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The fundamental laws of physics have a profound geometric character. This fact can only be appreciated by familiarising oneself with modern differential geometry, to which this module gives an introduction. The topics covered include manifolds, in particular geometric objects that can be placed on manifolds, such as vector fields, differential forms and more generally tensors. The module also discusses symmetries, including the notion of infinitesimal symmetries as captured by Killing vector fields. Integration of differential forms, including the Stokes’ theorem is covered as well. The final part of the module gives an introduction to Lie Groups and Lie algebras, building up to the description of Lie algebras in terms of left-invariant vector fields on the group manifold.  

This module provides an introduction to the modern theory of gravitation: Einstein's general theory of relativity.

Topics to be covered include:

• Specifying geometry
• Special relativity
• Equivalence principle
• General relativity
• Schwarzschild solution
• Schwarzschild black hole

This module will develop the ideas behind general relativity (GR) to an advanced level. You will explore solutions to these equations including black holes and cosmological solutions.

You will also have the opportunity to study more advanced topics including modified gravity models (eg models with extra dimensions) that are at the forefront of current research.

General relativity predicts the existence of black holes which are regions of space-time into which objects can be sent but from which no classical objects can escape. This module develops techniques to systematically study black holes and their properties, including horizons and singularities. Astrophysical processes involving black holes are discussed, and there is a brief introduction to black hole radiation discovered by Hawking. 

This module facilitates an understanding of Friedmann models and hot big bang. It encompasses the study of thermal history, freezout, relics, recombination, last scattering; dark matter candidates.

Other topics will include inflation, fluctuations from inflation, structure formation, gravitational lensing CMB anisotropies, and dark energy.

This module provides an introduction to the theoretical and conceptual foundations of quantum field theory, which is a highly versatile and important subject in modern theoretical and mathematical physics. After a short review of some elementary aspects of classical field theory, the first part of this module introduces the crucial concept of field quantisation and develops perturbative methods leading to the famous Feynman diagrams. The more advanced component of this module includes the study of renormalisation techniques for quantum field theories and a discussion of physical applications to quantum electrodynamics and the standard model of particle physics. 

The paradigm of Quantum Information Science (QIS) is that quantum devices made of systems such as atoms and photons, can outperform the present-day technology in key applications ranging from computing power and communication security to precision measurements. Quantum information processing and the measurement and control of individual quantum systems are central topics in QIS, lying at the intersection of quantum mechanics with ‘classical’ disciplines such as information theory, probability and statistics, computer science and control engineering.

This course gives an introduction to QIS, emphasising the differences and similarities between the classical and the quantum theories. After a short review of the necessary probabilistic notions, the first part introduces the operational framework of quantum theory involving the fundamental concepts of states, measurements, quantum channels, instruments. This includes some of the influential results in the field such as entanglement and quantum teleportation, Bell's theorem and the quantum no-cloning theorem. The second part covers at least two topics from: quantum Markovian evolutions, quantum statistics, continuous variable systems.

The dissertation is an extended piece of research, in an area covered by the taught modules but on a topic linked to a contemporary research forefront. The study will be largely self-directed, with oversight and support provided by a supervisor from the School of Mathematical Sciences or the School of Physics and Astronomy.

The topic could be based on a research investigation, a review of research literature, or a combination of these. You can choose among a range of topics proposed by supervisors or propose an original topic yourself. The dissertation offers an excellent introduction to exciting research topics, insights into how research is conducted, and a solid basis for pursuing a PhD.