Explore the fundamental properties of probability, including probability measures. independent random variables and elements of Brownian motion.

Assess ideas of continuity and linear algebra, with a particular emphasis on vector spaces and linear maps.

Deepen your understanding of Fourier analysis and how the Fourier transform works.

Become familiar with the stochastic calculus in relation to non-linear filtering and modern numerical methods for solving the filtering problem.

Discover why Markov processes are widely used to model random evolutions and how the theory connects with many other subjects in mathematics.

Understand the curvature of a surface in 3-dimensional space and expand your knowledge of topological surfaces.

Build your understanding of algebraic curves and explore concepts including Meromorphic differentials and Abel's theorem.

Engage with algebraic topology and advance your knowledge of homology theory, fundamental groups and the Galois correspondence for covering spaces.

Learn to use algebraic and geometric ideas together, studying basic concepts from both perspectives and applying them to numerous examples.

Appreciate geodesics and curvature and the relationship between them and use these ideas to analyse how local geometric conditions can lead to global topological constraints.

Explore the foundations and examples of manifolds, including smooth manifolds, quotients, tangent spaces and vector bundles.

Delve into key aspects of differential topology, which is concerned with the topology of smooth manifolds.

Enhance your understanding of the theory of almost complex manifolds and complex manifolds, considering examples including Kähler manifolds and complex projective manifolds.

Further your appreciation of algebra and explore the basics of homological algebra, complexes and exact sequences.

Familiarise yourself with key concepts on this introduction to some of the more advanced topics in the theory of groups.

Examine Galois theory and why the formula for the solution to a quadratic equation is well-known, while no formula is possible for quintics.

Study the elementary theory of graphs and analyse why graphs are used in many areas of mathematics and other fields.

Explore the representations of groups and the character of a group representation.

Discover how computer theorem provers are maturing and improve your understanding of this area of mathematics by studying the Lean theorem prover.

Develop your knowledge further with this introduction to commutative algebra that looks into the prime and maximal ideals, the nilradical and the Jacobson radical.

Explore semisimple complex Lie algebras and the classification of irreducible representations.

Analyse a range of advanced algebra topics, incorporating aspects of number theory, geometry and topology.

Examine the properties of natural numbers and assess various theorems relating to number theory.

Gain an appreciation for algebraic number theory and advance your understanding of quadratic fields.

Explore the different types of elliptic curves and investigate the Mordell-Weil theorem.

Develop an understanding of partial differential equations (PDE) appearing in physics and geometry, and classical techniques to study them analytically.

Analyse the way in which we reason formally about mathematical structures and look at the ZFC axioms to develop the notion of cardinality.

Analyse geometric notions associated with domains in the plane and their boundaries, and how they are transformed under holomorphic mappings.