Transforming Numbers - Why is Arithmetic Geometry Geometry?

Explore the fundamental question of why arithmetic geometry is considered geometry in this conference talk that examines Grothendieck's vision of algebraic geometry's "arithmetic aspect" through geometric intuition, concepts, and techniques. Delve into the prominent mathematical development that has created new scenarios for understanding key ideas and tensions within pure mathematics, particularly the relationships between geometry and number theory, and between continuous and discrete mathematics. Investigate how mathematical concepts like number and space can be transformed through innovative mathematical practices and immersion in new structures, with particular focus on how rich hybrid structures such as schemes can bridge and integrate the traditional antithesis between discrete and continuous mathematics. Examine the complex evolution of ideas about space and geometry throughout human history, with special attention to the acceleration brought by 20th century mathematical abstractions. Gain insights from three distinguished mathematicians as they reflect on the meandering development of these fundamental concepts and attempt to answer the central question of arithmetic geometry's geometric nature. Learn about the transformative power of new mathematical structures and how they reshape our understanding of basic mathematical ideas through the lens of Grothendieck's revolutionary approach to algebraic geometry.