Explore recent developments in unique continuation properties for solutions of elliptic equations in this 52-minute conference talk by Eugenia Malinnikova from Stanford University. Delve into unexpected uniqueness properties of discrete harmonic functions and their connection to Anderson localization in the Anderson-Bernoulli model for two and three dimensions. Discover the solution to the Landis conjecture regarding the decay of real-valued solutions of the Schrodinger equation with bounded potential. Gain insights from joint works with Buhovsky, Logunov, Sodin, Nadirashvili, and Nazarov as part of the Colloque des sciences mathématiques du Québec/CSMQ series at the Centre de recherches mathématiques.