Eyvindur Ari Palsson
The two main themes of my research program are multilinear operators and point configurations. Many concepts in mathematics, from boundary value problems in partial differential equations and mathematical physics to finite point configurations in geometric combinatorics, are fundamentally tied to various operator bounds. This frequently leads to the estimation of linear and multilinear operators using techniques from harmonic analysis, often combined with geometric and combinatorial principles.
A particularly rich source of inspiration is a famous question of Erdös where he asked about the least number of distinct distances determined by points in the plane. Instead of distance, which is a simple pattern, I mainly study multipoint configuration versions of this problem, both in the original discrete setting, as well as in a continuous setting where geometric measure theory plays a key role. These point configuration problems have direct connections to big data.
Geometric averaging operators, such as averaging a function over a sphere, play a key role in many of my investigations and are of independent interest. A particularly important class are multilinear analogues of linear generalized Radon transforms. In addition to applications to geometric measure theory and combinatorics, further investigation of these multilinear generalized Radon transforms led to applications to restriction theory, partial differential equations, Sobolev trace inequalities and multilinear analogues of Stein's spherical maximal operator theorem.
My work on multilinear singular integral operators goes all the way back to my thesis. I used time-frequency analysis to study both Lp estimates, variational estimates and sparse estimates for such operators. These operators are frequently motivated by, and have potential applications to, both non-linear partial differential equations and ergodic theory.