Topological Solitons and their Moduli Spaces

在非线性场论中，拓扑孤子是光滑的、局域的、有限能量解。由于拓扑约束，例如缠绕数或非平凡的陈氏类，孤子数是守恒的。这些孤子最初源于物理，产生了有趣的数学对象，可以用微分几何和代数拓扑学来研究。PHD项目主要集中在双曲空间上的阿贝尔涡以及Cp^1块和Rp^2块，其中区域是一般的二维流形。导出了孤子模空间的几何性质。例如，在某些特殊情况下，可以计算和研究度量和Ricci曲率的显式公式。在其他情况下，可以计算总体积、直径和总曲率等全局信息。此外，还将研究不同类型的拓扑孤子的动力学。最广为人知的动力学是测地线流。最近，一种被称为Ricci磁测地线运动的新的动力学类型被发现。必要的计算将主要是分析性的。然而，得到的公式可能会变得很长，因此使用合适的符号计算机代数包(如Maple)是必不可少的。参考文献：[1]N.S.Manton和P.Sutcliffe，拓扑孤立子，剑桥大学出版社，2004年.

"Topological solitons are smooth, localised, finite energy solutions in non-linear field theories. The soliton number is conserved due to a topological constraint, such as a winding number or a non-trivial Chern class. Originally motivated from physics, these solitons give rise to interesting mathematical objects which can be studied using differential geometry and algebraic topology.The PhD project mainly focusses on Abelian vortices on hyperbolic space as well as CP^1 lumps and RP^2 lumps where the domain is a general two dimensional manifold. Geometric properties of the soliton moduli space will be derived. Forexample, in some special cases, explicit formulas for metric and Ricci curvature can be calculated and studied. In other cases, global information such as the total volume, diameter and the total curvature can be computed. Furthermore,different types of dynamics of topological solitons will be studied. The most well-known dynamics is geodesic flow. Recently, a novel type of dynamics, known as Ricci magnetic geodesic motion, has been discovered. The necessary computations will mainly be analytical. However, the resulting formulas can become lengthy so that the use of a suitable symbolic computer algebra package such as Maple is essential. Some numerical calculations may alsobecome necessary.References:[1] N.S. Manton and P.M. Sutcliffe, Topological Solitons, Cambridge University Press, 2004."