ERC Starting Grant for Ryomei Iwasa

The prestigious ERC Starting Grants are aimed at talented young researchers who want to establish their research teams. Each grant is designed to create ideal conditions for an already promising research career with the opportunity for in-depth independent research.

Ryomei is presently working as a postdoc at Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay. He is a former postdoc of the SYM and GeoTop centres and has chosen the University of Copenhagen as his host institution for his ERC Starting Grant. He will come back to join the department again in April 2025.

Ryomei completed his PhD in 2018 at the University of Tokyo under the supervision of Thomohide Terasoma. His thesis was titled "Homology pro stability for Tor-unital pro rings". He was then employed as a postdoc in Copenhagen for four years, with Lars Hesselholt and Jesper Grodal as supervisors. In 2020 he won the prestigious Marie Skłodowska-Curie fellowship. After Copenhagen, he became a postdoc at Université Paris-Saclay but has frequently been a guest here at MATH.

Ryomei has received 1.5 million EUR for a five-year research project, starting 1 April 2025, during which he will hire two PhD students (2025 and 2026) and two postdocs (e.g. 2025 and 2027).

The project proposal

The project is called “Motivic Stable Homotopy Theory: a New Foundation and a Bridge to 𝑝-Adic and Complex Geometry” (MOSHOT).

The project is centred on the field of algebraic geometry and involves homotopy theory and analytic geometry. The overall goal is to unveil the underlying principles of a large variety of cohomology theories in algebraic and analytic geometry and develop robust foundations that facilitate the study of those cohomology theories from the vantage point of homotopy theory.

This will be achieved through innovations of motivic stable homotopy theory beyond the current technical limitations of 𝔸1-homotopy invariance. In addition, its interdisciplinary perspective will be advanced, especially in relation to 𝑝-adic geometry and complex geometry. The research proposal consists of five main objectives, which are organically related to each other.

• The first objective is to establish a six functor formalism, which would be the most important challenge in non-𝔸1-invariant motivic stable homotopy theory.
• The second objective is to investigate the kernel of the 𝔸1-localization and aims to describe it in terms of 𝑝-adic or rational Hodge realization, following the principle of trace methods of algebraic K-theory. In particular, in the 𝑝-adic context, this will lead to the 𝑝-adic rigidity, which will conclusively connect motivic homotopy theory with 𝑝-adic geometry.
• The third objective is to find out the potential of unstable motivic homotopy theory and develop calculation techniques.
• The fourth objective is to establish a general and universal construction of motivic filtration of localizing invariants, such as algebraic K-theory and topological cyclic homology.
• The last objective is to explore the analogue in complex geometry, which is an interesting unexplored subject that will pave the way for further developments of motivic homotopy theory for a broader range of analytic geometry.