Quanta Magazine reported the scientific research results of Associate Professor Zhengyi Zhou

On November 7, 2023, the Quanta Magzine reported the research results of Zhengyi Zhou.

 A fundamental problem in contact topology is understanding the tight contact structures (a special structure on odd-dimensional manifolds). Classical results of Gromov and Eliashberg tell us that contact manifolds with a symplectic filling, i.e.,those can arise as a convex boundary of a symplectic manifold, must be tight. Therefore, a basic question is understanding the distance between tightness and existences of fillings. Tight contact structures without fillings were constructed on some manifolds previously. The question that Zhengyi Zhou and three other researchers investigated was whether there is such a structure on the simplest manifold, namely the standard sphere. A contact structure on a one-dimensional sphere is trivially fillable. In dimension 3, Eliashberg proved that there is only one tight contact structure on the sphere, which is also fillable. In their joint work, Zhou and his coauthors constructed tight contact structures on spheres of dimension at least 5 that are not fillable. As a corollary, they proved that in dimensions 7 and above, the existence of a tight contact structure on a manifold is equivalent to the existence of a tight but not strongly fillable contact structure, which illustrates a great difference, and in a certain sense the optimal difference, between tight structures and those that are strongly fillable.

 Zhengyi Zhou's research area is symplectic topology and contact topology. He uses the theory of pseudo-holomorphic curves, in particular, symplectic field theory, to study the rigidity phenomena in symplectic and contact topology, especially problems related to symplectic fillings, and symplectic cobordisms. Major works include proving Wolf Prize laureate Eliashberg's conjecture on the nonexistence of Liouville fillings for real projective spaces; constructing the first non-topological obstruction to the existence of Weinstein fillings; with Moreno, constructing a series of contact invariants from rational symplectic field theory; and with Bowden, Gironella, and Moreno, constructing tight contact structures on high-dimensional spheres that are not fillable.

 Quanta Magazine is an editorially independent online publication of the Simons Foundation, focusing on recent breakthroughs in physics, math, biology, and computer science, with the aim of popularizing science and describing the fundamentals and discovery of the latest advances in a particular research.