学术报告（苏治铜 2025.5.20）

摘要： In this talk, we introduce a decomposition lemma in the convex integration scheme. The lemma allows error terms to be expressed using fewer rank-one symmetric matrices than n(n+1)/2 within constructing flexible C^(1,α ) solutions to a system of nonlinear PDEs in dimension n ≥ 2, which can be viewed as a kind of truncation of the codimension one local isometric embedding equation in Nash-Kuiper Theorem. This leads to flexible solutions with higher Hölder regularity, and consequently, improved very weak solutions to certain geometric equations.

Our arguments involve applications of several results from algebraic geometry and topology, including theorem on vectorfields on spheres, the intersection of projective varieties, and projective duality. We also use an elliptic method ingeniously that avoids loss of differentiability. Consequently, our improvements on exponent involves the Radon-Hurwitz number, which exhibits an 8-fold periodicity on n that is related to Bott periodicity. This work is inspired by Cao-Székelyhidi'19, and is a joint work with Weijun Zhang.