"The proposed research program has three main objectives. The first and second objectives are to seek extreme precisions in optical atomic spectroscopy and optical clocks, and to use this quest as a mean of exploration in atomic physics. The third objective is to explore new possibilities that stem from extreme precision. These goals will be pursued via three complementary activities: #1: Search for extreme precisions with an Hg optical lattice clock. #2: Explore and exploit the rich Hg system, which is essentially unexplored in the cold and ultra-cold regime. #3: Identify new applications of clocks with extreme precision to Earth science. Clocks can measure directly the gravitational potential via Einstein’s gravitational redshift, leading to the idea of “clock-based geodesy”.The 2 first activities are experimental and build on an existing setup, where we demonstrated the feasibility of an Hg optical lattice clock. Hg is chosen for its potential to surpass competing systems. We will investigate the unexplored physics of the Hg clock. This includes interactions between Hg atoms, lattice-induced light shifts, and sensitivity to external fields which are specific to the atomic species. Beyond, we will explore the fundamental limits of the optical lattice scheme. We will exploit other remarkable features of Hg associated to the high atomic number and the diversity of stable isotopes. These features enable tests of fundamental physical laws, ultra-precise measurements of isotope shifts, measurement of collisional properties toward evaporative cooling and quantum gases of Hg, investigation of forbidden transitions promising for measuring the nuclear anapole moment of Hg.The third activity is theoretical and is aimed at initiating collaborations with experts in modelling Earth gravity. With this expertise, we will identify the most promising and realistic approaches for clocks and emerging remote comparison methods to contribute to geodesy, hydrology, oceanography, etc."