Brad DeLong on Theories of Economic History
Theories of Economic History - I Theories of Economic History - II Theories of Economic History - III Theories of Economic History - IV Theories of Economic History - Slides
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Consciousness as Research
On Consciousness Research Consciousness Science On Consciousness Research Consciousness Science
On Quotient Spaces in Relation to Music with Dmitri Tymoczko
Dmitri Tymoczko Dmitri Tymoczko On Math On Topology On Symmetry On Math On Geometry of Music On Quotient Space and Symmetry in Music Music ~***~ AI Dmitri Tymoczko uses quotient spaces to model how musicians and listeners abstract away from specific musical details —such as which octave a note is in, the order of notes in a chord, or the specific transposition of a chord. By "gluing together" points in a high-dimensional space that represent equivalent musical situations, he creates complex, low-dimensional geometric shapes called orbifolds . These spaces, often called OPTIC spaces (Octave, Permutation, Transposition, Inversion, Cardinality), help visualize and analyze musical voice leading and harmonic relationships. [ 1 , 2 , 3 , 4 , 5 , 6 ] Here is how Tymoczko uses quotient spaces to model music: 1. Abstracting Musical Information (The OPTIC Relations) Tymoczko forms quotient spaces by applying five standard equivalence relations, often referred to in col...
Automated Theorem Proving - Mel Nathanson
How Harmonic's " Mathematical Superintelligence" System Aristotle Solved Some of My Problems from Mel Nathanson on Research Seminars (04/30/2026): https://researchseminars.org/seminar/New_York_Number_Theory_Seminar : Several open problems from my paper ``Problems and results on intersections of product sets and sumsets in semigroups" (arXiv:2604.04781) were recently solved by Aristotle, a formal reasoning agent developed by Harmonic. The solutions were published on April 20, 2026, in the paper ``Global product intersections sets in semigroups" by Wouter Van Doorn, Pietro Monticone, and Quanyu Tang (arxiv:2604:18869). This talk will describe product intersection sets, the open problems solved by Aristotle, what parts of the solution were in the original paper, and what exactly was the new idea that Aristotle invented to solve the problems. Source Paper from Mel Nathanson Problems and Results on Intersections of Product Sets and Sumsets in Semigroups Melvyn B. N...