The synthetic study by real algebraic geometry, from sub-Riemannian geometry to tropical geometry

对于与次黎曼几何和热带几何有关的外微分系统的积分曲线上的奇异曲面，我们实现了奇点与实代数几何的分类，并完成了它们的一般规范形。从Legendre对偶和控制论出发，我们将切向簇的奇点分类为有框架的曲线曲面的奇点，发展了映射的开度的概念，并将其应用于切向簇到一般子流形的奇点的分类。我们从非线性控制理论和实代数群的表示理论两个方面研究了G2次黎曼几何，并对相关的奇点进行了分类。此外，我们还发展了D4几何的三重性和D4奇点理论。我们已经完成了所有上述主题的论文，这些论文都已经在国际学术期刊上发表或正在提交。

For singular surfaces associated to integral curves of exterior differential systems related to sub-Riemannian geometry and tropical geometry, we have realized the classification of singularities from real algebraic geometry, and we have completed their generic normal forms. From Legendre duality and control theory, we have classified singularities of tangential varieties to singularities of framed curves and surfaces, we have developed the notion of opening of mappings, and we have applied it to the classification for singularities of tangential varieties to general submanifolds. We have studied G2 sub-Riemannian geometry from non-linear control theory and the representation theory of real algebraic groups, and we have classified the related singularities. Moreover, we have developed the triality of D4 geometry and D4 singularity theory. We have completed the papers for all above themes, all of which already appeared or are under submission in international academic journals.