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Explore the mathematical theory of projections in Euclidean space through this comprehensive graduate-level course taught by Professor Lawrence D. Guth at MIT. Delve into the fundamental question of how geometric objects relate to their shadows when projected onto various subspaces, examining whether concentrated projections indicate concentration in the original object. Master key concepts including the exceptional set problem, a central challenge in projection theory that was recently solved in 2024 by Orponen-Shmerkin-Ren-Wang after being open since the 1960s. Study the connections between projection theory and diverse mathematical fields including number theory, homogeneous dynamics, harmonic analysis, and combinatorics. Learn fundamental methods such as the Fourier method, the large sieve technique, and smoothing applications. Investigate classical results including the Szemeredi-Trotter theorem, sum-product theory, and the Bourgain-Katz-Tao projection theorem. Examine advanced topics like random walks on finite groups, homogeneous dynamics, and sharp projection theorems including Beck's theorem and AD regular cases. Gain insight into recent breakthrough developments and understand how projection theory serves as a bridge connecting geometry, analysis, and number theory through the study of shadows and their relationship to original mathematical objects.
