The main topic of my research is nodal geometry. In the beginning of the 19-th century Napoleon Bonaparte set a prize for the best mathematical explanation of Chladni’s resonance experiments. Nodal sets observed in these experiments fascinated scientists for years, but our mathematical understanding of zeros of solutions of elliptic differential equations, such as nodal sets, remains quite incomplete. During the last two years we have made progress towards proving conjectures of Yau and Nadirashvili on nodal sets and introduced new methods for their study. We believe these methods will prove useful for other problems in partial differential equations and will bring new insights into nodal geometry.

Awards and Achievements