【报告摘要】 Elliptic and more generally abelian functions were discovered in the 19th century through the study of the integration of algebraic differential forms. In the middle of the 20th century, Manin showed how to use the associated differential equations to prove a function field version of the Mordell conjecture about rational points on algebraic curves. It has been observed that Manin's construction is related to the work of Fuchs, Gauss, and Picard on linear differential operators. Over the years, Manin's construction has been reinterpreted in various ways , most notably using differential algebra. In this lecture, in addition to this history, I will describe some of my own work (joint with Dupuy and Freitag) applying methods from mathematical logic allowing us to relate the classical analytic theory of these differential equations with the more algebraic formulation to give new, effective finiteness theorems for related problems in Hodge theory and function field arithmetic.

【报告人简介】 Thomas Scanlon is a professor in the Department of Mathematics at the University of California, Berkeley. His main research areas are model theory and its applications in number theory and algebraic dynamics. He obtained his PhD degree from Harvard University under the supervision of Ehud Hrushovski in 1997. Professor Scanlon was an invited speaker at the 2006 International Congress of Mathematicians. His work was published in prestigious journals including J. Amer. Math. Soc., Ann. Math., Inventiones Math., etc.. He also served as an associate editor of J. Amer. Math. Soc., APAL, etc..