Hilbert’s 10th Problem and Decidability in Algebra and Number Theory
Date:
Abstract: In 1900, David Hilbert posed a question at an international mathematics conference in Paris: Is there an algorithm that can determine whether a given polynomial equation with integer coefficients has an integer solution? The question became known as Hilbert’s 10th Problem. Several decades later, it became increasingly clear that such an algorithm may never exist. This marked the start of a research area on the intersection of logic, algebra, and number theory: to determine which classes of problems from number theory and algebra are decidable (i.e. solvable by an algorithm) and which are undecidable.
During this talk, I will discuss the history of Hilbert’s 10th Problem and highlight some recent developments. The focus of the talk will be on how questions surrounding Hilbert’s 10th Problem give rise to interesting problems in number theory, and conversely, how classical theorems from algebra, number theory, and quadratic form theory have been used to investigate questions surrounding decidability in number theory.
This talk was given at the Algebra Colloquium of Charles University’s Faculty of Mathematics and Physics. You may find the slides for this talk here.
A shortened version of this talk was given at the Belgian Mathematical Society’s Young Scholar Day on December 20, 2023. You may find the slides for this talk here