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Pseudodifferential calculus and Laplacians associated with Riemannian metrics on noncommutative tori : 비가환 원환면 위에서의 의미분 연산과 기만 거리와 연관된 라플라시안
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.advisor | Raphael Ponge | - |
| dc.contributor.author | 하현수 | - |
| dc.date.accessioned | 2018-11-12T00:59:27Z | - |
| dc.date.available | 2018-11-12T00:59:27Z | - |
| dc.date.issued | 2018-08 | - |
| dc.identifier.other | 000000152163 | - |
| dc.identifier.uri | https://hdl.handle.net/10371/143239 | - |
| dc.description | 학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2018. 8. Raphael Ponge. | - |
| dc.description.abstract | This thesis consists of the following three chapters | - |
| dc.description.tableofcontents | Abstract i
1 Introduction 1 1.1 Oscillating integrals and pseudodifferential operators on noncommutative tori . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Pseudodifferential calculus on noncommutative tori . . . . . . 2 1.3 Laplacians associated with Riemannian metrics on noncommutative tori . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Oscillating integrals and PDOs on noncommutative tori 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Noncommutative Tori . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Noncommutative tori . . . . . . . . . . . . . . . . . . . 7 2.2.2 Dynamics on A_x0012_ . . . . . . . . . . . . . . . . . . . . . . 11 2.2.3 Distributions on A_x0012_ . . . . . . . . . . . . . . . . . . . . 18 2.2.4 Differential operators . . . . . . . . . . . . . . . . . . . 20 2.3 Classes of Symbols on Noncommutative Tori . . . . . . . . . . 23 2.3.1 Standard symbols . . . . . . . . . . . . . . . . . . . . . 23 2.3.2 Homogeneous and classical symbols . . . . . . . . . . . 26 2.4 Amplitudes and Oscillating Integrals . . . . . . . . . . . . . . 30 2.4.1 Spaces of amplitudes . . . . . . . . . . . . . . . . . . . 30 2.4.2 A_x0012_-Valued oscillating integrals . . . . . . . . . . . . . . 33 2.5 Pseudodifferential Operators on Noncommutative Tori . . . . 42 2.5.1 PDOs associated with amplitudes . . . . . . . . . . . . 42 2.5.2 PDOs associated with symbols . . . . . . . . . . . . . 44 2.5.3 Smoothing operators . . . . . . . . . . . . . . . . . . . 50 2.A Integration in Locally Convex Spaces . . . . . . . . . . . . . . 53 2.A.1 Riemann integration . . . . . . . . . . . . . . . . . . . 53 2.A.2 Lebesgue integration . . . . . . . . . . . . . . . . . . . 56 2.B Differentiable Maps with Values in Locally Convex Spaces . . 61 2.B.1 Differentiation . . . . . . . . . . . . . . . . . . . . . . . 61 2.B.2 Differentiation under the integral sign . . . . . . . . . . 68 2.B.3 Fourier transform and Schwartz's class . . . . . . . . . 70 3 Pseudodifferential calculus on noncommutative tori 75 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2 Composition of PDOs. Amplitudes . . . . . . . . . . . . . . . 78 3.3 Composition of PDOs. Symbols . . . . . . . . . . . . . . . . . 87 3.4 Adjoints of PDOs. Action on A 1 _x0012_ . . . . . . . . . . . . . . . . 94 3.5 Sobolev Spaces on Noncommutative Tori . . . . . . . . . . . . 98 3.6 Boundedness and Sobolev Mapping Properties . . . . . . . . . 108 3.7 Ellipticity and Parametrices . . . . . . . . . . . . . . . . . . . 111 3.8 Spectral Theory of Elliptic Operators . . . . . . . . . . . . . . 117 3.8.1 Fredholm properties . . . . . . . . . . . . . . . . . . . 117 3.8.2 Spectra of positive order elliptic PDOs . . . . . . . . . 120 3.9 Trace-Class and Schatten-Classes Properties of PDOs . . . . . 124 3.9.1 Schatten classes . . . . . . . . . . . . . . . . . . . . . . 124 3.9.2 Schatten-classes properties of PDOs . . . . . . . . . . 127 4 Laplacians associated with Riemannian metrics on noncommutative tori 130 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.2 Riemannian metrics on Noncommutative Tori . . . . . . . . . 131 4.2.1 Matrices over noncommutative tori . . . . . . . . . . . 131 4.2.2 Riemannian metrics on noncommutative tori . . . . . . 134 4.3 Laplacians on Noncommutative Tori . . . . . . . . . . . . . . 136 Abstract (in Korean) 150 | - |
| dc.language.iso | en | - |
| dc.publisher | 서울대학교 대학원 | - |
| dc.subject.ddc | 510 | - |
| dc.title | Pseudodifferential calculus and Laplacians associated with Riemannian metrics on noncommutative tori | - |
| dc.title.alternative | 비가환 원환면 위에서의 의미분 연산과 기만 거리와 연관된 라플라시안 | - |
| dc.type | Thesis | - |
| dc.contributor.AlternativeAuthor | Ha Hyunsu | - |
| dc.description.degree | Doctor | - |
| dc.contributor.affiliation | 자연과학대학 수리과학부 | - |
| dc.date.awarded | 2018-08 | - |
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