Automorphic Forms, Crystallization in the Plane, and Arthur’s Unitarity Conjecture

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This award centers around two central mathematical problems. The first comes from mathematical physics and discrete geometry, and is known as “Wigner crystallization” after the physicist Eugene Wigner’s landmark 1934 paper. Here one wishes to give a rigorous proof explaining why electrons (or, more generally, certain molecules) arrange themselves in a crystalline, honeycomb-shaped grid. This is related to the sphere packing problem (which asks to find exactly how densely space can be filled by equal-sized, non-overlapping balls), and the PI and collaborators plan to use tools from number theory to attack the problem. The second problem is an important case of the “Unitary dual problem”, which more generally asks to determine all the possible incarnations of a symmetry group which preserve distances (known as “unitary representations”). Conjectures of James Arthur predict exotic, special types of unitary representations, which are of special interest because they appear to be related to number theory (and also to string theory, in many cases). The bulk of the funding from the award will support graduate students working on aspects of these problems.The proposed plan of attack for Wigner crystallization is to prove that the hexagonal lattice is universally optimal in the plane, in that it minimizes potential energy for any completely monotonic function of distance-squared. Such a result was proven by the PI and collaborators in 8 and 24 dimensions, and is known (by work of Mircea Petrache and Sylvia Serfaty) to imply Wigner crystallization. This will require developing more machinery from classical modular forms, which were the key ingredient in 8 and 24 dimensions, since the problem is quite different in two dimensions. Arthur’s conjecture will also be attacked via automorphic forms, to construct realizations of Arthur representations (when possible) using Eisenstein series. Other tools will be the unitarity algorithm which has now been implemented as part of the Atlas of Lie Groups software package, as well as important results of Adams-Barbasch-Vogan on the structure of Arthur packets.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.