Geometric Representation Theory and W-algebras

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In the early nineteenth century it was impossible to draw a distinction between mathematicians and physicists, since the greatest scientists worked on every important problem in these fields. One of the most influential polymaths of the era was Emmy Noether, who developed all of the foundational theories which inspired this research project. Her greatest contribution to physics was probably Noether's first theorem, which says that if you want to understand the conservation laws of the universe then it suffices to understand the symmetries of the universe. Conservation laws are the most fundamental laws of physics, giving us clues about the nature of matter, and the shape of space, and so Noether's theorem started a wave of discovery which has been growing and growing for over a hundred years, as mathematicians and physicist seek to understand the symmetries of the universe. Today mathematicians and physicists are much easier to distinguish, however the subjects are still deeply intertwined. In modern day mathematical language, the study of symmetries is called representation theory and the goal of this project is to understand how Noether's algebraic structures can be expressed as symmetries. To rephrase this, my objective is to understand the representations of certain important families of algebras.The ancient Greeks believed that all matter could be built up from indivisible pieces - the word "atom" literally means "indivisible" - and in the language of modern particle physics it is well-understood that all matter in the universe can be built up from the fundamental particles. In precisely the same way, the representations I seek to understand are also built from fundamental building blocks, known as irreducible representations. Can we describe these irreducible representations explicitly? Can we determine their structure and calculate their dimensions? In this research project we will answer these fundamental, elusive questions by relating each representation to an important geometric space, known as a symplectic leaf of a Poisson variety.Some of the most important unanswered questions in this field pertain to algebras which we call "modular": this is because the underlying number system is not linear, like the real number line, but is circular like the numbers on the face of a clock. Questions in modular representation theory tend to be significantly harder due to the added complexity of the geometry and the arithmetic.By working with tools on the interface between abstract algebra and geometry this project will make substantial exciting progress in some of the most challenging problems in modular representation theory, showing that Noether's wave of discovery is still growing on the ocean of mathematics.