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Explore a comprehensive lecture on iterated beta integrals, a type of iterated integrals on branched universal abelian coverings of the projective line. Delve into the recent work conducted by Nobuo Sato and Minoru Hirose at Nagoya University, examining how these integrals generalize hyperlogarithms and beta integrals. Discover the unique property of translation invariance and its implications for equalities between iterated integrals on translation-equivalent coverings. Investigate how this property leads to non-trivial equalities between special values of hyperlogarithms for genus-zero pairs of translation-equivalent coverings. Examine famous formulas such as Zagier's 2-3-2 formula for multiple zeta values, its analog for multiple t-values, and the 2-1 formula for multiple zeta star values. Learn about the classification of all genus-zero pairs and the discovery of new interesting cases, including the resolution of Charlton's conjecture concerning the area of a family of Lawson surfaces.
