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Donaldson Question:"Tamed to Compatible"


Academic Report

Title: Donaldson Question:"Tamed to Compatible"

Reporter: Prof. WANG Hongyu

Time: Oct 18, 2019 (Friday) PM 14:40-15:20

Location: A1101# room, Innovation Park Building

Contact: Prof. LEI Fengchun (tel:84708360)

Abstract: In this talk, we show that on any tamed closed almost complex fourmanifold (M; J) whose dimension of J-anti-invariant cohomology is equal to self-dual second Betti number minus one, there exists a new symplectic form compatible with the given almost complex structure J. In particular, if the self-dual second Betti number is one, we give an armative answer to Donaldson question for tamed closed almost complex four-manifolds that is a conjecture in joint paper of Tosatti, Weinkove and Yau. Our approach is along the lines used by Buchdahl to give a unied proof of the Kodaira conjecture. Thus, our main result gives an armative answer to the Kodaira conjecture in symplectic version.

The brief introduction to the reporter: Wang Hongyu, Professor, School of mathematics science, Yangzhou University, is mainly engaged in differential geometry, partial differential equations and low dimensional topology. In recent years, he is mainly engaged in the research of metric geometry, symplectic geometry and nonlinear development equations. A number of research results have been achieved independently or in cooperation with others. Mainly: solve the 10th problem in Yau problem set, which is part of Chern Hopf guess. In cooperation with Ding Weiyue of Peking University and Wang Youde of Chinese Academy of Sciences, we have studied Schr ö dinger flow, studied the generalized Heisenberg model with value in Hermite symmetric space and the third-order nonlinear Schr ö dinger equation with tight Hermite Lie algebra, given the one-to-one correspondence between them, and constructed the specific periodic solution. We have also proved the existence of the global solution of Schr ö dinger flow. Sex. In this paper, the Yang mills field and its equations are studied systematically and deeply, the module space geometry of the classical Yang mills field on R4 is discussed, and infinite non-minimum solutions are constructed for the Yang mills equation, that is, infinite non-minimum solutions are constructed for the unstable Yang mills equation, that is, the existence theorem for the unstable Yang mills field is established. This result is included in the Graduate School of physics of American University. Book.