Module MATH4201: ANALYSIS IV

Department: MATHEMATICAL SCIENCES

MATH4201: ANALYSIS IV

Prerequisites

• Complex Analysis II (MATH2011) and Analysis in Many Variables II (MATH2031)

Corequisites

• None.

Excluded Combination of Modules

• Analysis III (MATH3011).

Aims

• To provide the student with basic ideas about topology and integration and to introduce the student to the theory and application of function spaces.

Content

• Metric Spaces.
• Integration in n-dimensional real space.
• Harmonic functions.
• Compactness in function spaces.
• Conformal mapping.
• Reading material on a topic in one of the following areas: analytic continuation along arcs, monodromy theorem, simple Riemann surfaces, Picard's theorem.

Learning Outcomes

• By the end of the module students will:
• be able to solve novel and/or complex problems in Analysis.
• have a systematic and coherent understanding of theoretical mathematics in the field of Analysis.
• have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Metric spaces: compactness.
• Completeness.
• Contraction mappings.
• Picard theorem.
• Function Spaces: Stone-Weierstrass theorem.
• Arzela-Ascoli theorem.
• Normal families.
• Integration: Riemann integral.
• Darboux theorem.
• Sets of measure zero.
• Harmonic functions: maximum principle.
• Dirichlet problem and poisson formula.
• Conformal mapping: Schwarz lemma.
• Reflection principle.
• Schwarz-Chistoffel formula.
• Riemann mapping theorem.
• Knowledge and understanding of a topic in the following areas: analytic continuation (continuation along arcs, monodromy theorem, simple Riemann surfaces, Picard's theorem).

• In addition students will have highly specialised and advanced mathematical skills in the following areas which will be used with minimal guidance: Spatial awareness.
• Ability to read independently to acquire knowledge and understanding in the area of analytic continuation.

• Students will have enhanced problem solving skills.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on student's progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University