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 must now be confronted with each other for exploiting decisive progress on both sides (e.g., in arithmetic of moduli of higher genus curves, and for Goodwillie-Weiss manifold calculus).
Main Conference
Feb-March, 2028 RIMSCombinatorial arithmetic geometry
In arithmetic homotopy theory, the origin of the combinatorial method can be found in the use of braids and Lie algebra techniques in Galois-Teichmüller theory (Ihara program, Oda's question) in the study of the absolute Galois group of rational numbers, of KZ associators and QTQH algebras, and geometric relations between periods of mixed Tate motives. It then reaches a more systematic level by exploiting Thurston's complex of Teichmüller structures (or, equivalently, the Deligne-Mumford topological stratification of the moduli stacks of curves).
These arithmetic-geometric and geometric approaches have recently been renewed via some categorical encounters and the refinement of structuring questions:
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- Combinatorial anabelian geometry (via log-compatification and semi-graphs of anabelioids) allows combinatorial descriptions of arithmetic and geometric results in higher dimensions (e.g., hyperplane configurations of curves);
- Operads and Quillen model categories (via little 2-discs operads, Drinfeld-Kohno Lie algebra, and chord diagrams) exploit and reveal finer categorical structures with new applications (Hairy graph complex for GT, Kontsevich graph complex for Kashiwara-Vergne);
- Interactions of topology and number theory (with a focus on Graph homology, Topology and geometry of moduli spaces, Grothendieck-Teichmuller group, Johnson homomorphisms, Knots and Primes, see also Satellite)
Each of these directions has resulted in decisive results and progress in both Galois and topological flavors: some BGT combinatorial models for Galois, the anabelian revisiting of some group-theoretic Grothendieck-Teichmüller theory constructions, progress in geometric deformation-quantization, applications to Goodwillie-Weiss manifold calculus, the construction of universal invariants of knots,... In this panorama, homological aspects will act as a complementary guiding beacon.
The goal of this conference is to provide a mature and structured report on these progress for a broad audience so as to act as a catalyst in exploiting further the combinatorial properties of homotopic arithmetic-geometry.
Mini-courses
Combinatorial anabelian geometry
March, 2028 RIMSLecturers: A. Minamide (Zen Univ., JP), S. Tsujimura (RIMS Kyoto, JP)
Graph complexes, mixed Hodge structures, and operads
March, 2028 RIMSLecturers: B. Fresse (Lille, FR), G. Horel (Univ. SPN, FR), T. Willwacher (ETH Zurich, CH)
Workshops
IUT Summit 2028
TBA, 2028Org.: To be announced
Anabelian Geometry Series (opus 5)
March, 2028 TBAOrg.: Y. Ozeki (Kanagawa, JP), Y. Taguchi (IS Tokyo, JP)
Anabelian and p-adic Geometry
March, 2028 RIMSOrg.: A. Mézard (IMJ-PRG, FR), G. Yamashita (RIMS Kyoto, JP), S. Yasuda (Hokkaido, JP)
Satellite Event
Interactions of Topology and Number Theory
March, 2028 TBAOrg.: H. Nakamura (Osaka Univ., JP), T. Sakasai (Univ. Tokyo, JP)
※ 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.