Mathematics seminars

Zander Kelley presents joint work with Xin Lyu separating randomized and deterministic three-player communication in the number-on-forehead model. Earlier work by Kelley, Lovett and Meka gave an explicit function with an efficient randomized protocol but a deterministic lower bound of Ω(n^(1/3)). The argument studies whether its yes-instances can be covered efficiently by small cylinder intersections. A sifting lemma finds a denser induced subgraph inside a bipartite graph with large grid norm. An improved version raises the communication lower bound to Ω(n^(1/2)). The key structural result covers small cylinder intersections by a few reasonably small slice functions; the talk compares this simplification with Szemerédi’s triangle removal lemma.

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.

Gregorio Baldi surveys recent results on how the Hodge locus is distributed in mixed variations of Hodge structures, including the pure setting. The presentation also examines concrete applications of these results to moduli spaces.

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.