The aim of the module is to provide the students with a solid background in partial differential equations at a research-led level. The module will formally introduce the students to the concept of variational formulation of partial differential equations and their applications to inverse problems, reflecting the research expertise of the department of Mathematics and Statistics.

The following syllabus is indicative of the content to be addressed in this module.  1. Metric spaces:  - neighborhoods, open sets, closure of a set (adherent points)  - convergence in a metric space  - compact and connected sets.  2. Introduction to functional analysis:  - bounded linear operators  - dual vector spaces  - adjoint operators  - compact operators.  3. Hilbert spaces:  - Projection on closed convex sets/subspaces  - Riesz representation theorem  - Lax-Milgram theorem  - The Fredholm alternative.  4.Variational formulation of partial differential equations (PDEs):  - Measure theory, Lebesgue integration and Lp spaces  - Weak derivatives and Sobolev spaces  - Introduction to weak solutions to PDEs and boundary value problems.  5. Applications to Inverse Problems:  - Formulation of an inverse problem, including Calderòn's inverse conductivity problem, optical tomography and related inverse problems of physical interest. - Applications of inverse problems to imaging.  

On successful completion of this module, students will be able to: 1. explain in professional reports the concepts of metric spaces and bounded linear operators that are fundamental in functional analysis; 2. examine the properties of bounded and compact linear operators on both finite and infinite dimensional spaces; 3. formulate and analyse boundary value problems in their variational form; 4. apply the knowledge of weak derivatives to formulate an inverse problem in its weak formulation.

On successful completion of this module, students will be able to: 1. advocate the key role of partial differential equations in real-world applications; 2. value the importance of a rigorous mathematical reasoning in applied mathematics.

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The module will be taught by introducing an environment for the students that is experiential, collaborative and challenge-driven, beyond the conventional lecture format. Students will develop advanced skills in the analysis and solution of partial differential equations and boundary value problems of physical interest in real-world applications, including inverse problems, as well as modern and cutting-edge mathematical tools with important applications in imaging. The module, embedded in a number of research-led undergraduate programmes, will provide the students with research-led skills in mathematical analysis and partial differential equations that reflect the CHALLENGE-DRIVEN & EXPERIENTIAL nature of the module and the programmes. Through continuous assessment, the module will stimulate collaboration between peers. The module, embedded in a number of COLLABORATIVE & CROSS-DISCIPLINARY programmes, will encourage students to think critically and to collaborate across the scientific disciplines of mathematics and physics. It will offer authentic learning experiences drawn from real-world examples. The module will provide a learning experience for the students that will stimulate the students to stay CURIOUS (think logically, analytically and critically and demonstrate strong problem-solving skills); to be COURAGEOUS (can undertake complex problems with diligent organisation and resilience, and communicate abstract concepts and highly technical problems effectively in a wide variety of contexts); to be ARTICULATE (can work and communicate effectively and efficiently on trans- and inter-disciplinary projects and teams); to be AGILE (can take a high-level view of a problem and formulate relevant models and modelling strategies that are flexible and generalisable to a wide variety of settings); to be RESPONSIBLE (able to challenge and question the appropriate use of a rigorous analytical techniques, modelling, and the responsible implementation of these).