Group theory is a fascinating subject in its own right, but it becomes doubly fascinating in its applications. The applications in physics which I am particularly interested in are in particle physics, quantum mechanics, field theory and solid state physics. The latter includes crystallography as well as the analysis of models and computational methods. Group theory has some small applications in musics, art and architecture. Overt symmetry is often beautiful but a beauty that arises from a hidden or at least nonobvious symmetry is sublime.
Category theory is a unification of many branches of mathematics.
What is beautiful is to find theorems in various branches of mathematics that are all merely instances of one theorem in category theory. Optics and photography. I have had a general interest in photography and started studying optics to help me understand photography better. But the more I study optics the more interesting I find it to be in its own right and now my interest in photography seems to be an adjunct to my interest in optics. Linguistics in all of its various aspects.