NTU Math Special Talk
 
演講者：Dr. Kuo, Ting-Jung (TIMS, National Taiwan University)
講  題：Painlevé VI equation and its application
時  間：2016年2月22日 (星期一) 15:30~16:20
地  點：臺灣大學天數館440室
摘  要：Painlevé VI equation is a 2nd order nonlinear complex ODE with four parameters and denoted by PVI(α,β,γ,δ). For PVI((1/8),((-1)/8),(1/8),(3/8)) (the Hitchin equation), Hitchin in 1995 proved a beautiful theorem so called the Hitchin theorem which gives explicit expression for a class of solutions (namely completely reducible solutions) to PVI((1/8),((-1)/8),(1/8),(3/8)). In this talk, we first give a complete classification of solutions to the Hitchin equation study the problem: How many singular points of a solution λ(t) to PVI((1/8),((-1)/8),(1/8),(3/8)) might have in C∖{0,1}? Here t₀∈C∖{0,1} is called a singular point of λ(t) if λ(t₀)∈{0,1,t₀,∞}. Based on this classification theorem, we proved that there are only three solutions that are smooth on C\{0,1}. Second, we will generalize the Hitchin theorem to the case PVI((1/2)(n+(1/2))²,((-1)/8),(1/8),(3/8)) where n∈N. As an important application of this generalization, we proved the pre-modular form obtained by Chai, Lin and Wang has simple zeros only and then confirm the Dahmen-Beukers conjecture.