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Watch a 59-minute mathematics lecture exploring how motivic homotopy theory enables enumerative geometry calculations over arbitrary base fields, yielding additional arithmetic and geometric insights. Delve into classic examples like Bézout's theorem and counting lines on cubic surfaces, examining solutions over complex and real numbers before generalizing to arbitrary fields using A1 degree from motivic homotopy theory. Learn about tropical geometry, particularly tropical plane curves, and their application in proving Bézout's theorem for curves over arbitrary fields. Explore tropical correspondence theorems from collaborative research with Jaramillo Puentes and ongoing work with Jaramillo Puentes-Markwig-Röhrle. Access supplementary lecture notes and problem sets to reinforce understanding of these advanced mathematical concepts. Part of the 2024 Graduate Summer School program at PCMI focusing on Motivic Homotopy Theory, this lecture requires foundational knowledge in algebraic geometry, algebraic topology, and homotopy theory.
