought to look great in any standards-compliant moder https://cluster-site.onrender.com/posts/dynamics-in-jordan-algebras/ Sun, 02 Nov 2025 10:25:51 +0000 https://cluster-site.onrender.com/posts/dynamics-in-jordan-algebras/ • The n-Category Café Skip to the Main Content Note:These pages make extensive use of the latest XHTML and CSSStandards. • They ought to look great in any standards-compliant moder https://cluster-site.onrender.com/posts/applied-category-theory-2026/ Mon, 27 Oct 2025 16:56:16 +0000 https://cluster-site.onrender.com/posts/applied-category-theory-2026/ • The n-Category Café Skip to the Main Content Note:These pages make extensive use of the latest XHTML and CSSStandards. • They ought to look great in any standards-compliant moder https://cluster-site.onrender.com/posts/a-complex-qutrit-inside-an-octonionic-one/ Mon, 06 Oct 2025 17:24:31 +0000 https://cluster-site.onrender.com/posts/a-complex-qutrit-inside-an-octonionic-one/ • The n-Category Café Skip to the Main Content Note:These pages make extensive use of the latest XHTML and CSSStandards. • They ought to look great in any standards-compliant moder https://cluster-site.onrender.com/posts/good-reads-the-princeton-companion-to-mathematics/ Thu, 25 Sep 2025 01:33:44 +0000 https://cluster-site.onrender.com/posts/good-reads-the-princeton-companion-to-mathematics/ • Good Reads: The Princeton Companion to Mathematics Next up onGood Reads: ThePrinceton Companion to Mathematics,edited by Fields medalistTimothy Gowers.This book is an exceptional https://cluster-site.onrender.com/posts/what-is-a-good-quantum-encoding-part-1/ Thu, 25 Sep 2025 01:33:44 +0000 https://cluster-site.onrender.com/posts/what-is-a-good-quantum-encoding-part-1/ • What is a Good Quantum Encoding? • Part 1 Over the past couple of years, I’ve been learning a little about the world of quantum machine learning (QML) and the sorts of things peo https://cluster-site.onrender.com/posts/a-shadow-of-triality/ Sat, 13 Sep 2025 12:06:12 +0000 https://cluster-site.onrender.com/posts/a-shadow-of-triality/ • The n-Category Café Skip to the Main Content Note:These pages make extensive use of the latest XHTML and CSSStandards. • They ought to look great in any standards-compliant moder https://cluster-site.onrender.com/posts/equivalence-via-surjections/ Fri, 29 Aug 2025 22:23:54 +0000 https://cluster-site.onrender.com/posts/equivalence-via-surjections/ • The n-Category Café Skip to the Main Content Note:These pages make extensive use of the latest XHTML and CSSStandards. • They ought to look great in any standards-compliant moder https://cluster-site.onrender.com/posts/burrito-monads-arrow-kitchens-and-freyd-category-recipes/ Thu, 28 Aug 2025 16:02:51 +0000 https://cluster-site.onrender.com/posts/burrito-monads-arrow-kitchens-and-freyd-category-recipes/ • Burrito Monads, Arrow Kitchens, and Freyd Category Recipes Posted by Tom Leinster Guest post by Khyathi Komalan and Andrew Krenz From Lawvere’s Hegelian taco to Baez’s layer cake https://cluster-site.onrender.com/posts/magnitude-enriched-categories-and-llms/ Sun, 02 Feb 2025 04:33:03 +0000 https://cluster-site.onrender.com/posts/magnitude-enriched-categories-and-llms/ • Math3ma celebrates 10‑year anniversary, reflecting on growth from niche blog to influential math resource. • Author began as study tool, now shares graduate‑level insights with g https://cluster-site.onrender.com/posts/new-video-podcast-faq/ Tue, 28 Mar 2023 17:05:56 +0000 https://cluster-site.onrender.com/posts/new-video-podcast-faq/ • Launch of new video podcast ‘fAQ’ featuring casual chats on quantum physics and AI. • Co-hosts: author and Adam Green, a biologist and former Khan Academy academic director. • Ta https://cluster-site.onrender.com/posts/what-is-superposition-really/ Tue, 28 Mar 2023 17:05:56 +0000 https://cluster-site.onrender.com/posts/what-is-superposition-really/ • Superposition is simply a linear combination of states, like adding vectors in linear algebra. • The concept can be visualized without math using everyday analogies, such as over https://cluster-site.onrender.com/posts/modeling-sequences-with-quantum-states/ Thu, 16 Mar 2023 03:18:37 +0000 https://cluster-site.onrender.com/posts/modeling-sequences-with-quantum-states/ • Introduces quantum-inspired methods for modeling sequential data via tensor networks. • Uses bipartite graph representations of matrices to visualize linear maps. • Applies quant https://cluster-site.onrender.com/posts/understanding-entanglement-with-svd/ Thu, 16 Mar 2023 03:18:37 +0000 https://cluster-site.onrender.com/posts/understanding-entanglement-with-svd/ • SVD decomposes any matrix M into UDV†, revealing singular values and vectors. • Singular values, arranged in D, indicate the importance of each concept in M. • Left and right sin https://cluster-site.onrender.com/posts/what-is-quantum-technology/ Thu, 16 Mar 2023 03:18:37 +0000 https://cluster-site.onrender.com/posts/what-is-quantum-technology/ • SandboxAQ spun out of Alphabet, focusing on AI and quantum technologies. • Founder Jack Hidary explains what quantum technology means in new YouTube videos. • The team emphasizes https://cluster-site.onrender.com/posts/symposium-at-the-masters-university/ Tue, 14 Mar 2023 23:30:26 +0000 https://cluster-site.onrender.com/posts/symposium-at-the-masters-university/ • Symposium at The Master’s University, Santa Clarita, California, June 9-10, 2024. • Math3ma Institute hosts event, inviting scholars, industry, and government scientists. • NASA https://cluster-site.onrender.com/posts/a-new-perspective-of-entropy/ Tue, 10 Jan 2023 19:24:41 +0000 https://cluster-site.onrender.com/posts/a-new-perspective-of-entropy/ • Introduces a novel link between Shannon entropy and topological simplices via operad derivations. • Explores how entropy functions satisfy a Leibniz-like rule on simplicial struc https://cluster-site.onrender.com/posts/the-yoneda-lemma/ Sun, 01 Jan 2023 04:24:44 +0000 https://cluster-site.onrender.com/posts/the-yoneda-lemma/ • Yoneda lemma states that an object is fully determined by its hom‑functor relationships. • Yoneda embedding maps each object X to functor hom(-,X), fully faithful. • Morphisms f: https://cluster-site.onrender.com/posts/introducing-the-math3ma-institute/ Tue, 22 Feb 2022 17:33:28 +0000 https://cluster-site.onrender.com/posts/introducing-the-math3ma-institute/ • Math3ma expands into a new research hub, the Math3ma Institute, on TMU campus and online. • Visiting professor at TMU collaborates with math, science, engineering faculty to laun https://cluster-site.onrender.com/posts/math3ma-behind-the-scenes-3b1b-podcast/ Mon, 01 Nov 2021 18:14:54 +0000 https://cluster-site.onrender.com/posts/math3ma-behind-the-scenes-3b1b-podcast/ • Grant Sanderson shares early inspirations that sparked his passion for math and physics. • He reflects on his graduate school journey and key learning moments. • The conversation https://cluster-site.onrender.com/posts/language-statistics-category-theory-part-3/ Mon, 27 Sep 2021 18:25:00 +0000 https://cluster-site.onrender.com/posts/language-statistics-category-theory-part-3/ • Introduces enriched category theory framework for modeling language expressions and their relationships. • Builds on Part 2’s set assignment to words, extending to statistical co https://cluster-site.onrender.com/posts/monotone-convergence-theorem/ Mon, 16 Aug 2021 14:43:34 +0000 https://cluster-site.onrender.com/posts/monotone-convergence-theorem/ • The Monotone Convergence Theorem (MCT) allows interchange of limit and integral for increasing nonnegative functions. • MCT requires pointwise a.e. convergence, measurability, an https://cluster-site.onrender.com/posts/dominated-convergence-theorem/ Mon, 16 Aug 2021 14:35:13 +0000 https://cluster-site.onrender.com/posts/dominated-convergence-theorem/ • DCT states limit and integral commute under pointwise a.e. convergence with integrable dominating function. • Requires measurable functions and existence of integrable g bounding https://cluster-site.onrender.com/posts/the-borel-cantelli-lemma/ Mon, 16 Aug 2021 14:22:00 +0000 https://cluster-site.onrender.com/posts/the-borel-cantelli-lemma/ { ‘summary’: [ ‘• The Borel‑Cantelli Lemma applies to any finite‑measure space, not just probability spaces.’, ‘• It states that if the sum of measures of sets (E_n) converges https://cluster-site.onrender.com/posts/entropy--algebra--topology/ Wed, 28 Jul 2021 14:35:51 +0000 https://cluster-site.onrender.com/posts/entropy--algebra--topology/ • Introduces function d(x) = -x log x, a derivation reminiscent of Leibniz rule. • d relates to Shannon entropy, but is nonlinear, so H(p) ≠ d(∑ p_i). • Paper shows Shannon entropy https://cluster-site.onrender.com/posts/language-statistics-category-theory-part-2/ Wed, 21 Jul 2021 12:22:49 +0000 https://cluster-site.onrender.com/posts/language-statistics-category-theory-part-2/ • Language modeled as category L: objects are English expressions, morphisms are substring inclusions. • Category L captures syntax but lacks semantic depth, prompting richer categ https://cluster-site.onrender.com/posts/language-statistics-category-theory-part-1/ Wed, 07 Jul 2021 20:18:57 +0000 https://cluster-site.onrender.com/posts/language-statistics-category-theory-part-1/ • Authors propose a new preprint exploring math behind large language models. • Question: how to model transition from probability distributions on text to syntax and semantics. •