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.
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