Biography

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Recent Research

https://hal.science/hal-05464059/document

Hopf Problem of Six-Sphere

We construct a differential 2-form from any almost complex structure on a smooth manifold. If this 2-form is not vanishing, then the almost complex structure is not integrable. We show that for the classic almost complex structure J_0 on S^6 from the multiplication in Octonions, its 2-form is not vanish therefore this gives a new proof for non-integrabilty of J0. Therefore this 2-form is not trivial, nor empty. This 2-form suggests a relation between the integrability of almost complex structures and the three cohomology groups H^1(S^6) = 0, H^2(S^6) = 0 and H^3(S^6) = 0 of the underlying manifold S^6, comparing with H^1(M^n)-H^3(M^n) of complex manifolds M^n (2 < n ≤ 6) : M = CP^3 , M = S^3 × S^3, M = S^1 × S^3, and M=CP^2 . Applying this 2-form to S^6 Hopf problem, we prove that all almost complex structures on S^6 are not integrable. Therefore S^6 is not a complex manifold. We study Hopf problem by employing theories and measures of Algebra, Geometry, Topology and Analysis (AGTA).
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Visiting Fellow, Department of Mathematics
Princeton University, Princeton, NJ 08544-1000 USA

Visiting Professor, Department of Mathematics
Cornell University, Ithaca, NY14853-4201 USA
https://math.cornell.edu/jun-michael-ling

Member, The Mathematical Sciences Research Institute (MSRI), Berkeley, California

Calculus III
https://classes.cornell.edu/browse/roster/SP23/class/MATH/2130

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