{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/50692"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/50692","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"J - holomorphic curves and their applications","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.","abstract_html":"This thesis covers four results: 1. We prove an analog of Whitney&#x27;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 &lt;1, y1^2 + y2^2 &lt; 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 &gt;= 1 and n &gt;= 2, we show that D_R^2 * D^(n-2)(r) and D^2 * D^(n-2)(r) are not symplectomorphic.","abstract_has_math":false,"creators":["Wong, Yat Sen"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Tumanov, Alexander","Kerman, Ely","D’Angelo, John","Tolman, Susan"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-09-16T17:25:20Z","date_published":"2014-09-16T17:25:20Z","updated_at":"2026-07-22T22:25:41Z","subjects":["J-holomorphic curve","symplectic embedding","symplectomorphism"],"languages":["en"],"rights":["Copyright 2014 Yat Sen Wong"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/50692","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Tumanov, Alexander","Kerman, Ely","D’Angelo, John","Tolman, Susan"]},{"key":"dc:creator","label":"Author","values":["Wong, Yat Sen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-09-16T17:25:20Z","2014-08","2014-09-16"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["J-holomorphic curve","symplectic embedding","symplectomorphism"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2014 Yat Sen Wong"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/50692"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-06-11T13:32:45Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 3 WONG_YATSEN.pdf: 330480 bytes, checksum: f4bc51e7fa980d8637ad39d7ef27591c (MD5) WONG_YATSEN.zip: 27493 bytes, checksum: 4cc485cd461b58c919b555c722a3b8f6 (MD5) WONG_YATSEN.pdf: 330489 bytes, checksum: e963616d737b8955190e7464483927fa (MD5)","Made available in DSpace on 2014-09-16T17:25:20Z (GMT). No. of bitstreams: 3 Yat Sen_Wong.pdf: 330465 bytes, checksum: 3a881f2580e4b45ce958b44fb5a4ccef (MD5) WONG_YATSEN.zip: 27491 bytes, checksum: 5baa65d8bfaa0af310e1487f73ec7b32 (MD5) license.txt: 4060 bytes, checksum: e979ec359afc71a6caa8d6448e37a375 (MD5)"]},{"key":"dc:title","label":"Title","values":["J - holomorphic curves and their applications"]}]}],"canonical_facts":{"dc:contributor":["Tumanov, Alexander","Kerman, Ely","D’Angelo, John","Tolman, Susan"],"dc:creator":["Wong, Yat Sen"],"dc:date":["2014-09-16T17:25:20Z","2014-08","2014-09-16"],"dc:description":["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.","Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-06-11T13:32:45Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 3 WONG_YATSEN.pdf: 330480 bytes, checksum: f4bc51e7fa980d8637ad39d7ef27591c (MD5) WONG_YATSEN.zip: 27493 bytes, checksum: 4cc485cd461b58c919b555c722a3b8f6 (MD5) WONG_YATSEN.pdf: 330489 bytes, checksum: e963616d737b8955190e7464483927fa (MD5)","Made available in DSpace on 2014-09-16T17:25:20Z (GMT). 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