Defects in Triply Periodic Minimal Surfaces

三周期最小表面（tpms）是一种奇妙的几何物体，它结合了局部最小区域的肥皂泡之美和重复图案的晶体。它们在生物学和材料科学中被广泛观察到。提出的项目旨在引入晶体缺陷，即周期模式的不完善，到tpms。该项目于2018年6月启动。它的灵感来自于对材料科学的观察。该项目的主要成果是使用Traizet的节点开放技术严格构建平面缺陷。此外，通过有意打破对称性，近20年来首次明确构建了令人惊讶的3属新tpms。到目前为止的结果大大扩展了我们对tpms的认识，拓宽了我在数学和物理方面的研究范围。我提议把这个项目再延长一年。鉴于迄今为止的成就，扩展将更多地关注严格的数学结构。特别是，我计划通过粘接Scherk塔来构建线缺陷，并研究粘接螺旋的可能性。这些项目再次受到生物系统中观察到的结构的推动。这些结果不仅可以进一步提高我们对TPMSs的理解，还可以进一步提高我们对极小曲面模空间的理解。

Triply periodic minimal surfaces (TPMSs) are fantastic geometric objects that combine the beauty of soap bubbles with locally minimized areas and crystals with repeating patterns. They are widely observed in biology and material science. The proposed project aims to introduce crystallographic defects, i.e. imperfections of the periodic patterns, into TPMSs.The project started in June 2018. It was motivated by observations in material sciences. The major achievement of the project is the rigorous constructions of planar defects using Traizet's node-opening technique. Moreover, by intentionally breaking symmetries, surprising new TPMSs of genus three were explicitly constructed for the first time in almost 20 years. The results so far significantly expanded our knowledge of TPMSs and widened my scope of research both in mathematics and physics.I propose to renew the project for another year. In view of the achievement so far, the extension will focus more on rigorous mathematical constructions. In particular, I plan to construct line defects by gluing Scherk towers and investigate the possibility of gluing helicoids. These projects are again motivated by structures observed in biological systems. The results are expected to further improve our understanding not only on TPMSs but also on the moduli space of minimal surfaces in general.