Pure Mathematics Course list
Courses marked with (*) are core courses, of which at least three must be taken.
Please note: the modules listed here are for the current academic year. The programme is substantially the same from year to year, but modules are subject to change depending on your year of entry.
The Taught Course Centre (TCC) is a collaboration between the Mathematics Departments at the Universities of Bath, Bristol, Imperial, Oxford and Warwick. The lectures are open to all postgraduate students and are taking place in the room 6M42 (Huxley Building). The link to TCC can be found here.
M5P18 Fourier Analysis and Theory of Distributions
Dr S. Boegli
Spaces of test functions and distributions, Fourier Transform (discrete and continuous), Bessel’s, Parseval’s Theorems, Laplace transform of a distribution, Solution of classical PDE’s via Fourier transform, Basic Sobolev Inequalities, Sobolev spaces.
M5P19 Measure and Integration
Dr I. Krasovsky
Rings and algebras of sets, construction of a measure. Measurable functions and their properties, Egorov's theorem, convergence in measure. Lebesgue integral, its elementary properties, integral and sequences, Fubini theorem. Differentiation and integration: monotone functions, functions of bounded variation, absolutely continuous functions, signed measures. Lebesgue-Stiltjes measures. Lp spaces.
M5P41* Analytic Methods in Partial Differential Equations
Dr G. Holzegel
The main object of this module is to introduce several fundamental techniques of analysis for the study of partial differential equations.
The topic will include Fourier analysis, distributions, differential operators, pseudo-differential operators. There will be a review of Sobolev spaces, embedding theorems, potentials. We will apply it to study L2 properties, almost orthogonality, and the regularity of wave (hyperbolic) equations as well as elliptic and parabolic equations.
M5P6 Probability Theory
Prof B. Zegarlinski
Prerequisites: Measure and Integration (M3/4P19, Term 1)
A rigorous approach to the fundamental properties of probability. Probability measures. Random variables Independence. Sums of independent random variables; weak and strong laws of large numbers. Weak convergence, characteristic functions, central limit theorem. Elements of Brownian motion. Martingales.
M5P60 Geometric Complex Analysis
Dr A. De Zotti
Complex analysis is the study of the functions of complex numbers. It is employed in a wide range of topics, including dynamical systems, algebraic geometry, number theory, and quantum field theory, to name a few. On the other hand, as the separate real and imaginary parts of any analytic function satisfy the Laplace equation, complex analysis is widely employed in the study of two-dimensional problems in physics such as hydrodynamics, thermodynamics, Ferromagnetism, and percolations.
While you become familiar with basics of functions of a complex variable in the complex analysis course, here we look at the subject from a more geometric viewpoint. We shall look at geometric notions associated with domains in the plane and their boundaries, and how they are transformed under holomorphic mappings. In turn, the behavior of conformal maps is highly dependent on the shape of their domain of definition. Below is a rough guide to the syllabus.
Part 1) Elements of holomorphic mappings: Poincare metric, Schwarz-Pick lemma, Riemann mapping theorem, growth and distortion estimates, normal families, canonical mappings of multiply connected regions.
Part 2) Elements of Potential theory: Dirichlet problem, Green's function, logarithmic potential, Harnack inequality, harmonic measure.
Part 3) Elements of quasi-conformal mappings and elliptic PDEs, Beltrami equation, singular integral operators, measurable Riemann mapping theorem.
M5P67* Stochastic Calculus with Applications to non-Linear Filtering
Professor D. Crisan
Prerequisites: Ordinary differential equations, partial differential equations, real analysis, probability theory.
The course offers a bespoke introduction to stochastic calculus required to cover the classical theoretical results of nonlinear filtering as well as some modern numerical methods for solving the filtering problem. The first part of the course will equip the students with the necessary knowledge (e.g., Ito Calculus, Stochastic Integration by Parts, Girsanov’s theorem) and skills (solving linear stochastic differential equation, analysing continuous martingales, etc) to handle a variety of applications. The focus will be on the use of stochastic calculus to the theory and numerical solution of nonlinear filtering.
1. Martingales on Continuous Time (Doob Meyer decomposition, L_p bounds, Brownian motion, exponential martingales, semi-martingales, local martingales, Novikov’s condition)
2. Stochastic Calculus (Ito’s isometry, chain rule, integration by parts)
3. Stochastic Differential Equations (well posedness, linear SDEs, the Ornstein-Uhlenbeck process, Girsanov's Theorem)
4. Stochastic Filtering (definition, mathematical model for the signal process and the observation process)
5. The Filtering Equations (well-posedness, the innovation process, the Kalman-Bucy filter)
6. Numerical Methods (the Extended Kalman-filter, Sequential Monte-Carlo methods).
M5P7 Functional Analysis
Prof B. Zegarlinski
This module brings together ideas of continuity and linear algebra. It concerns vector spaces with a distance, and involves linear maps; the vector spaces are often spaces of functions.
Vector spaces. Existence of a Hamel basis. Normed vector spaces. Banach spaces. Finite dimensional spaces. Isomorphism. Separability. The Hilbert space. The Riesz-Fisher Theorem. The Hahn-Banach Theorem. Principle of Uniform Boundedness. Dual spaces. Operators, compact operators. Hermitian operators and the Spectral Theorem.
M5P70 Markov Processes
Prof X-M Li
Markov processes are widely used to model random evolutions with the Markov property `given the present, the future is independent of the past’. The theory connects with many other subjects in mathematics and has vast applications. This course is an introduction to Markov processes. We aim to build intuitions and foundations for further studies in stochastic analysis and in stochastic modelling.
The module is largely self-contained, but it would be useful for students to also take Measure and Integration (M345P19). A good knowledge of real analysis would be helpful (M2PM1).
It is related to:
Applied probability (M345S4), Random Dynamical Systems and Ergodic Theory (M4PA40), Probability theory (M345P6), Stochastic Calculus with Applications to non-Linear Filtering (M45P67), Stochastic Differential Equations (M45A51),
Stochastic simulation (M4S9*), Ergodic Theory (M4PA36), Computational Stochastic Processes (M4A44), and many Mathematical Finance modules.
Contents: 1. Discrete time and finite state Markov chains : Chapman-
irreducible, Perron-Froebenius theorem for stochastic matrices, recurrent and
2. Discrete time Markov processes on general state space. Conditional
expectations, Chapman-Kolmogorov equation, Feller property, strong Feller
property, Kolmogorov's theorem, stopping times, strong Markov, stationary
process, weak convergence and Prohorov's theorem, Existence of invariant
measures : Krylov-Bogolubov method, Lyapunov method. Ergodicity by
contraction method and Doeblin's criterion. Structures of invariant measures,ergodic theorems.
M5P20 Geometry I: Algebraic Curves
Plane algebraic curves including inflection points, singular and non-singular points, rational parametrisation, Weierstrass form and the Group Law on non-singular cubics. Abstract complex manifolds of dimension 1 (Riemann surfaces); elliptic curves as quotients of C by a lattice. Elliptic integrals and Abel’s theorem.
M5P21 Geometry II: Algebraic Topology
Dr C. Urech
Homotopies of maps and spaces. Fundamental group. Covering spaces, Van Kampen (only sketch of proof). Homology: singular and simplicial (following Hatcher’s notion of Delta-complex). Mayer-Vietoris (sketch proof) and long exact sequence of a pair. Calculations on topological surfaces. Brouwer fixed point theorem.
M5P33* Algebraic Geometry
Dr G. Fernandez da Silva
Pre-requisites: M4P55 Commutative Algebra
Algebraic geometry is the study of the space of solutions to polynomial equations in several variables. In this course, you will learn to use algebraic and geometric ideas together, studying some of the basic concepts from both perspectives and applying them to numerous examples.
Affine varieties, projective varieties. The Nullstellensatz.
Morphisms and rational maps between varieties. Chevalley's theorem, completeness of projective varieties.
Dimension. Regular and singular points.
Examples of algebraic varieties.
M5P5 Geometry of Curves and Surfaces
The main object of this module is to understand what is the curvature of a surface in 3-dimensional space.
Topological surfaces: Defintion of an atlas; the prototype definition of a surface; examples. The topology of a surface; the Hausdorff condition, the genuine definition of a surface. Orientability, compactness. Subdivisions and the Euler characteristic. Cut-and-paste technique, the classification of compact surfaces. Connected sums of surfaces. Smooth surfaces: Definition of a smooth atlas, a smooth surface and of smooth maps into and out of smooth surfaces. Surfaces in R3, tangents, normals and orientability. The first fundamental form, lengths and areas, isometries. The second fundamental form, principal curvatures and directions. The definition of a geodesic, existence and uniqueness, geodesics and co-ordinates. Gaussian curvature, definition and geometric interpretation, Gauss curvature is intrinsic, surfaces with constant Gauss curvature. The Gauss-Bonnet theorem. (Not examinable and in brief) Abstract Riemannian surfaces, metrics. Mean curvature and minimal surfaces, including the definition of mean curvature, its geometric interpretation, the definition of minimal surfaces and some examples.
M5P51* Riemannian Geometry
Dr H. Arguz
Prerequisites: Geometry of Curves and Surfaces (M4/4P5) and Manifolds (M4P52).
The main aim of this module is to understand geodesics and curvature and the relationship between them. Using these ideas we will show how local geometric conditions can lead to global topological constraints. Theory of (embedded) surfaces: Gauss map, second fundamental form, curvature and Gauss Theorem Egregium. Riemannian manifolds: Levi-Civita connection, geodesics, (Riemann) curvature, Jacobi fields. Isometric immersions and second fundamental form. Completeness: Hopf-Rinow Theorem and Hadamard Theorem. Constant curvature. Variations of energy: Bonnet-Myers Theorem and Synge Theorem.
Dr S. Feyzbakhsh
Smooth manifolds, quotients, smooth maps, submanifolds, rank of a smooth map, tangent spaces, vector fields, vector bundles, differential forms, the exterior derivative, orientations, integration on manifolds (with boundary) and Stokes' Theorem. This module focuses on foundations as well as examples
M5P54* Differential Topology
Dr S.A. Filippini
Prerequisites: Fundamental group and covering spaces from Algebraic Topology (M4P21) and vector fields and differential forms, derivatives and pull-backs of smooth maps, exterior differentiation and integration from Manifolds (M4P52).
Differential topology is concerned with the topology of smooth manifolds.
The first part of the module deals with de Rham cohomology, a form of cohomology defined in terms of differential forms. We will prove the Mayer-Vietoris exact sequence, Künneth formula and Poincaré duality in this context, and discuss degrees of maps between manifolds.
The second part of the module introduces singular homology and cohomology, the relation to de Rham cohomology via de Rham's theorem, and the general form of Poincaré duality. Time permitting, there will also be a brief introduction to Morse theory.
M5P57* Complex Manifolds
Prerequisite: Manifolds (M4P52). Some useful overlap with Differential Topology (M4P54).
Complex and almost complex manifolds, integrability. Examples such as the Hopf manifold, projective space, projective varieties. Hermitian metrics, Chern connection. Various equivalent formulations of the Kaehler condition. Hodge decomposition for Kaehler manifolds. Line bundles and Kodaira embedding. Statement of GAGA. Basic Kodaira-Spencer deformation theory.
Algebra and Discrete Mathematics
M5P10 Group Theory
Prof A. Ivanov
An introduction to some of the more advanced topics in the theory of groups. Composition series, Jordan-Hölder theorem, Sylow’s theorems, nilpotent and soluble groups. Permutation groups. Types of simple groups. Automorphisms. Free groups, Generators and relations. Free products.
M5P11 Galois Theory
Professor A. Corti
The formula for the solution to a quadratic equation is well-known. There are similar formulae for cubic and quartic equations, but no formula is possible for quintics. The module explains why this happens. Irreducible polynomials. Field extensions, degrees and the tower law. Extending isomorphisms. Normal field extensions, splitting fields, separable extensions. The theorem of the primitive Element. Groups of automorphisms, fixed fields. The fundamental theorem of Galois theory. The solubility of polynomials of degree at most 4. The insolubility of quintic equations.
M5P12 Group Representation Theory
Dr T. Schedler
Representations of groups: definitions and basic properties. Maschke's theorem, Schur's lemma. Representations of abelian groups. Tensor products of representations. The character of a group representation. Class functions. Character tables and orthogonality relations. Finite-dimensional algebras and modules. Group algebras. Matrix algebras and semi-simplicity. Representations of quivers.
M5P17 Algebraic Combinatorics
Dr J. Fawcett
An introduction to a variety of combinatorial techniques that have wide applications to other areas of mathematics.
Elementary coding theory. The Hamming metric, linear codes and Hamming codes. Combinatorial structures: block designs, affine and projective planes. Construction of examples using finite fields and vector spaces. Steiner systems from the Golay code. Basic theory of incidence matrices. Strongly regular graphs: examples, basic theory, and relationship with codes and designs.
The Mathieu group and their relationship with codes and strongly regular graphs.
M5P46* Lie Algebras
Dr A. Pal
The semisimple complex Lie Algebras: root systems, Weyl groups, Dynkin diagrams, classification. Cartan and Borel subalgebras. Classification of irreducible representations.
M5P55* Commutative Algebra
Prof A. Skorobogatov
Prime and maximal ideals, nilradical, Jacobson radical, localization. Modules. Primary decomposition of ideals. Applications to rings of regular functions of affine algebraic varieties. Artinian and Noetherian rings, discrete valuation rings, Dedekind domains. Krull dimension, transcendence degree. Completions and local rings. Graded rings and their Poincaré series.
M5P61* Infinite Groups
Dr J. Britnell
Free groups. Group presentations, Tietze transformations, the word problem. Residually finite groups. Cayley graphs, actions on graphs, the Nielsen--Schreier Theorem. Free products, the Table-Tennis Lemma, amalgams. HNN extensions, the Higman Embedding Theorem, the Novikov--Boone Theorem. Geometry of groups, hyperbolic groups.
M5P63* Algebra IV
This course is a selection of topics in advanced algebra. It will be useful for the students who want to specialise in algebra, number theory, geometry or topology.
Co-requisites: Algebra 3 (M3P8) and Galois Theory (M3P11). Group Theory (M3P10) and Group Representations (M3P12) will be useful but are not obligatory.
Projective, injective and flat modules.
Modules over principal ideal domains.
Abelian categories, resolutions and derived functors.
Group homology and cohomology.
M5P65 Mathematical Logic
Prof D. M. Evans
The module is concerned with some of the foundational issues of mathematics. In propositional and predicate logic, we analyse the way in which we reason formally about mathematical structures. In set theory, we will look at the ZFC axioms and use these to develop the notion of cardinality. These topics have applications to other areas of mathematics: formal logic has applications via model theory and ZFC provides an essential toolkit for handling infinite objects.
In addition to the material below, this M4 version of the module will have additional extension material for self-study. This will require a deeper understanding of the subject than the corresponding M3 module.
Propositional logic: Formulas and logical validity; a formal system; soundness and completeness.
Predicate logic: First-order languages and structures; satisfaction and truth of formulas; the formal system; Goedel’s completeness theorem; the compactness theorem; the Loewenheim-Skolem theorem.
Set theory: The axioms of ZF set theory; ordinals; cardinality; the Axiom of Choice.
M5P72 Modular Representation Theory
Prof M. Liebeck
Modular representation theory is the study of representations of finite groups over fields of characteristic other than zero. It has many applications and connections both inside and outside group theory. The subject was pioneered
by Richard Brauer, whose methods were mainly character-theoretic. Later on, the theory was revolutionised by J. A. Green, whose techniques were different to those of Brauer, the main goal being to understand the modules, rather than just their characters. This course will combine both approaches. The material covered will include Brauer characters, defect groups, blocks, decomposition numbers as well as projective and injective modules. For much of the theory, our guiding example will be the family of finite simple groups PSL(2,p), where p is a prime number.
-- Modular character theory: p-singular and p-regular elements, Brauer characters.
-- Decomposition numbers and the decomposition matrix. Counting irreducible modules.
-- Irreducible, projective and injective modules. Projective indecomposable modules.
-- Group algebras, blocks and defect groups.
-- Vertices and sources; the Green correspondence.
M5P8 Algebra 3
Dr D. Helm
Rings, integral domains, unique factorization domains. Modules, ideals homomorphisms, quotient rings, submodules quotient modules. Fields, maximal ideals, prime ideals, principal ideal domains. Euclidean domains, rings of polynomials, Gauss’s lemma, Eisenstein’s criterion. Field extensions. Noetherian rings and Hilbert’s basis theorem. Dual vector space, tensor algebra and Hom. Basics of homological algebra, complexes and exact sequences.
M5P14 Number theory
Prof T. Gee
The module is concerned with properties of natural numbers, and in particular of prime numbers, which can be proved by elementary methods. Fermat-Euler theorem, Lagrange's theorem. Wilson's theorem. Arithmetic functions, multiplicative functions, perfect numbers, Möbius inversion, Dirichlet Convolution. Primitive roots, Gauss's theorem, indices. Quadratic residues, Euler's criterion, Gauss's lemma, law of quadratic reciprocity, Jacobi symbol. Sums of squares. Distribution of quadratic residues and non-residues. Irrationality, Liouville's theorem, construction of a transcendental number. Diophantine equations. Pell's equation, Thue's Theorem, Mordell's equation.
M5P15 Algebraic Number Theory
Dr A. Caraiani
An introduction to algebraic number theory, with emphasis on quadratic fields. In such fields the familiar unique factorisation enjoyed by the integers may fail, but the extent of the failure is measured by the class group. The following topics will be treated with an emphasis on quadratic fields . Field extensions, minimum polynomial, algebraic numbers, conjugates and discriminants, Gaussian integers, algebraic integers, integral basis, quadratic fields, cyclotomic fields, norm of an algebraic number, existence of factorisation. Factorisation in Ideals, Z -basis, maximal ideals, prime ideals, unique factorisation theorem of ideals and consequences, relationship between factorisation of numbers and of ideals, norm of an ideal. Ideal classes, finiteness of class number, computations of class number. Fractional ideals, Minkowski’s theorem on linear forms, Ramification, characterisation of units of cyclotomic fields, a special case of Fermat’s last theorem.
M5P32* Elliptic Curves
Prof T. Gee
The p -adic numbers. Curves of genus 0 over Q . Cubic curves and curves of genus 1. The group law on a cubic curve. Elliptic curves over p -adic fields and over Q . Torsion points and reduction mod p . The weak Mordell-Weil theorem. Heights. The (full) Mordell-Weil theorem.
M5P58* Modular Forms
Dr D. Helm
The action of SL(2,R) on the upper half plane. Congruence subgroups. Fundamental domains. Modular curves. Cusps. Modular forms. Expansions at cusps. Examples: Eisenstein series. The Delta function. Finite-dimensionality of spaces of modular forms. Hecke operators. Eigenforms. Petersson inner product. Non-examinable: relation to elliptic curves. Fermat's Last Theorem.
M5PA23 Dynamical Systems
The theory of Dynamical Systems is an important area of mathematics which aims at describing objects whose state changes over time. For instance, the solar system comprising the sun and all planets is a dynamical system, and dynamical systems can be found in many other areas such as finance, physics, biology and social sciences. This course provides a rigorous treatment of the foundations of discrete-time dynamical systems, which includes the following subjects:
- Periodic orbits
- Topological and symbolic dynamics
- Chaos theory
- Invariant manifolds
- Statistical properties of dynamical systems
M5PA24 Bifurcation Theory
Dr D. Turaev
This module serves as an introduction to bifurcation theory, concerning the study of how the behaviour of dynamical systems (ODEs, maps) changes when parameters are varied.
The following topics will be covered:
1) Bifurcations on a line and on a plane.
2) Centre manifold theorem; local bifurcations of equilibrium states.
3) Local bifurcations of periodic orbits – folds and cusps.
4) Homoclinic loops: cases with simple dynamics, Shilnikov chaos, Lorenz attractor.
5) Saddle-node bifurcations: destruction of a torus, intermittency, blue-sky catastrophe.
6) Routes to chaos and homoclinic tangency.
M5PA38 Advanced Dynamical Systems*
Prof J. Lamb
This reading course deals with topics in dynamical systems at an advanced level, touching upon current frontline research. Each year a selection will be made of material from the area of local bifurcation theory, global bifurcation theory, ergodic theory of dynamical systems or dynamical systems methods for PDEs/FDEs.
In this year's course we will give an introduction to the theory of homeomorphisms and diffeomorphisms of the circle. It is one of the oldest topics in dynamical systems due to its connections with celestial mechanics.
This topic has remained very active from the early 20th century until now, with several successive breakthroughs: combinatorial and topological study in the beginning, then study of the differentiable structure of the orbits leading to KAM theory, and different aspects of renormalization until very recently. We will try to cover as much of the theory as possible, thus giving the students a glimpse to various active fields and to important techniques in Dynamical Systems. A first course in dynamical systems is sufficient background for this course.
Assessment: Students taking the course for credit are to prepare an essay (counting for 60%) and give an oral presentation about their work (counting for 40%).
M5PA40 Random Dynamical Systems and Ergodic Theory
Professor J. Lamb
Ergodic theory has strong links to analysis, probability theory, (random and deterministic) dynamical systems, number theory, differential and difference equations and can be motivated from many different angles and applications. In contrast to topological dynamics, Ergodic theory focusses on a probabilistic description of dynamical systems, and hence, a proper background of probability and measure theory is required to understand even the basic material in ergodic theory. For this reason, the first part of the course will concentrate on a self-contained review of the required background; this can take up to three weeks and might be skipped if not necessary. The second part of the course will focus on selected topics in ergodic theory. The course will be organised as a reading course; there will weekly meetings, where selected material will be presented and discussed within the group; this will guide the independent study. The students will do a project in the second part of the course, which should be submitted by the end of the term, so that the project does not come into conflict with the exams. The project will count towards 60% of the mark. There will also be a thirty-minute regular oral exam, which consists of two parts, each of which will contribute 20% to the mark. The first part of the regular oral exam will concern a discussion about the project: the student will have five minutes time to explain the project, after which there will questions related to the project (up to ten minutes). The second half of the exam will consist of questions about the material of the course.
The core content of the course is given as follows:
1. Review of Probability/Measure/Integration Theory (in particular Carathéodory Theorem, Lebesgue integration, conditional expectations, Banach–Alaoglu Theorem, Lebesgue Density Theorem, Central Limit Theorems, Radon–Nikodym Theorem),
2. Invariant measures and Krylov–Bogolubov Theorem,
3. Poincaré recurrence,
4. Ergodic theorems (such as Birkhoff Ergodic Theorem, Maximal Ergodic Theorem),
5. Decay of correlations,
6. Detailed discussion of examples (such as circle maps, maps with critical points, hyperbolic toral automorphisms, Bernoulli shifts),
7. Ergodicity via Fourier series
9. Markov chains and ergodicity/mixing of Markov measures,
10. Characterisation of weakly mixing by means of ergodicity of two-point motions.
M5PA48 Dynamics of Games
Professor S. van Strien
Recently there has been quite a lot of interest in modeling learning through studying the dynamics of games. The settings to which these models may be applied is wide-ranging, from ecology and sociology to business, as actively pursued by companies like Google. Examples include
(i) optimization of strategies of populations in ecology and biology
(ii) strategies of people in a competitive environment, like online auctions or (financial) markets.
(iii) learning models used by technology companies
This module is aimed at discussing a number of dynamical models in which learning evolves over time, and which have a game theoretic background. The module will take a dynamical systems perspective. Topics will include replicator dynamics and best response dynamics.
M5PA50 Introduction to Riemann Surfaces and Conformal Dynamics
Dr F. Bianchi
This elementary course starts with introducing surfaces that come from special group actions (Fuchsian /Kleinian groups). It turns out that on such surfaces one can develop a beautiful and powerful theory of iterationsof conformal maps, related to the famous Julia and Mandelbrot sets. In this theory many parts of modern mathematics come together: geometry, analysis and combinatorics.
Topics: Riemann surfaces. The theory of iterated conformal maps. Properties of Julia, Fatou and Mandelbrot sets.
Part 1: Discrete groups, complex Mobius transformations, Riemann surfaces, hyperbolic metrics, fundamental domains.
Part 2: Normal families of maps and equicontinuity, iterations of conformal mappings, periodic points and local normal forms. Fatou/Julia invariant sets, post-critical set.
1) Kleinian Groups by Berbard Maskit,
2) The Geometry of Discrete Groups by Alan F. Beardon,
3) Dynamics in one complex variable by John Milnor,
4) Riemann surfaces, dynamics and geometry, lecture notes by Curtis McMullen.