Institute for Advanced Study, School of Mathematics

Charles Bordenave revisits von Neumann and Goldstine's proposal to model numerical computing errors as probabilistic noise. This perspective helps explain large errors in computed eigenvalues of badly conditioned matrices. Toeplitz matrices provide a case study: they occur throughout mathematics, physics, and engineering, are highly sensitive to small perturbations, and possess a rich spectral theory that supports precise descriptions under random perturbations. The talk is based on joint work with Mireille Capitaine and François Chapon.

Jeroen Zuiddam examines whether repeated tasks permit economies of scale, focusing on fast matrix multiplication and graph Shannon capacity as unresolved direct-sum problems. The talk introduces Strassen's asymptotic-spectrum duality, originally developed for asymptotic tensor rank and later applied to Shannon capacity and other problems. It then explores approximation by sequences of easier instances: suitable notions of convergence, construction of the sequences, asymptotic-spectrum distance, and approaches using polynomials and algebraic geometry. Based on joint work with David de Boer, Pjotr Buys, Sven Polak, Matthias Christandl, Koen Hoeberechts, Harold Nieuwboer, and Péter Vrana.

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.