Enveloping algebras of infinite-dimensional Lie algebras

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Mathematicians are interested in symmetry, and often model symmetry through an algebraic structure called a ring. Most rings encountered "in nature" are noncommutative: the order of operations matters. In the real world the order of operations also matters: putting on your socks before putting on your shoes gives a different result than putting on your shoes before your socks! Less frivolously, the order also matters when moving in three-dimensional space, which is why most graphics software (such as video games, and also medical imaging software) uses a noncommutative ring called the quaternions to do calculations.The symmetries of a geometric object are often modelled through an object called a Lie algebra. Lie algebras, in turn, are associated with noncommutative rings called enveloping algebras. Lie algebras are often studied through their representations, which echo the symmetry encoded in the Lie algebra. The properties of the Lie algebra and the enveloping algebra tend to depend, subtly and powerfully, on the structure of representations of the Lie algebra.The usual geometric objects that mathematicians study have finitely many dimensions: for example, the space we move around in is three-dimensional. In order to do the delicate and complicated calculations involved in quantum mechanics, however, physicists need to study spaces that have infinitely many dimensions. Their symmetries are encoded in infinite-dimensional Lie algebras.A famous infinite-dimensional Lie algebra is called the Virasoro algebra, which is renowned in mathematics and physics. It may be viewed as a mathematical model of statistical mechanics, and so is of deep importance to physics. Infinite-dimensional Lie algebras and their enveloping algebras are famously difficult to understand. For example, it has been known for almost 100 years that the enveloping algebras of finite-dimensional Lie algebras have a property called 'noetherian', named for the German mathematician Emmy Noether. Rings that are noetherian are relatively well-behaved; those that are not noetherian are more exotic. However, nobody knows if it is even possible for the enveloping algebra of an infinite-dimensional Lie algebra to be noetherian. This question was first asked in print 45 years ago, and very little progress had been made on it until I proved, in 2013, that the enveloping algebra of the Virasoro Lie algebra is not noetherian. This proof used the geometry of representations of the Virasoro algebra and so demonstrated the power of geometric techniques to understand algebraic problems.The main objective of this proposal is to prove that it is not possible for an infinite-dimensional Lie algebra to have a noetherian enveloping algebra. I will do this through a variety of methods, many focused on understanding the geometry of families of representations of infinite-dimensional Lie algebras. Understanding this will have applications to physics as well as other areas of mathematics.