INVARIANT ALGEBRAS IN HYPERBOLIC GEOMETRY

在数学中，突破往往来自于改变视角，将问题嵌入到新的背景中，用新的眼光看待它。通过这样做，数学领域之间的联系似乎遥远的出现，往往是新的，富有成效的联系建立。这些反过来又提供了新的强大工具来解决长期存在的问题，这些问题到目前为止一直抵制其他方法。该项目首次将自守李代数和模形式结合在一起，并源于意识到在上半平面上构建不变代数揭示了向量值模形式背后的新结构，实际上为它们配备了代数结构。这一观察开启了一个全新的、完全未经探索的、令人兴奋的研究，将两个不同的世界联系起来。但是自守代数首先是什么呢？为了回答这个问题，首先考虑等变向量。等变向量是具有理想对称性质的映射。它们出现在数学的许多领域，包括动力系统、量子物理、数论和可积系统。所有这些向量的集合在加法下是闭的，这是一个广泛使用的事实。但在某些情况下，这个集合在乘法下也是封闭的，这是一个尚未充分利用的事实。在这种情况下，等变向量的集合称为自守代数。自守代数最著名的非平凡例子是卡茨-穆迪代数，它在数学和数学物理的许多分支中无处不在。一个较早的，也许不太为人所知的例子，是昂萨格在他的工作介绍了李代数的晶体统计，而最近的例子包括自守李代数相关联的有限对称群的柏拉图固体，作为研究的背景下，可积系统。所有这些例子都源于球面几何。它们是球面上的映射，其对称性对应于球面本身的旋转。然而，另一个著名的非欧几何，双曲几何，有更丰富的对称结构，包括模群，这是解释了许多美丽和优雅的数学。双曲几何中的自守代数将探索这种新的，迷人的联系。

In mathematics often breakthroughs come from changing perspective and embedding a problem into a new context, looking at it with new eyes. By doing so, connections amongst areas of mathematics seemingly distant appear and often new, fruitful links are established. These, in turn, provide new powerful tools to solve long-standing problems, which resisted other methods so far. This project brings for the first time together automorphic Lie algebras and modular forms and stems from realising that constructing invariant algebras on the upper half plane uncovers a new structure behind vector-valued modular forms, and in fact equip them with an algebra structure. This observation opens a new, entirely unexplored, exciting research, connecting two different worlds. But what are automorphic algebras in the first instance? To answer this question, consider first equivariant vectors. Equivariant vectors are maps with ideal symmetry properties. They appear in many areas of mathematics including dynamical systems, quantum physics, number theory and integrable systems. The set of all such vectors is closed under addition, a fact that is extensively used. But in certain instances, this set is also closed under a multiplication, a fact that is not yet fully exploited. In such a case, the set of equivariant vectors is called an automorphic algebra. The most famous nontrivial examples of automorphic algebras are the Kac-Moody algebras, which are ubiquitous in many branches of mathematics and mathematical physics. An earlier, perhaps less known, example, is the Lie algebra introduced by Onsager in his work on crystal statistics, while more recent examples include automorphic Lie algebras associated the finite symmetry groups of Platonic solids, as studied in the context of integrable systems. All these examples germinate from spherical geometry. They are maps on the sphere which have symmetries corresponding to rotations of the sphere itself. However, the other famous non-Euclidean geometry, hyperbolic geometry, has a much richer structure of symmetries, including the modular group, which is accountable for a lot of beautiful and elegant mathematics. Automorphic algebras in hyperbolic geometry will explore this new, fascinating connection.