Speaker : 1. 張聖容 院士 (美國普林斯頓大學 Eugene Higgins 數學講座教授) 2. 于靖 院士 (國立臺灣大學數學系講座教授)
Title : 1. Conformal Invariants: Geometric and Analytic Aspects 2. A Start with the Linear Independence over Rationals
Time : 2014-06-28 (Sat) 09:10 - 11:30
Place : 2nd Conference room, H.S.S. Building of Academia Sinica
Abstract: 1. Conformal invariants play important roles in the study of many problems in geometry and mathematical physics. In the talk, I will survey properties of a class of integral conformal invariants and their connections to geometric quantities in the ADS/CFT setting. Special emphasis will put be on the tool of fully non-linear PDE in the study of these invariants. 2. Logarithms of algebraic numbers have the mysterious property (Alan Baker 1966) that if they are linearly independent over the rational numbers then they are in fact linearly independent over all the algebraic numbers. This was a major conjecture after Hermite-Lindemann first confirmed the transcendence of these logarithms in late 19th century. While Baker's theorem is still the only verified incidence of this linear independence phenomenon in the classical world, in the last 30 years we have discovered that analogous theorems do hold quite frequently for families of natural transcendental quantities in the positive characteristic world. We are led to conjecture that this "independence" assertion should be true also for many classical families of special values, in particular for (higher degree) polylogarithms of algebraic numbers, as well as for the family of multiple zeta values of any given weight. We will explain the "motivic" transcendence theory behind the scene which may eventually help solving these wide open problems.