Webtagr - Technology News Summarizer

Formalizing Analysis I in Lean: An Interactive Learning Project

2025-05-31

The author is formalizing their 20-year-old real analysis textbook, "Analysis I," using the Lean proof assistant. This isn't a simple translation; it involves converting definitions, theorems, and exercises into Lean code. Readers can complete the exercises by filling in 'sorries' in the code, learning Lean and the Mathlib library along the way. The project currently includes several translated sections, strategically transitioning from a 'handmade' construction of natural numbers to the Mathlib standard library. The author invites volunteers to test and improve the project.

(terrytao.wordpress.com)

Development real analysis

Interactive Math Proof Assistant Built with Python and SymPy

2025-05-13

A developer has built an interactive mathematical proof assistant using Python and the SymPy library. It semi-automatically proves asymptotic estimates involving scalar functions. Mimicking the Lean proof assistant, the tool supports linear and log-linear arithmetic, allowing users to guide the proof process by supplying high-level tactics. Currently running in Python's interactive mode, a graphical user interface is planned for the future. The developer intends to extend the tool to handle a broader range of mathematical tasks, such as estimating function space norms.

(terrytao.wordpress.com)

Development math proof SymPy

Rethinking Orders of Infinity with Nonstandard Analysis: An Algebraic Approach

2025-05-04

This paper explores a novel approach to studying asymptotic notation and orders of infinity using nonstandard analysis. Traditional analysis relies on complex epsilon-delta arguments to handle orders of infinity. However, nonstandard analysis cleverly hides many quantifiers through the introduction of ultrafilters, transforming the problem into one with a more algebraic nature. The paper demonstrates that within the nonstandard framework, orders of infinity form a totally ordered vector space and possess a completeness property reminiscent of the completeness of real numbers. This algebraic approach simplifies computations with asymptotic notation, especially in symbolic computation, but sacrifices the ability to extract explicit constants.

(terrytao.wordpress.com)

Development nonstandard analysis asymptotic notation orders of infinity

Automating Asymptotic Estimate Verification: A Python Tool

2025-05-02

This post describes a Python tool for automatically verifying asymptotic estimates, particularly those involving a finite number of positive real numbers combined using arithmetic operations like addition, multiplication, division, exponentiation, and min/max. The tool uses case splitting and linear programming to automatically determine if an inequality holds, providing a proof or counterexample. The author illustrates the tool's usefulness with personal examples and discusses future improvements, such as handling more complex expressions and integration into existing mathematical software platforms.

(terrytao.wordpress.com)

Development asymptotic estimates automatic verification

Decomposing Factorials into Large Factors: Progress on an Old Conjecture

2025-03-28

A new paper studies the problem of factoring a factorial into factors as large as possible. Erdős and others proposed a conjecture about this, but the proof was lost. This paper, using clever applications of the prime number theorem and approximate factorization, provides new upper and lower bounds, partially solving this long-standing problem and offering new avenues to fully resolve the remaining conjectures.

(terrytao.wordpress.com)

Development combinatorics

Breakthrough: 3D Kakeya Conjecture Solved

2025-03-02

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

How Math Lit Up America: The Landscape Function and the LED Energy Revolution

2025-02-24

US residential electricity consumption has slightly decreased in recent years, primarily due to improvements in lighting efficiency, specifically the widespread adoption of LED light bulbs. Behind this energy revolution is an unexpected driver: a breakthrough in pure mathematics—the landscape function. Initially a purely mathematical discovery, this function is now central to efficient LED design. Through numerical simulations, the landscape function has helped researchers overcome the "green gap" (the lack of efficient green LEDs), accelerating LED R&D and saving US consumers billions of dollars in energy costs.

(terrytao.wordpress.com)

Tech

Unlocking Spherical Trigonometry with Quaternions

2025-01-30

This article leverages the algebraic properties of quaternions to derive a 'master equation' for spherical trigonometry, elegantly proving the spherical law of cosines, the spherical law of sines, and Napier's rules. The author cleverly connects quaternions to the relationships between sides and angles of spherical triangles, using rotations and inner products to derive concise and elegant formulas. Applications to practical problems like calculating sunrise and sunset times are discussed, showcasing the power of quaternions in geometric problems.

(terrytao.wordpress.com)

Misc quaternions spherical trigonometry

Tao's New Paper: Delving into Eigenvalue Distribution of GUE and its Minors

2024-12-22

In his latest arXiv preprint, renowned mathematician Terence Tao delves into the distribution of eigenvalues of the Gaussian Unitary Ensemble (GUE) and its minors at fixed indices. Employing determinantal processes and sophisticated analytical techniques, the paper establishes several estimates regarding eigenvalue gaps, addressing previously unanswered questions and paving the way for future work on the limiting behavior of 'hives' with GUE boundary conditions. This research significantly contributes to the understanding of random matrix models and related fields.

(terrytao.wordpress.com)

Development random matrices eigenvalue distribution