MMath&Phys Mathematics and Physics

Topology provides the tools to study properties of shapes that are not affected by manipulations such as bending, stretching, or twisting. Such properties include compactness of the shape, whether the shape is connected, and how many holes the shape has. Topology is important in fields like algebraic geometry, functional analysis, theoretical physics, and has found successful applications in data analysis and computing.

Topological spaces also provide a foundation for analysis in settings where metrics are either not available or not immediately apparent, such as that of linear functionals on Banach spaces and bounded linear operators. The theory of linear operators is fundamental in areas such as partial differential equations, quantum physics, and representation theory. 

This unit will introduce the concept of a topological space; describe how shapes such as Klein bottles, Möbius bands, and tori can be thought of as topological spaces; cover various ways in which topological spaces can be distinguished; and cover some more advanced concepts in topology. It will then use some key topological ideas to develop the theory of Banach spaces and the operators between them; to equip spaces of operators with topologies; and to understand the spectra of such operators, concluding with applications to other parts of mathematics. 

To introduce the theory of topological spaces and continuous functions. To develop ability working with Banach spaces and their operators. To study applications of and connections between these topics. 

• Define and identify topologies on sets, continuous functions on topological spaces, and homeomorphisms between topological spaces.
• Construct topological spaces using subspace, quotient and product topologies.
• Distinguish topological spaces by their connectedness, compactness, convergence, and separation properties.
• Use the fundamental group and homology groups to classify topological spaces.
• Recognise Banach spaces and Hilbert spaces, and deduce and apply properties of Banach spaces and Hilbert spaces.
• Analyse spaces, functionals, and operators using strong and weak topologies.
• Use spectra to classify and compare linear operators.
• Apply the theory of linear operators to other areas of mathematics.

In addition to delivery of content in the two lectures per week, tutorials will provide an opportunity for students' work to be discussed and for feedback on their understanding to be given. Feedback on assessments will be provided. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Coursework: On returned scripts within a week of submission