Mathematical  Culture and Thought

Mathematical Culture and Thought

Grothendieck’s Double Limit Theorem and Model Theory

Document Type : Survey

Author
Department of Science‎, ‎Arak University of Technology‎, ‎Iran; and ‎School of Mathematics‎, ‎IPM‎, ‎Ira‏n
Abstract
Alexander Grothendieck is undoubtedly one of the most influential mathematicians of the 20th century who revolutionized modern algebraic geometry‎. ‎This article intends to study the story of one of Grothendieck's first results in another field of mathematics‎, ‎namely functional analysis‎, ‎which has recently been discovered to have a surprising connection with model theory‎. ‎We will show that the fundamental theorem of stability theory‎, ‎which was proved by Shelah in the 1970s‎, ‎is in fact a consequence of Grotendieck's double limit theorem‎, ‎which was proved in his doctoral thesis in the 1950s‎. ‎We will also discuss the model-theoretic meaning of Grothendieck's theorem and reinforce this theorem for generically stable types‎. ‎Finally‎, ‎we will discuss some possible applications and results in future work‎.
Keywords

Subjects


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Volume 43, Issue 2 - Serial Number 75
November 2024
Pages 247-259

  • Receive Date 01 March 2023
  • Revise Date 26 March 2023
  • Accept Date 26 March 2023
  • Publish Date 21 November 2024