Symmetries of quasiplatonic Riemann surfaces

Generalised Fermat Hypermaps and Galois Orbits

We consider families of quasiplatonic Riemann surfaces characterised by the fact that -- as in the case of Fermat curves of exponent $n$ -- their underlying regular (Walsh) hypermap is the complete bipartite graph $ K_{n,n} $, where $ n $ is an odd prime power. We will show that all these surfaces, regarded as algebraic curves, are defined over abelian number fields. We will determine the orbits under the action of the absolute Galois group, their minimal fields of definition, and in some easier cases also their defining equations. The paper relies on group-- and graph--theoretic results by G.…

Symmetries of partial differential equations

We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in C^n. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise upper bound of the dimension of this Lie group for some specific systems of partial differential equations.

四元数单位球上切片黎曼几何的李理论

Lie theory of the slice Riemannian geometry on the quaternionic unit ball

本文研究四元数单位球上的切片黎曼度量，证明其几何与李群Sp(1,1)紧密相关，计算其等距群，并与四元数庞加莱几何对比。

具有对称性的极小曲面

Minimal surfaces with symmetries

本文研究了有限群或离散无限群在黎曼曲面与欧氏空间上作用时$G$-等变共形极小浸入的存在性条件，并推广至具有限总高斯曲率完备端的极小曲面情形。