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).
Main Conference
Apr-June, 2027 RIMSHomotopy, rationality, and geometry
Unifying fundamental groups and Galois groups is the core of the homotopic approach in arithmetic geometry. Rationality issues like the search of rational points, or of non-rational points, typically on varieties, but also on stacks, over various ground fields, have benefited from the homotopic approach. Recent progress both in classical arithmetic geometry and in anabelian arithmetic geometry have made new geometric contexts to appear, as can be seen in the following situations that offer a basis for confronting the various viewpoints:
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- The moduli situation, where spaces parametrize geometric objects and reflect their structure and one goal is to investigate their fields of definition (e.g., Hurwitz spaces and components, SL2-torsors and rationally connected varieties, for the inverse Galois problem);
- The refinement of anabelian algorithms, which have revealed the significance of new types of structures (e.g., monoids and quasi-tripods via mono-anabelian reconstructions), new anabelian models of central objects in number theory (e.g., Belyi-Galois-Teichmüller subgroups that approximate the absolute Galois group of rational numbers), and new scopes of applications (e.g., for Diophantine questions).
At a finer level, progress has crystallized around the following structuring topics: rationality obstructions (the Chabauty-Kim method, the section conjecture, the Brauer-Manin/Colliot-Thélène approach to the Noether program), rationality estimates (heights and IUT, the Heath-Brown-Serre dimension growth conjecture, the Batyrev-Manin conjecture, variations around the Malle conjecture), and non-rationality (e.g., variations on the Hilbert specialization property).
In addition to reporting on the overall progress on these themes, this conference will illustrate how the homotopy approach shapes new interfaces with algebraic geometry (e.g., Berkovich analytic geometry or Diophantine geometry) and number theory (revisiting of classical programs such as Ihara’s and Greenberg’s on abelian varieties and Jacobians).
Mini-courses
Variations around the Hilbert property
Apr-June, 2027 RIMSLecturers: A. Behajaina (Lille, FR), C. Demarche (IMJ-PRG, FR), A. Fehm (TU Dresden, DE), A. Javanpeykar (Radboud, NL), D. Neftin (Technion, IL)
Anabelian reconstructions principles
Apr-June, 2027 RIMSLecturers: B. Collas (RIMS, JP), K. Sawada (RIMS, JP), A. Topaz (Alberta, CA), S. Tsujimura (RIMS, JP)
Workshops
Berkovich geometry & Homotopy type
Apr-June, 2027 RIMSOrg.: E. Lepage (IMJ-PRG, FR) & J. Poineau (Caen, FR)
Counting results in geometric Galois theory
Apr-June, 2027 RIMSOrg.: T. Yasuda (Osaka, JP), E. Boughattas (Rennes, FR).
Arithmetic invariants from homotopy Galois theory
Apr-June, 2027 RIMSOrg.: S. Philip (Stockholm, SE), N. Yamaguchi (IS Tokyo, JP)
Satellite Event
Rational obstruction & Noether program
Apr-June, 2027 KyotoLecturers: M. Florence (IMJ-PRG, FR), A. Hoshi (Niigata, JP), D. Izquierdo (IMJ-PRG, FR)
※ Program details, exact event dates and registration information will be announced later. In the meantime, save the date and explore a selection of activities on related topics.