Real Submanifolds and Holomorphic Mappings in Geometric Function Theory

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Abstract: The long-term goal of this project is to study holomorphicmappings and real submanifolds arising in complex analysis andholomorphic dynamical systems. A large part of the PI's proposedresearch has its roots in Birkhoff's fixed point theorem, the KAM(Kolmogorov-Arnold-Moser) theory, and the Siegel-Bruno-Yoccoztheory. On the other hand, new research obtained by the method ofcomplex analysis will give new insights into the classicalarea-preserving maps and reversible maps, and reversible orHamiltonian systems. The PI's proposed research involves severalproblems in connection with real analytic surfaces intwo-dimensional complex space, formally linearizable reversible orarea-preserving real analytic maps, and singular Levi-flathypersurfaces in the complex projective space. The differential equations concerning the motion of N mass points, a model of the solar system,in the three-dimensional space attracting each other according to the Newton's law form a Hamiltonian system.The reversibility of a dynamical system could just be the time-symmetry that is important in practical matters. The periodic orbits of certain area-preserving mappings correspond to the periodic motion in the Hamiltonian system of the restricted three-body problem, and such studyof the existence of such periodic orbits goes back at least to work of Poincar\'e and Birkhoff on the celestial mechanics about a century ago. The PI's work aims to understand the existence of periodic orbits in such Hamiltonian or time-symmetric systems, both over the real numbers and over the complex numbers.The PI is active in undergraduate teaching, andhas also organized a graduate student seminar dealing with his research area.