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Category: Math

Beyond Reality: From Jordan Algebras to the Leech Lattice in an Exotic Spacetime

2025-03-17
Beyond Reality: From Jordan Algebras to the Leech Lattice in an Exotic Spacetime

This article explores the deep connections between Jordan algebras, octonions, and the Leech lattice. Starting with Pascual Jordan's work in the 1930s on the algebraic properties of Hermitian matrices, it introduces formally real Jordan algebras and their classification, including a special 27-dimensional exceptional Jordan algebra. Building on this, the article explains how projective spaces are constructed from Jordan algebras, focusing on the octonionic projective plane generated by the exceptional Jordan algebra. Finally, it delves into an exotic spacetime constructed from octonionic Hermitian matrices and a unique integral unimodular lattice within it—the Leech lattice. A surprising finding is that this lattice exhibits two distinct orbits under the action of the E6 group, unlike typical understanding.

(cp4space.hatsya.com)
Math Jordan algebras octonions Leech lattice

Breakthrough: 3D Kakeya Conjecture Solved

2025-03-02
Breakthrough: 3D Kakeya Conjecture Solved

A major breakthrough in geometric measure theory! Hong Wang and Joshua Zahl's preprint resolves the infamous three-dimensional Kakeya set conjecture. The conjecture asserts that a Kakeya set—a subset containing a unit line segment in every direction—must have Minkowski and Hausdorff dimension equal to three. The proof, spanning 127 pages, uses an iterative induction argument cleverly handling 'sticky' and 'non-sticky' cases. This landmark result builds on decades of work, incorporating previous findings and novel ideas, marking a significant milestone in geometric measure theory.

(terrytao.wordpress.com)
Math Kakeya Conjecture Geometric Measure Theory Mathematical Breakthrough

Modular Forms: Unveiling Hidden Symmetries and Infinite Possibilities

2025-02-24
Modular Forms: Unveiling Hidden Symmetries and Infinite Possibilities

Mathematicians have discovered that modular forms, a special type of function, possess infinite symmetries stemming from their unique transformation properties on the complex plane. These transformations replicate the fundamental domain to the entire upper half-plane, relating copies through specific rules. While seemingly simple geometric operations, they hold immense power. Hecke's theory revealed that modular forms reside in specific spaces, allowing us to leverage their infinite symmetries to tackle problems like representing integers as sums of four squares. By converting sequences into generating functions, if the function is a modular form, coefficients can be precisely calculated, unlocking infinite possibilities. This provides a powerful tool for solving numerous problems in mathematics and physics.

(www.quantamagazine.org)
Math modular forms symmetry

The Monstrous Function That Broke Calculus

2025-01-24
The Monstrous Function That Broke Calculus

In the 19th century, Karl Weierstrass unveiled a function that sent shockwaves through the mathematical community. This function, continuous everywhere but differentiable nowhere, resembled an infinitely jagged sawtooth, defying intuition and challenging the very foundations of calculus. Its seemingly paradoxical properties forced mathematicians to rigorously redefine continuity and differentiability, ultimately leading to the development of modern analysis. This 'mathematical monster' not only holds theoretical significance but also finds practical applications in fields like Brownian motion, showcasing the boundless possibilities within mathematics.

(www.quantamagazine.org)
Math Weierstrass function mathematical analysis

Century-Old Math Problem Solved: Proving the Irrationality of ζ(3)

2025-01-09
Century-Old Math Problem Solved: Proving the Irrationality of ζ(3)

This article recounts the legendary story of mathematician Roger Apéry's 1978 proof that ζ(3) (the Riemann zeta function at 3) is irrational. His proof was met with skepticism and even caused chaos at the conference where it was presented. However, Apéry was ultimately proven correct. For years, mathematicians struggled to expand Apéry's method with little progress. Recently, Calegari, Dimitrov, and Tang developed a more powerful method, proving the irrationality of a series of zeta-like values, including ζ(3), solving a decades-old problem. This breakthrough lies not only in its result but also in the generality of its approach, providing new tools for future irrationality proofs.

(www.quantamagazine.org)
Math irrational numbers

Infinity's Size: Mathematicians Get Closer to Answering How Many Real Numbers Exist

2025-01-09
Infinity's Size: Mathematicians Get Closer to Answering How Many Real Numbers Exist

For decades, mathematicians believed determining the total number of real numbers was an unsolvable problem. A new proof suggests otherwise. The article details how mathematicians Asperó and Schindler proved that two axioms previously considered competing foundations for infinite mathematics actually imply each other. This finding strengthens the case against the continuum hypothesis and indicates an extra size of infinity exists between the two that, 143 years ago, were hypothesized to be the first and second infinitely large numbers. While this result has generated excitement and debate within the mathematical community, the arguments surrounding the sizes of infinite sets are far from settled.

(www.quantamagazine.org)
Math continuum hypothesis infinity