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Explore the foundations of modern algebraic geometry in this 53-minute lecture delivered by Ben Webster from the University of Waterloo as part of the Fields Institute's Shared Graduate Courses Program. The lecture, scheduled for January 6, 2025, serves as the first installment in the Modern Algebraic Geometry series, introducing graduate students to key concepts and methodologies in this mathematical field. Gain insights into the contemporary approaches to studying geometric objects defined by polynomial equations and learn how these techniques connect to other areas of mathematics.