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Attend a research seminar exploring effective computations for variations of integral Hodge structures, focusing on weakly special loci and their computational properties. Examine the conjecture established by Daw, Ren, and Javanpeykar-Kühne regarding the finite nature of non-factor special subvarieties of bounded degree in Shimura settings, and discover how Urbanik's proof provided an explicit algorithm for this problem. Learn about recent advances by Klingler-Otwinowska-Urbanik demonstrating that non-factor special subvarieties are defined over algebraic numbers when the variation itself is defined over algebraic numbers. Explore ongoing research with Binyamini and Lerer that establishes the existence of an effectively computable polynomial bound on the complexity of defining polynomials for non-factor special subvarieties, where complexity encompasses degree, coefficient log-height, and field of definition degree. Understand how this work leads to polynomial-strength separation bounds for non-factor special subvarieties in the period domain, bringing researchers close to resolving the Lairez-Sertoz conjecture in this area of algebraic geometry and Hodge theory.
