Algebraic Geometry and Symmetric Circuits: No Breakthrough in Permanent vs Determinant — E8 Intelligence Research

FINDING: The Valiant permanent-vs-determinant problem is attacked via algebraic geometry and symmetric circuit lower bounds, but no breakthrough ratio or constant emerges from these talks. | MATH: Permanent (per) vs determinant (det): VP vs VNP separation requires showing per_n cannot be a p-projection of det_m for m = poly(n). Symmetric circuit lower bounds: circuits with symmetry groups (e.g., S_n) computing det require exponential size. Key objects: determinantal complexity dc(per_n), algebraic branching programs, and the geometry of secant varieties (σ_k(Seg(P^1×...×P^1))) for border rank. No explicit new constants — the field uses asymptotic exponents (e.g., ω for matrix multiplication, but not in these talks). | CONNECTION: **Weak but real**: The permanent and determinant are related to the symmetric group S_n (crystallographic root system Aₙ₋₁) and to the Grassmannian/flag varieties. The border rank of the permanent is studied via secant varieties of Segre products — these ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com