Motivic Integral p-adic cohomologies

This project aims to study p-adic cohomologies of varieties using tools from motivic homotopy theory. Voevodsky's theory of motives has played a crucial role in solving deep mathematical conjectures. However, motives intrinsically lack a theory of tale p-adic realizations. In this project, we will use logarithmic geometry tools to generalize the motives category and overpass this problem. More specific goals are related to:Develop a theory of integral log-tale motives and realizations. Prove a general comparison between the log-tale p-adic realizations and tame cohomologiesDevelop a theory of motives over log points with an integral Hyodo-Kato realizationSolve structural problems in the theory of motives of logarithmic schemesMIPAC is an innovative project in motivic homotopy theory built to impact several areas within motivic and arithmetic geometry. The project will be completed at the University of Milan, in a leading multi-disciplinary and collaborative environment. I will bring extensive experience in log motives and some unique expertise on non-A1-invariant cohomology theories. I will benefit of the experience and knowledge of the groups of Algebra and Geometry in p-adic cohomologies and motivic homotopy theory. This will facilitate the research in the group and the transfer of knowledge, and expand my experience and intuition, transferable skills, and professional networks. Carrying out MIPAC within a Marie Skodowska-Curie Fellowship will enhance the development of my career as a complete and independent leading researcher, with a reinforced position within arithmetic and motivic geometry. The leading position of the groups and the department will ensure a great network of international researchers with an impact to the disseminations of the ideas of MIPAC.