Webtagr - Technology News Summarizer

Quadratic Forms Beyond Arithmetic: Four Decades of Algebraic Advances

2025-07-21

This article reviews major advances in the algebraic theory of quadratic forms over the last four decades, focusing on how the introduction of algebro-geometric methods revolutionized the field. Tracing the concept's origins from early work in ancient Babylon and Greece to landmark theorems by Fermat and Lagrange, it highlights the solution of the Milnor conjectures and novel approaches to studying quadratic forms using algebro-geometric tools such as quadric hypersurfaces and algebraic cycles. The article also explores field invariants associated with quadratic forms (the u-invariant and Pythagoras numbers), and discusses open questions concerning dimensions and splitting patterns of quadratic forms.

Formalizing a Linear Algebra Proof with Lean

2025-01-24

This article details the author's experience formalizing a simple theorem about the linear independence of eigenvectors in linear algebra using the Lean proof assistant. The article explains Lean's syntax, the use of the Mathlib library, and how automation tools simplify the proving process. The authors explore improving and generalizing the theorem and introduce Mathlib's version control and community collaboration. Finally, the article looks ahead to the role of proof assistants and AI in future mathematical research.

Treewidth: A Key Parameter in Graph Theory and Its Applications

2025-01-14

This article delves into treewidth, a crucial parameter in graph theory. Defined using tree decompositions, treewidth characterizes graph structure and is closely related to algorithmic complexity. The article explores multiple equivalent definitions of treewidth, its structural properties, and computational methods. It then details its broad applications in sparse numerical linear algebra, Bayesian inference, game theory, low-dimensional topology, network science, and algebraic geometry. The author also discusses advances in related width parameters and how treewidth can improve the efficiency of graph algorithms.