标题： 空间与数的对称性——阿贝尔几何 显示英文标题

标题： Symmetries of spaces and numbers -- anabelian geometry

摘要： “数与几何空间能否从其对称性中重建？” 这个问题是阿贝尔几何的核心，这一理论是在国际社区的共同努力下发展起来的，有许多变体，日本算术学派是其中的核心。如果答案是肯定的，它说明了同伦方法在算术几何中的普遍性。 从简单的例子开始，我们首先介绍该理论的动机和指导原则，然后展示其最具结构性的结果和当代趋势。 结果是，读者将看到一个丰富而多样的数学图景，它依赖于理论和显式方法，从数论延伸到拓扑学。 显示英文摘要

摘要： ``Can number and geometric spaces be reconstructed from their symmetries?'' This question, which is at the heart of anabelian geometry, a theory built on the collaborative efforts of an international community in many variants and with the Japanese arithmetic school as a core, illustrates, in the case of a positive answer, the universality of the homotopic method in arithmetic geometry. Starting with elementary examples, we first introduce the motivations and guiding principles of the theory, then present its most structuring results and its contemporary trends. As a result, the reader is presented with a rich and diverse landscape of mathematics, which thrives on theoretical and explicit methods, and runs from number theory to topology.