Objective The proposed project concerns p-adic Hodge Theory, a major area of arithmetic algebraic geometry; it owes its existence to the 1968 observation of John Tate that the well-known Hodge decomposition of the singular cohomology of a complex manifold should have a p-adic analogue, in which the singular cohomology is replaced by the p-adic etale cohomology.There are currently two distinct approaches to the study of this Hodge-Tate decomposition. The first, developed mainly by Bloch, Kato, and Tsuji, uses algebraic K-theory, syntomic complexes, and p-adic vanishing cycles, while the second, more in line with Tate's original ideas, was developed by Faltings using his almost mathematics and purity theorems.The project will resolve a number of outstanding open problems in the field, including the relation between these distinct methods, the resolution of a 1983 conjecture of Bloch on vanishing cycles, and the development of integral results keeping track of p-torsion. It is a particularly timely moment to carry out such a project as the ER has recently developed a new integral p-adic cohomology theory with Bhatt and Scholze, while the Supervisor has recently put Faltings' machinery on a rigorous base in an extended joint work with Gros. The conjunction of the complementary backgrounds of the ER and Supervisor will be central to the project. Specifically, the ER's recent work with Bhatt and Scholze, as well as his expertise in K-theory, topological cyclic homology, de Rham--Witt methods, etc., will be merged with the Supervisor's detailed understanding of the aforementioned works of Bloch, Kato, Tsuji, and Faltings. The theories of perfectoid spaces and pro-etale cohomology, as introduced by Scholze, will play a fundamental role. Fields of science natural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Keywords p-adic Hodge theory vanishing cycles K-theory Programme(s) H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions Main Programme H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility Topic(s) MSCA-IF-2015-EF - Marie Skłodowska-Curie Individual Fellowships (IF-EF) Call for proposal H2020-MSCA-IF-2015 See other projects for this call Funding Scheme MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF) Coordinator INSTITUT DES HAUTES ETUDES SCIENTIFIQUES Net EU contribution € 173 076,00 Address ROUTE DE CHARTRES 35 91440 Bures Sur Yvette France See on map SME The organization defined itself as SME (small and medium-sized enterprise) at the time the Grant Agreement was signed. Yes Region Ile-de-France Ile-de-France Essonne Activity type Research Organisations Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 173 076,00