Algebraic Geometry

France’s main investigator-led call supports fundamental, exploratory and applied research across scientific disciplines. Five instruments cover young investigators, single teams and collaborative projects. Step-one submission or registration closes on 13 October 2026 at 17:00 Paris time. ANR–NSF and ANR–FRQ collaborations are suspended for this edition.

Clay Research Fellowships support exceptional recent PhD mathematicians, including candidates completing their theses. Fellowships normally run until five years after the doctorate and provide salary and research expenses. They may be held internationally subject to employment and immigration constraints. The institute is seeking nominations for 2027; materials should arrive by 15 October 2026 for full consideration. This is a nomination process, rather than an open self-application.

The inaugural award recognises influential mathematical contributions to materials science during the six years preceding the award. Nominations close on 15 October 2026. The recipient gives a plenary lecture at the 2027 SIAM materials-science conference.

Chris Skinner introduces the Birch and Swinnerton-Dyer conjecture and surveys major advances toward it before and after the turn of the millennium. The presentation revisits the substance of his plenary lecture at the 2025 Clay Research Conference on the Millennium Prize Problems.

Linquan Ma introduces recent progress on singularities in mixed characteristic. Starting from singular points of algebraic varieties, such as the cusp y² = x³, the talk explains their algebraic study through quotient rings and the mixed-characteristic setting of integer-coefficient polynomial rings. A central tool is the big Cohen–Macaulay algebra: a typically non-Noetherian ring with strong homological properties whose existence connects to advances in p-adic Hodge theory. The presentation also discusses applications of this singularity theory to birational geometry.

Philip Engel explains a counterexample to the integral Hodge conjecture for very general abelian varieties of dimension at least four. Degenerations of principally polarized abelian varieties are associated with regular matroids, and a new matroid invariant obstructs algebraicity of the minimal curve class on a very general fiber. Combining this obstruction with a theorem of Voisin and the intermediate Jacobian gives stable irrationality of a very general cubic threefold. The work is joint with Olivier de Gaay Fortman and Stefan Schreieder.

James McKernan discusses the log canonical threshold in birational geometry and Mori theory, and its extension to the classification of foliations on algebraic varieties. He introduces a foliated version of the threshold, proposes that it satisfies the ascending chain condition, and verifies this in dimension two. The approach introduces an interpolated canonical divisor and establishes that the minimal model program can be run for these divisors.

Shikha Bhutani proves the outstanding positive-characteristic cases of the Borisov–Alexeev–Borisov conjecture for geometrically integral ε-klt del Pezzo surfaces over arbitrary fields. Earlier work by Bernasconi and Martin established boundedness except in characteristics 2, 3 and 5, reducing those cases to a uniform estimate for irregularity. The talk establishes that estimate and thereby completes the boundedness result across all positive characteristics.

Tree-level open string amplitudes are integrals over the moduli space of n points on the projective line, and their field theory limit is the Cachazo-He-Yuan formula for tr(\Phi^3) amplitudes. Joint work with Freddy Cachazo, Alfredo Guevara and Sebastián Mizera (CEGM) replaces the line by P^(k-1), giving amplitude-like objects whose Feynman diagrams are higher-dimensional polyhedral complexes rather than trees. After reviewing this framework, I will discuss recent progress on new factorization patterns which appear and linear zero-preserving deformations, which produce nonlinear sigma model amplitudes at k=2 by a result of Arkani-Hamed, Cao, Dong, Figueiredo and He, and new factorization behavior at k greater than or equal to 3.