报告题目: Gårding Polynomials and Liouville Rigidity for Hessian PDE

时间：2026年7月9日上午10:00-11:00

地点：文北楼102

报告人：方浩 (University of Iowa)

摘要：In this talk I will discuss a structural approach to Liouville rigidity for fully nonlinear Hessian-type equations in both the real and complex setting. Using the Harvey--Lawson viewpoint of subequations, we study such equations through admissible sets in the space of symmetric or Hermitian forms.

I will explain a purely geometric criterion for Liouville rigidity: bounded entire viscosity solutions with mild H\"older regularity must be constant if and only if the associated admissible set satisfies a suitable geometric reduction property. This unifies and extends several known Liouville theorems, including results for complex Hessian equations and certain mixed degenerate Monge--Amp\`ere-type equations, while weakening the regularity assumptions in earlier works. This is jointwork with Biao Ma and Jinyang Wu.

I will then describe a new algebraic framework, developed jointly with Biao Ma, based on Gårding polynomials and ideal Gårding polynomials. These classes properly extend real stable polynomials and provide a systematic way to construct Liouville admissible sets. In particular, they yield new spectral and anisotropic examples of fully nonlinear geometric PDEs beyond the classical symmetric-cone setting.

邀请人：马辉