關於酉單值群拉梅方程的一些新結果

New Results on Lame Equations with Unitary Monodromy

吳博生(Po-Sheng Wu)

指導教授 ： 王金龍

國立臺灣大學/理學院/數學研究所/碩士(2021年)

https://doi.org/10.6342/NTU202101949

摘要

本論文利用酉單值群拉梅方程和球狀環面之間的一個對應來研究拉梅方程。最新的研究顯示球狀環面可以透過環面上的沃羅諾伊圖和德勞內三角化來研究。我們將利用該結果來計算給定在給定的酉單值群之下對應的拉梅方程的個數，以及決定從酉單值群拉梅方程的模空間至其所在之橢圓曲線的遺忘映射的結構。我們亦將推廣此結果至數個奇點的情況。

關鍵字

拉梅方程 ； 平均場方程 ； 微分幾何 ； 代數幾何 ； 常微分方程

並列摘要

It is known that generalized Lame equations with unitary monodromy are corresponding to spherical tori with conical singularities. A recent study shows that one can study the geometry of moduli spaces of spherical surfaces with Voronoi diagrams and Delaunay triangulations. In this thesis we will count the number of Lame equation with given unitary monodromy, and formulate a general conjecture to the number of Lame equations with unitary monodromy given the underlying elliptic curve. Moreover, we will generalize some of the previous results to multiple singularities.

並列關鍵字

Lame equation ； mean field equation ； algebraic geometry ； differential geometry ； ordinary differential equation

參考文獻

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[2] Sander R. Dahmen, Counting Integral Lame Equations by Means of Dessins d’Enfants,Trans. Amer. Math. Soc. 359 (2017), no.2, p. 909-922.

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[5] C. Y. Cheng, C. L. Wang and P. S. Wu, Generalized Lame Equation with Even Singu-larities, preprint 2021.