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Title:J - holomorphic curves and their applications
Author(s):Wong, Yat Sen
Director of Research:Tumanov, Alexander
Doctoral Committee Chair(s):Kerman, Ely
Doctoral Committee Member(s):Tumanov, Alexander; D’Angelo, John; Tolman, Susan
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):J-holomorphic curve
symplectic embedding
Abstract:This thesis covers four results: 1. We prove an analog of Whitney's embedding theorem for J-holomorphic discs. 2. For zj = xj + i*yj in C and let D_R^2 = {(z1, z2) in C^2 : x1^2 + x2^2 <1, y1^2 + y2^2 < 1} be the real bi-disc in C^2. We find the sharp lower bound for R such that D_R^2 admits a symplectic embedding into D(R) * C, the complex cylinder with base radius R. The sharp lower bound for R is shown to be 2/sqrt(pi). As a consequence, we know that D_R^2 and D^2 are not symplectomorphic. 3. We extend the second result by showing that if T is an orthogonal matrix on R^4 = C^2, then TD^2 is symplectomorphic to D^2 if and only if T is unitary or conjugate to unitary. 4. A high dimensional case of the second result: for r >= 1 and n >= 2, we show that D_R^2 * D^(n-2)(r) and D^2 * D^(n-2)(r) are not symplectomorphic.
Issue Date:2014-09-16
Rights Information:Copyright 2014 Yat Sen Wong
Date Available in IDEALS:2014-09-16
Date Deposited:2014-08

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