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).

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).

Scientific committee: B. Collas (RIMS, JP), P. Dèbes (Lille, FR), C. Demarche (IMJ-PRG, FR), A. Fehm (TU Dresden, DE) & S. Mochizuki (RIMS, JP)

※ This event is part of the special year ``Arithmetic, Homotopy, and Geometry 2027-28''.