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Explore the mathematical foundations of direct-sum problems through asymptotic spectrum and approximation approaches in this computer science and discrete mathematics seminar. Learn about the fundamental question of computational cost when performing tasks repeatedly and whether economies of scale can be achieved, examining central problems like fast matrix multiplication in algebraic complexity theory and Shannon capacity in graph theory. Discover Strassen's asymptotic spectrum duality approach originally developed for matrix multiplication studies and its recent applications across various mathematical areas. Investigate modern approximation and limit approaches to these challenging problems, including techniques for analyzing "hard" graphs by constructing sequences of easier-to-analyze graphs that converge to them. Understand different notions of convergence and construction methods for such sequences, with examples including asymptotic spectrum distance from Strassen's duality and polynomial-based approaches through algebraic geometry. Gain insights into ongoing research addressing these long-standing open problems in mathematics and computer science through collaborative work spanning multiple institutions and research areas.
