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Explore the theoretical foundations of homomorphism indistinguishability in this 58-minute conference talk from the Simons Institute. Delve into Lovász's 1967 theorem establishing that two graphs are isomorphic if and only if they are homomorphism indistinguishable over all graphs, meaning they have equal numbers of homomorphisms from any graph F. Examine how this concept has evolved to characterize natural relaxations of graph isomorphism across diverse fields including quantum information theory, algebraic graph theory, convex optimization, and category theory. Discover recent applications in analyzing graph learning architectures and understand the development of a comprehensive theory addressing two fundamental questions: determining the distinguishing power of homomorphism counts from graphs in a given class, and establishing the computational complexity of deciding homomorphism indistinguishability over restricted graph families. Learn how this framework provides crucial insights into both the descriptive and computational aspects of graph isomorphism relaxations in modern theoretical computer science.
