Mathematics. Course List

Transcription

1 Mathematics 1 Mathematics Course List Code Course Title Unit I. Ph.D. Programme (Full-time and Part-time); II. M.Phil. Programme (Full-time and Part-time); and III. M.Sc. Programme (Full-time and Part-time) MAT 5001, 5002 Guided Studies I, II 3 each MAT 5011, 5012 Real Analysis I, II 3 each MAT 5031, 5032 Complex Analysis I, II 3 each MAT 5051, 5052 Abstract Algebra I, II 3 each MAT 5070 Topology of Manifolds 3 MAT 600T M.Phil. Thesis Project 6 MAT 601T M.Phil. Thesis Project (Part-time) 4 *MAT 6010 Topics in Mathematics 3 *MAT 6020 Topics in Differential Geometry 3 *MAT 6030 Topics in Algebra 3 *MAT 6040 Topics in Partial Differential Equations 3 *MAT 6050 Topics in Complex Analysis 3 *MAT 6060 Topics in Number Theory 3 *MAT 6070 Topics in Topology 3 *MAT 6080 Topics in Functional Analysis 3 *MAT 6110 Topics in Applied Mathematics 3 *MAT 6120 Topics in Numerical Analysis 3 *MAT 6130 Topics in Optimization Theory 3 *MAT 6140 Topics in Applied Partial Differential Equations 3 MAT 6011, 6012 Topics in Mathematics I, II 3 each MAT 6021, 6022 Topics in Differential Geometry I, II 3 each MAT 6031, 6032 Topics in Algebra I, II 3 each MAT 6041, 6042 Topics in Partial Differential Equations I, II 3 each MAT 6051, 6052 Topics in Complex Analysis I, II 3 each MAT 6061, 6062 Topics in Number Theory I, II 3 each MAT 6071, 6072 Topics in Topology I, II 3 each MAT 6081, 6082 Topics in Functional Analysis I, II 3 each MAT 6111, 6112 Topics in Applied Mathematics I, II 3 each MAT 6121, 6122 Topics in Numerical Analysis I, II 3 each MAT 6131, 6132 Topics in Optimization Theory I, II 3 each MAT 6141, 6142 Topics in Applied Partial Differential 3 each Equations I, II MAT 700T Ph.D. Thesis Project 6 MAT 701T Ph.D. Thesis Project (Part-time) 4 MAT 7010 Graduate Seminars 3 * Courses offered in and before.

2 2 Postgraduate Programmes of Study Course Description MAT 5001 Guided Studies I 3 U; 3 STOT Series of projects in pure or applied mathematics focusing on a central theme. Prerequisite: permission of the instructor. MAT 5002 Guided Studies II 3 U; 3 STOT Series of projects in pure or applied mathematics focusing on a central theme. This course is a continuation of MAT Prerequisite: permission of the instructor. MAT 5011 Real Analysis I This course provides a solid foundation in the Lebsegue integration theory and basic techniques in analysis. Topics include σ-algebra of sets, measure theory, Lebsegue integration theory, convergence theorems, Lp-spaces and differentiation. Students taking this course are expected to have knowledge in advanced calculus and elementary analysis. MAT 5012 Real Analysis II This course provides more advanced topics in real analysis. Topics include signed measures, Hahn decomposition theorem, Lebesgue decomposition theorem, product measures, Fubini theorem, measure and topology, and Riesz representation theorem. Students taking this course are expected to have knowledge in MAT 5011 or equivalent. MAT 5031 Complex Analysis I This course is intended to provide a solid and advanced training in the basic techniques and theorem of complex analysis. Topics include: properties of holomorphic functions, complex integration, conformal mappings, singularities and residues. Students taking this course are expected to have knowledge in advanced calculus and elementary analysis. MAT 5032 Complex Analysis II This is a continuation of MAT 5031; advanced topics in complex analysis will be selected from: Schwarz lemma, Riemann mapping theorem, Picard theorem, Weierstrass theorem and Mittag Leffler theorem, analytic continuation and introduction to Riemann surfaces. Students taking this course are expected to have knowledge in advanced calculus and elementary analysis. MAT 5051 Abstract Algebra I This course is intended to provide a solid background knowledge in abstract algebra. Topics

3 Mathematics 3 include group theory, Sylow s theorems, structure of finitely generated abelian groups, rings and ideals, polynomial rings, principal ideal domain (PID), modules, fields, Galois theory. Students taking this course are expected to have knowledge in a first course in algebra. MAT 5052 Abstract Algebra II This is a continuation of MAT Topics include fields, cyclic extensions, separable extensions, integral ring extensions, integral Galois extensions, Noetherian rings and modules, localization, Hilbert basis theorem, primary decomposition. Students taking this course are expected to have knowledge in MAT 5051 or equivalent. MAT 5070 Topology of Manifolds This course is an introduction to several basic topological invariants for manifolds. Major topics are: differentiable manifolds and maps, Sard s theorem, degree of maps, fundamental group, covering space, homology group. Students taking this course are expected to have knowledge in elementary analysis. MAT 600T M.Phil. Thesis Project 6 U; 6 STOT Series of projects in pure or applied mathematics focusing on a central theme. A certain degree or originality is expected in these projects. Moreover, these projects preferably lead to a Master s thesis. This course may be taken again for credit with the instructor s consent. Prerequisite: permission of the instructor. MAT 601T M.Phil. Thesis Project (Part-time) 4 U; 4 STOT Same as MAT 600T. MAT 6011, 6012 Topics in Mathematics I, II Usually, more than one section with various topics selected from advanced pure mathematics will be offered. The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor. MAT 6021, 6022 Topics in Differential Geometry I, II Various topics selected from differential geometry. The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor. MAT 6031, 6032 Topics in Algebra I, II Various topics selected from algebra. The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor.

4 4 Postgraduate Programmes of Study MAT 6041, 6042 Topics in Partial Differential Equations I, II Various topics selected from partial differential equations. The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor. MAT 6051, 6052 Topics in Complex Analysis I, II Various topics selected from complex analysis (one or several variables). The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor. MAT 6061, 6062 Topics in Number Theory I, II Various topics selected from number theory. The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor. MAT 6071, 6072 Topics in Topology I, II Various topics selected from topology. The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor. MAT 6081, 6082 Topics in Functional Analysis I, II Various topics selected from functional analysis. The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor. MAT 6111, 6112 Topics in Applied Mathematics I, II Usually, more than one section with various topics selected from advanced applied mathematics will be offered. Prerequisite: permission of the instructor. MAT 6121, 6122 Topics in Numerical Analysis I, II Various topics selected from numerical analysis. The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor. MAT 6131, 6132 Topics in Optimization Theory I, II Various topics selected from optimization theory. The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor.

5 Mathematics 5 MAT 6141, 6142 Topics in Applied Partial Differential Equations I, II Various topics selected from applied partial differential equations. The selection of the topics depends on the field of interest of the instructor. Prerequisite: permission of the instructor. MAT 700T Ph.D. Thesis Project 6 U; 6 STOT Series of advanced projects in pure or applied mathematics focusing on a central theme. A high degree of originality is expected in these projects. Moreover, these projects preferably lead to a doctoral thesis. This course may be taken again for credit with instructor s consent. MAT 701T Ph.D. Thesis Project (Part-time) 4 U; 4 STOT Same as MAT 700T. MAT 7010 Graduate Seminars 3 U; 3 STOT Usually, more than one section will be offered. The contents of the seminars depend on the field of interest of the instructor. This course may be taken again for credit with instructor s consent. Prerequisite: permission of the instructor. Study Scheme I. Ph.D. Programme (Full-time and Part-time) Students are required to complete a minimum of 54 units of courses for full-time students and 56 units for part-time students for graduation. (a) For Full-time Students: Required Courses: Each Term of Attendance (totally 6 terms) 54 units MAT 700T and 7010 Total: 54 units (b) For Part-time Students: Required Courses: Each Term of Attendance (totally 8 terms) 56 units MAT 701T and 7010 Total: 56 units 2. Comprehensive/Qualifying Examination (applicable to students admitted in and thereafter only) By the end of the second term after the commencement of the Ph.D. programme, students are required to pass at least two of the following qualifying examinations in: i) real analysis;

6 6 Postgraduate Programmes of Study ii) complex analysis; iii) abstract algebra; and iv) applied mathematics before advancement to the Ph.D. candidature. Normally each of these examinations is held once per year. 3. Other Requirements (a) IT Proficiency Test. (For students admitted in and thereafter only. Please refer to Student IT Competence.) (b) Students are required to submit a research thesis and pass an oral examination for graduation. (c) An Improving Postgraduate Learning (IPL) course Observing Intellectual Property and Copyright Law during Research. (For students admitted in and thereafter only.) 4. Remarks (a) A student must satisfy the course requirements as for the M.Phil. programme. Students with Master s degree in Mathematics from a recognized institution may be granted exemption from the course requirements by the Graduate Panel on the basis of graduate-level courses already completed. (b) Students should complete 9 units of required courses (for full-time students) and 7 units of required courses (for part-time students) every term, unless with exemption given by the Graduate Panel. II. M.Phil. Programme (Full-time and Part-time) A. Applicable to students admitted in and thereafter Students are required to complete a minimum of 45 units of courses coded MAT 5xxx- 7xxx with at least 15 units (5 courses) from MAT , , of which: (i) at least 3 units should be from courses coded MAT 5011, 5031, 5051; and (ii) at least 6 units should be from courses coded MAT 5011, 5031, 5051, 6111, 6112, 6121, 6122 Exemption may be granted by the Graduate Panel on the recommendation of the advisor. The following is a general study scheme for an M.Phil. student: Full-time Part-time First year of attendance Course work 12 units 12 units 12 units Second year of attendance Course work 3 units 3 units 18 units 12 units Third year of attendance 18 units Total: 45 units 45 units 2. Other Requirements (a) IT Proficiency Test. (Please refer to Student IT Competence.) (b) Students are required to submit a research thesis and pass an oral examination for graduation. (c) An Improving Postgraduate Learning (IPL) course Observing Intellectual Property and Copyright Law during Research.

7 Mathematics 7 B. Applicable to students admitted in Students are required to complete a minimum of 45 units of courses coded MAT 5xxx- 7xxx with at least 15 units (5 courses) from MAT , , of which: (i) at least 3 units should be from courses coded MAT 5011, 5031, 5051; and (ii) at least 6 units should be from courses coded MAT 5011, 5031, 5051, 6110, 6111, 6112, 6120, 6121, 6122 Exemption may be granted by the Graduate Panel on the recommendation of the advisor. The following is a general study scheme for an M.Phil. student: Full-time Part-time First year of attendance Course work 12 units 12 units 12 units Second year of attendance Course work 3 units 3 units 18 units 12 units Third year of attendance 18 units Total: 45 units 45 units 2. Other Requirements (a) IT Proficiency Test. (Please refer to Student IT Competence.) (b) Students are required to submit a research thesis and pass an oral examination for graduation. III. M.Sc. Programme (Full-time and Part-time) Applicable to students admitted in and thereafter Students are required to complete 24 units, with at least 18 units (6 courses) in Mathematics coded MAT 5xxx-6xxx of which: (i) MAT 5001 is compulsory; and (ii) at least 6 units should be from the courses coded MAT 5011, 5031, Other Requirement IT Proficiency Test. (Please refer to Student IT Competence.)