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Explore how artificial intelligence can be applied to mathematical research in this conference talk examining the Erdős conjecture about independence sequences in trees. Learn about the mathematical background of independence sequences, where a_i represents the number of independent vertex sets of size i in a graph, and discover how this sequence's unimodality properties have been studied since the 1990s. Understand the progression from Erdős's original conjecture about unimodality in trees to the stronger log-concavity conjecture, which requires that a_i² ≥ a_{i-1}a_{i+1} for all indices. Follow the computational breakthroughs that led to finding the first counterexamples in 2023 with trees of 26 vertices, and examine the three known families of trees that violate log-concavity. Discover how the PatternBoost machine learning architecture was employed to systematically search for new counterexamples, resulting in the identification of tens of thousands of previously unknown cases with vertex counts ranging from 27 to 101. Gain insights into both the remarkable successes and intriguing limitations of this AI-driven approach to mathematical conjecture testing, demonstrating how modern computational methods can advance pure mathematical research in graph theory and combinatorics.
