An international seminar series on recent progress in four areas of arithmetic and anabelian geometry, with a focus on hot topics of the field. Each talk surveys the state of the art and highlights the latest development, offering a continuous view of the evolving landscape of mathematical sciences.
The Grothendieck–Teichmüller Lie algebra \(\mathfrak{grt}_1\), introduced by Drinfeld through the theory of associators, contains elements σ₃, σ₅, σ₇, … in each odd degree. The Deligne–Drinfeld conjecture predicts that \(\mathfrak{grt}_1\) is freely generated by these elements; equivalently, that the motivic Galois group of mixed Tate motives over ℤ exhausts the graded Grothendieck–Teichmüller group. Wilwacher (2026), building on work of Brown (2012) gives one inclusion.
By the Merkurjev–Suslin and Bloch–Kato (Rost–Voevodsky) theorems, the mod-ℓ Milnor K-ring of a field coincides with its mod-ℓ Galois cohomology ring. This talk discusses how a function field of transcendence degree at least two over an algebraically closed field can be recovered from its mod-ℓ Milnor K-ring in degrees one and two — a cohomological counterpart of Bogomolov's birational anabelian program.
In positive characteristic p, Milnor K-theory modulo p is no longer governed by Kummer theory but, via the Bloch–Gabber–Kato theorem, by logarithmic differential forms. The talk addresses the reconstruction of function fields from Milnor K-theory modulo the characteristic, the case left out by the ℓ-adic methods.
Bogomolov's program asserts that a function field K of transcendence degree at least two over an algebraically closed field is determined by small Galois-theoretic data, such as the pro-ℓ abelian-by-central quotient of its absolute Galois group; Pop established the first cases over the algebraic closure of a finite field. This talk presents the reconstruction theorem in full and its relation to the Milnor K-theoretic results.
Grothendieck's section conjecture predicts that, for a hyperbolic curve X over a finitely generated field K, the rational points of X correspond to the conjugacy classes of sections of the projection π₁(X) → GK. Over p-adic fields, Mochizuki's Hom-theorem and the birational results of Koenigsmann and Pop give strong evidence. This talk present the p-adic anabelian approach of the Kyoto school.
The Neukirch–Uchida theorem asserts that isomorphisms between absolute Galois groups of number fields arise from field isomorphisms, and Uchida showed that the maximal prosolvable quotients already suffice. Uchida's conjecture extends this from isomorphisms to open homomorphisms: every open homomorphism between Galois groups of solvably closed extensions of global fields should be induced by a field embedding.