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On the convexity of general inverse $\sigma_k$ equations and some applications

February 24, 2023 @ 9:00 am - 10:00 am

CMSA Algebraic Geometry in String Theory 02.24.2023

Algebraic Geometry in String Theory Seminar

Speaker: Chao-Ming Lin (University of California, Irvine)

Title: On the convexity of general inverse $\sigma_k$ equations and some applications

Abstract: In this talk, I will show my recent work on general inverse $\sigma_k$ equations and the deformed Hermitian-Yang-Mills equation (hereinafter the dHYM equation). First, I will show my recent results. This result states that if a level set of a general inverse $\sigma_k$ equation (after translation if needed) is contained in the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the Monge-Ampère equation, the J-equation, the dHYM equation, the special Lagrangian equation, etc. Second, I will introduce some semialgebraic sets and a special class of univariate polynomials and give a Positivstellensatz type result. These give a numerical criterion to verify whether the level set will be contained in the positive orthant. Last, as an application, I will prove one of the conjectures by Collins-Jacob-Yau when the dimension equals four. This conjecture states that under the supercritical phase assumption, if there exists a C-subsolution to the dHYM equation, then the dHYM equation is solvable.

Details

Date:
February 24, 2023
Time:
9:00 am - 10:00 am
Event Category:

Venue

Virtual