# 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.

## Analysis

### MATH97148 Fourier Analysis and Theory of Distributions

**Dr I. Krasovsky****Term 2**

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.

### MATH97149 Measure and Integration

**Dr H. Altman**

**Term 1**

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.

### MATH97172* Stochastic Calculus with Applications to non-Linear Filtering

**Professor ****D. Crisan**

**Term 2**

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).

### MATH97173 Functional Analysis

**Prof B. Zegarlinski**

**Term 1**

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.

### MATH97220 Markov Processes

**Term 2****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-

Kolmogorov equations,

irreducible, Perron-Froebenius theorem for stochastic matrices, recurrent and

transient.

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.

### MATH97167 Probability Theory

**Prof B. ZegarlinskiTerm 2**

*Prerequisites: Measure and Integration (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.

## Geometry

### MATH97150 Geometry 1: Algebraic Curves

**Prof J. Nicaise**

**Term 1**

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.

### MATH97151 Geometry 2: Algebraic Topology

**Dr S. Sivek**

**Term 2**

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.

### MATH97153* Algebraic Geometry

**Dr T. SchedlerTerm 2**

*Pre-requisites: Commutative Algebra (Term 1)*

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.

### MATH97160 Geometry of Curves and Surfaces

**Dr D. Cheraghi**

**Term 2**

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 R^{3}, 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.

### MATH97161* Riemannian Geometry

**Dr M. Taylor**

**Term 2**

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.

### MATH97162* Manifolds

**Prof P. Cascini**

**Term 1**

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

### MATH97163* Differential Topology

**Dr J. Jackson** **Term 2**

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.

### MATH97165* Complex Manifolds

**Prof M. Guaraco**

**Term 2**

*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

### MATH97141 Group Theory

**Prof A. IvanovTerm 1**

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.

### MATH97142 Galois Theory

**Professor A. CortiTerm 2**

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.

### MATH97143 Group Representation Theory

**Dr T. Schedler **

**Term 2**

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.

### MATH97164* Commutative Algebra

**Dr A. Pal**

**Term 1**

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.

### MATH97170* Algebra 4

**Prof A. SkorobogatovTerm 2**

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.

### MATH97174 Algebra 3

**Dr D. Helm****Term 1**

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.

### MATH97226 Graph Theory

**Dr R. Barham** **Term 2**

Standard definitions and basic results about graphs. Common graph constructions: complete graphs, complete bipartite graphs, cycle graphs. Matchings and König's Theorem. Connectivity and Menger's Theorem. Extremal graph theory. The theorems of Mantel and Turán. Hamilton cycles, and conditions for their existence. Ramsey Theory for graphs, with applications. The Probabilistic Method and random graphs. Evolution of random graphs.

## Number Theory

### MATH97144 Number theory

**Dr D. Helm**

**Term 1**

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.

### MATH97145 Algebraic Number Theory

**Dr M. TamiozzoTerm 2**

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.

### MATH97152* Elliptic Curves

**Dr A. Pozzi**

**Term 2**

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.

## Dynamics

### MATH97219* Random Dynamical Systems and Ergodic Theory

**Professor J. Lamb**

**Term 2**

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*8. *Mixing,*9. *Markov chains and ergodicity/mixing of Markov measures,*10. *Characterisation of weakly mixing by means of ergodicity of two-point motions.

### MATH97242 Bifurcation Theory

**Prof D. TuraevTerm 2**

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.

### MATH97245 Dynamics of Games

**Prof S. van Strien**

**Term 1**

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.

### MATH97284 Dynamical Systems

**Prof J.Lamb****Term 1**

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