Geometric representation theory and crystals

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Symmetry is a fundamental area of mathematics. Mathematicians are particularly interested in continuous collections of symmetries, such as the group of rotations of a sphere. We call these objects Lie groups. An important problem is to understand how these Lie groups can arise as linear operators on vector spaces, these are called representations. One of the great achievements in mathematics in the late 19th and early 20th century was the classification of simple Lie groups and their representations.Much of this interest in Lie groups is motivated by theoretical physics. Lie groups and their representations have been used heavily since the 1930s in quantum mechanics to help describe and classify elementary particles. In recent years, there has been a resurgence of interaction between quantum field theory and representation theory, leading to important advances on both sides.In the past 20 years, mathematicians have developed geometric constructions in representation theory. In these constructions, we have a geometric object (an algebraic variety) whose topology encodes a representation of a Lie group. These geometric constructions are very beautiful and lead to deeper structures, such as categorification and canonical bases.There are two sources of these geometric constructions, which were invented roughly simultaneously. The first one comes from the theory of geometric Langlands duality, which was developed by Drinfeld as a geometric version of the famous Langlands conjectures, which have guided much of modern number theory. This first construction uses geometric objects called affine Grassmannians. The second construction was developed by Lusztig and Nakajima and involves geometric objects called quiver varieties. The first construction is more natural, but harder to work with; the second construction is more hands-on, but more ad-hoc.The existence of these two different, seemingly unrelated, geometric constructions was very mysterious and lead many mathematicians to the following question:What is the relationship between these two geometric constructions? In 2012, Webster, Weekes, Yacobi and I proposed an answer to this question through the mechanism of symplectic duality, a subtle relationship between certain classes of geometric objects, called conical symplectic resolutions. Symplectic duality has a beautiful origin in theoretical physics, more specifically from N = 4, 3-dimensional supersymmetric quantum field theories. To such a theory, we can consider all possible lowest energy states, which is called the moduli space of vacua. This moduli space has a few pieces, one of which is called the Higgs branch and another of which is called the Coulomb branch. These two branches of vacua form a symplectic dual pair.My current research focuses on exploring this symplectic duality and its consequences for categorification and special bases.