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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 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Jholomorphic curve
symplectic embedding symplectomorphism 
Abstract:  This thesis covers four results: 1. We prove an analog of Whitney's embedding theorem for Jholomorphic 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 bidisc 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^(n2)(r) and D^2 * D^(n2)(r) are not symplectomorphic. 
Issue Date:  20140916 
URI:  http://hdl.handle.net/2142/50692 
Rights Information:  Copyright 2014 Yat Sen Wong 
Date Available in IDEALS:  20140916 
Date Deposited:  201408 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois