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Explore the theory of total positivity in this comprehensive lecture, delving into its applications across various mathematical fields. Learn about the concept of totally positive matrices and Lusztig's groundbreaking work in extending this theory to split real reductive groups and flag manifolds. Discover how total positivity has been further generalized to Kac-Moody groups and its significant impact on cluster algebras, higher Teichmuller theory, and the theory of amplituhedron in physics. Examine the remarkable combinatorial, geometric, and representation-theoretic aspects of total positivity, drawing from recent research conducted by the speaker and collaborator Huanchen Bao. Gain insights into the mathematical foundations and practical applications of this fascinating theory over the course of this hour-long presentation.
