Director of Research (if dissertation) or Advisor (if thesis)
Tumanov, Alexander
Doctoral Committee Chair(s)
Kerman, Ely
Committee Member(s)
Tumanov, Alexander
D’Angelo, John
Tolman, Susan
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
J-holomorphic curve
symplectic embedding
symplectomorphism
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.
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