Theme of the Research Project
Arithmetic homotopy geometry (AHG) exploits arithmetic and geometric invariants via homotopy theory and especially the theory of fundamental groups (in its étale, Galois, motivic, and topological versions). It thrives and revisits leading questions and constructions on both ends of the number theory-geometry spectrum, with the following structuring topics: (A) Homotopy, Rationality, and Geometry, (B) The Homology-Homotopy Frontier, and (C) Combinatorial Arithmetic Geometry.
The ``Arithmetic Homotopy Geometry'' Project is a year-long program whose goal is to shape the lines of arithmetic homotopy geometry in terms of new leading questions and conjectures for the next-generation of arithmetic homotopy geometers.
Scientific Committee
- Benjamin Collas°, RIMS Kyoto University, JP
- Pierre Dèbes, Lille University, FR
- Benoit Fresse, Lille University, FR
- Yuichiro Hoshi, RIMS Kyoto University, JP
- Minhyong Kim, ICMS Edinburgh, UK
- Ariane Mézard, IMJ-PRG, FR
- Akio Tamagawa, RIMS Kyoto University, JP
Scientific Program & Seasons
Homotopy, rationality, and geometry
Scientific Committee: B. Collas (RIMS Kyoto, JP), P. Dèbes (Lille, FR), C. Demarche (IMJ-PRG, FR), A. Fehm (Dresden, DE), S. Mochizuki (RIMS Kyoto, JP)
Homotopic methods have deepened our understanding of rationality phenomena, from rational obstructions (non-abelian Chabauty, section conjecture, Brauer–Manin) to estimates in dimension growth and height estimates (IUT, Heath-Brown-Serre conjecture, Batyrev-Manin conjecture, Malle conjecture) and non-rationality (Hilbert specialization property).
Moduli situations (Hurwitz spaces and SL2-torsors) provide new contexts for geometry and arithmetic to interact. Anabelian algorithms reveal new structures (e.g., monoids and quasi-tripods) and new models for central objects of number theory (e.g., BGT subgroups wrt the absolute Galois group of rational numbers).
- Mini-courses
- Conference
- Workshops
- Satellite
The homology-homotopy frontier in arithmetic geometry
Scientific Committee: A. Betts (Cornell, US), B. Collas (RIMS Kyoto, JP), M. Kim (ICMS, UK), Y. Yatagawa (IS Tokyo, JP)
The two arithmetic universalities frameworks, (linear) motivic theory and (non-linear) anabelian homotopy theory now increasingly overlap: (a) nearly-abelian reconstructions (m-step reconstruction, abelian-by-central, section conjecture), (b) non-abelian Chabauty (pro-unipotent, m-step unipotent quotient, Selmer sections), and (c) monodromy method in local systems illustrate this new interface.
In these settings, derived perverse formalism, mixed Hodge structures, and p-adic Hodge techniques (esp. period maps) indicate special loci of interest for finer control of arithmetic properties; étale topological type provides an anabelian-motivic bridge for progress on both sides of the frontier.
- Mini-courses
- Conference
- Workshops
- Satellite
Combinatorial arithmetic geometry
Scientific Committee: A. Alekseev (Genève, CH), B. Collas (RIMS Kyoto, JP), B. Fresse (Lille, FR), Y. Hoshi (RIMS Kyoto, JP)
Original combinatorial methods (braid groups, Lie algebras, DM-compactification) have matured into categorical tools for refinement of structuring questions of arithmetic homotopy theory (e.g., Ihara program, Oda’s question): (a) combinatorial anabelian geometry
attacks higher dimensional arithmetic, (b) operads and Quillen model categories link Drinfeld-Kohno Lie algebra, hairy graph complex, and Galois-Teichmüller theory, all together for (c) further number theory-topology interactions (e.g., graph homology, Johnson homomorphisms, knots and primes, and associators).
These (anabelian) arithmetic and topological combinatorics are now confronted for decisive progress on both sides.
- Mini-courses
- Conference
- Workshops
- Satellite
※ Program details, event dates, and registration information will be announced soon. In the meantime, save the dates and explore a selection of activities on related topics.
Sponsors
This program is organized with the support of the Research Institute for Mathematics Sciences of Kyoto University (RIMS), the Centre National de la Recherche Scientifique (CNRS), the Japan Society for the Promotion of Science (JSPS-Kakenhi), and the AHGT Project.
