Geometric Variational Problems and Rearrangement Inequalities

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This proposal describes a research agenda on non-local shape optimization problems, as they arise in potential theory, biological aggregation, and harmonic analysis. Each project involves at least one student or postdoc. In potential theory, non-local energy functionals result from integrating a singular interaction potential, such as the Newtonian repulsion, over pairs of charged particles. More complex pair interactions of attractive-repulsive type are used in biological models to describe how the collective behaviour of herds and swarms emerges from adding up interactions between individuals. I am interested in shape optimization problems for these energy functionals under suitable geometric constraints. The long-term objective is to characterize not just the optimal configurations but all critical points of the energy functionals, and understand how the energy landscape shapes the resulting classical or and quantum dynamics over long time scales. In harmonic analysis, convolution-type functionals are used to capture information about sum sets and incidence geometry. The general principle is that large values of these functionals are associated with frequent coincidences and small sumsets; they indicate that the densities appearing in the functional share some common additive structure. The long-term objective is to reliably detect the presence of such structure in a variety of situations, and characterize its implications. I am interested in sharp inequalities of isoperimetric type that can be used to quantify the distance of near-maximizers to intervals, convex sets, or Bohr sets in the continuum, and to arithmetic progressions in the discrete setting. These inequalities become more powerful with increasing dimension, giving rise to new concentration inequalities that are awaiting detailed study. I propose 8 problems in these areas. The first two, (P1) and (P2), are extremal Capacitor problems. (P3) concerns minimization of non-local interaction energies that decay at infinity, which is a toy model for biological aggregation. (P4) and (P5) concern rearrangement inequalities on non-Euclidean spaces, specifically Gauss space, spheres, and orthogonal groups. The next three problems (P6)-(P8) touch on related questions in additive combinatorics. Finally, I plan to continue a long-standing collaboration on problems in Computer Communication Networks.