Algebra seminars

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.

Alexander Barg explores research questions inspired by Reed–Muller codes. The first concerns storage codes on triangle-free graphs, where neighboring vertices determine parity checks. Certain graphs admit codes approaching the maximum size of 2^n; whether Reed–Muller codes yield similar constructions remains open. The second extends the construction of Reed–Muller codes from cosets in an elementary abelian group to Coxeter groups, including permutation groups. Questions include analogues of the |u|u+v| construction and modern decoders, whether these codes share Reed–Muller codes' capacity achievement on the binary erasure channel, and a conjectured minimum-distance formula.

Youming Qiao develops connections between graphs and linear spaces of matrices. Tutte and Lovász linked graph perfect matchings to full-rank matrices; further correspondences relate independent sets to totally isotropic spaces, connectivity to orthogonal decompositions, and graph isomorphism to matrix-space isomorphism. These connections extend graph-theoretic questions and techniques to matrix spaces, including alternating paths, independence polynomials, Turán and Ramsey problems, expanders, and threshold phenomena. The methods connect to invariant theory, group theory, quantum information, and geometry. Based on joint work with Avi Wigderson, Yuval Wigderson, Gábor Ivanyos, Yinan Li, Chuanqi Zhang, and Markus Bläser.

Doron Puder studies probability distributions on finite groups obtained by evaluating a free-group word on uniformly random group elements. Earlier work connected such distributions on symmetric groups with the partially ordered set of finitely generated free-group subgroups. The talk introduces free groups and free-group algebras, then develops new connections between word distributions on matrix groups over finite fields and free-group algebras. It compares these findings with the symmetric-group case and presents related conjectures. Joint work with Danielle Ernst-West and Matan Seidel.

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.

Alexander Lubotzky examines whether infinite groups can be approximated by asymptotic homomorphisms into finite symmetric or unitary groups. He describes finitely presented groups that admit no unitary approximation in the Frobenius norm and several other norms. The argument links vanishing of higher cohomology to stability, then uses Garland’s method and high-dimensional expanders arising from Bruhat–Tits buildings. Certain non-residually finite central extensions of lattices in p-adic Lie groups are stable, preventing such approximations. These extensions come from a p-adic analogue of a result of Deligne. The talk explains the relevant notions and draws on joint work with M. De Chiffre, L. Glebsky, A. Thom and I. Oppenheim.

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.