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Explore fundamental questions in mathematics through this lecture that examines transformation problems across geometric, combinatorial, algebraic, and algorithmic domains. Begin with the classic "14-15 puzzle" attributed to Sam Loyd as a gateway to understanding how mathematical problems can be framed as configuration transformation challenges. Investigate the central question of whether and how one configuration can be transformed into another through specific types of moves. Delve into motion planning problems developed in collaboration with Herbert Edelsbrunner, examining the mathematical foundations underlying pathfinding and movement optimization. Analyze the Kovaldzhi and Brunck puzzle, which incorporates linear algebraic concepts and was studied in joint research with Gábor Tardos. Discover the connections between seemingly disparate mathematical areas through the unifying theme of transformation sequences. Encounter challenging open problems in the field that remain unsolved, providing insight into current research frontiers and potential areas for future mathematical investigation.
