Topology and Analysis III

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Description

Solving equations is a crucial aspect of working in mathematics, physics, engineering, and many other fields. These equations might be straightforward algebraic statements, or complicated systems of differential equations, but there are some fundamental questions common to all of these settings: does a solution exist? If so, is it unique? And if we know of the existence of some specific solution, how do we determine it explicitly or as accurately as possible? This course develops the foundations required to rigorously establish the existence of solutions to various equations, thereby laying the basis for the study of solutions. Through an understanding of the foundations of analysis, we obtain insight critical in numerous areas of application, such areas ranging across physics, engineering, economics and finance.

Objective

To introduce students to the foundational ideas and results in modern topology and analysis so as to equip them for further studies in these areas or to be able to use this knowledge in other fields such as differential equations, optimization, Fourier series and mathematical physics. At the end of this course, students should: understand how the notion of limit, introduced informally in Mathematics I is made rigourous and applied in the more general settings of normed vector spaces and metric spaces, understand what a metric space is and how notions such as continuity are interpreted in this framework and appreciate the importance of the notion of compactness, understand how to apply the Contraction Mapping Theorem to find the solutions to various mathematical problems.

Content

Topics covered are: sets, functions, metric spaces and normed linear spaces, compactness, connectedness, and completeness. Banach fixed point theorem and applications, uniform continuity and convergence. General topological spaces, generating topologies, topological invariants, quotient spaces. Introduction to Hilbert spaces and bounded operators on Hilbert spaces.

Mathai Varghese Lecturer for this course

Delivery

36 hours of lectures and tutorials

Assessment

Ongoing assessment 30%, exam 70%.

Graduate attributes

• Applying mathematical knowledge (1)
• Interpreting data and drawing conclusions (2)
• Formulating mathematical problems (3)
• Problem solving skills (4)
• Flexibility in working environment (5)

Linkage past

Prerequisite is a pass in Mathematics IB (MATHS 1012) or Mathematics IIM (MATHS 2004). It will be an advantage to have done Real Analysis II (PURE MTH 2003), although the necessary material from this course is revised during lectures.

Linkage present

This course lays the theoretical groundwork for studies in differential equations, optimization, Fourier series and mathematical physics.

Linkage future

For students wishing to study mathematical physics, the theory of differential equations, signal processing, or any analysis area in Pure Mathematics this course is essential.

Restrictions

None.

Recommended text

None.