Algebra

RIKEN iTHEMS invites independent research in algebra, geometry, analysis and related mathematical foundations, with interdisciplinary collaboration across natural and mathematical sciences. Senior researchers also coordinate projects and mentor colleagues. Positions are based at the Wako campus, Japan. Initial one-year contracts are renewable within role-specific limits. Employment starts on or after 1 April 2027, negotiable. Applications through Academic Jobs Online must arrive by 20 November 2026, Japan time.

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.

A Nichols algebra can be associated with an object of a braided tensor category through braid-group combinatorics or a universal characterization. Besides reconstructing quantum groups from their Cartan components, these algebras can generate new braided tensor categories with rich root-system structures. Simon Lentner connects this construction to conformal field theory, where braidings arise from monodromy of differential equations such as the Knizhnik–Zamolodchikov equation. The Kazhdan–Lusztig correspondence relates the representation theories in these settings to quantum groups. The talk describes a nonsemisimple form of this correspondence, conjectured by Feigin and collaborators and proved using Nichols algebras and categorical methods. It also presents recent work identifying the braided tensor category of affine sl(2) modules at admissible level through a quantum group of type sl(2|1).

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.

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.

Ryann Cartor introduces multivariate public-key cryptography, whose fast and compact post-quantum schemes use polynomial systems over finite fields. The tutorial surveys past and current multivariate cryptosystems before examining MinRank as a central cryptanalytic problem. MinRank attacks have broken prominent schemes in both multivariate and code-based cryptography. The material assumes basic algebra and connects these foundations to current research directions.

Eimear Byrne introduces q-matroids and rank-metric codes as vector-space analogues of classical matroids and Hamming-metric codes. The tutorial compares the connections between codes and matroids with those between rank-metric codes and q-matroids. It develops equivalent definitions of q-matroids, relates them to classical definitions, and explains how q-matroid invariants provide invariants for rank-metric codes.

Tenure-track appointment expected to start July 1, 2027. All areas of pure mathematics are considered, with particular interest in number theory, algebra and topology. Requires a PhD and excellence or strong potential in research and teaching; responsibilities include graduate supervision and undergraduate and graduate teaching. Salary: CAD 120,000–160,000 annually. Apply through this MathJobs posting with a cover letter, CV, research statement, a short account of significant research contributions and at least four references, including one addressing teaching. November 30, 2026 is the full-consideration date; review continues until filled. Canadians and permanent residents receive priority.

Jennifer Park studies definability in number fields in connection with Hilbert’s tenth problem over the rationals. An existential definition of the integers inside the rationals, or of the ring of integers inside a number field, would connect the corresponding rational decision problem to the classical undecidability theorem over the integers. The talk instead constructs a definition of a number field’s ring of integers using a single universal quantifier. This extends Koenigsmann’s universal definition of the integers inside the rationals and gives a particularly simple first-order description.

Michael Chapman examines the gap between the refutation of Connes’ embedding conjecture by MIP*=RE and the construction of non-hyperlinear groups. The talk introduces non-local games, their values and Tsirelson’s problem, then associates a finitely presented *-algebra to a synchronous game. It explains how representations of this algebra relate to different game values, and why the relevant von Neumann closure cannot embed into an ultraproduct of the hyperfinite II₁ factor. A route to a non-hyperlinear group would require an appropriate counterexample algebra to be a complex group ring. Chapman introduces linear constraint system games, their associated game groups, and explains how the MIP*=RE construction differs from such a game.