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Explore the fascinating world of fractional dimensions in mathematics through this lecture by Maxim Kontsevich from the Institut des Hautes Études Scientifiques. Delve into various theories of fractal dimensions arising in algebraic geometry, singularity theory, homological algebra, and supersymmetric 2D conformal theories. Discover how these diverse fields converge on a common list of possible fractal dimensions, with the intriguing smallest non-zero number being 1/3. Examine the critical locus of a holomorphic function in several variables as the main running example, viewed from multiple perspectives. Gain insights into how natural numbers traditionally used for cardinalities, ranks, and dimensions of mathematical structures are extended to fractional realms, opening up new avenues for understanding complex mathematical phenomena.
