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https://warwick.ac.uk/fac/sci/statistics/redirect?id=#MODULECODE#-moodle

In the ST118 page this will appear as: https://warwick.ac.uk/fac/sci/statistics/redirect?id=st118-moodle

Additional Information

Use the table below to populate the 'Additional Information' block in each module page. If the cell is left empty, the block will not appear.

Module Code Module Title Additional Information
MA117  

Prerequisites: No previous computing experience will be assumed, but students should have obtained a code to use the IT Services work area systems prior to this module. Information and assistance is available in the Student Computer Centre in the Library Road

Leads to: The following modules have this module listed as assumed knowledge or useful background:
MA124  

Assumed knowledge: None, other than that already covered in core first-year mathematics modules

Useful background: Prior experience with Python or other programming languages will be useful

Synergies: The lectures, course resources and assessment will make contact with material from other first-year mathematics modules, in particular:

Leads to: The following modules have this module listed as assumed knowledge or useful background:
MA132  

Assumed knowledge: Grade A in A-level Maths or equivalent.

Useful background: Some elementary knowledge of modular arithmetic, induction principle, set notation.

Synergies: Most later pure mathematics modules specifically:

Leads to: The following modules have this module listed as assumed knowledge or useful background:
MA133  

Assumed knowledge: A-level mathematics or equivalent, in particular Calculus

Useful background: Proficiency with Mechanics from Maths A-level, or having taken Physics A-level, useful but not essential, we will cover the necessary topics from first principles

Synergies: This module supports any module using differential or partial differential equations

Leads to: The following modules have this module listed as assumed knowledge or useful background:
MA138  

Assumed knowledge: Grade A in A-level Maths or equivalent.

Useful background: This module is a broad-based introduction to the language of mathematics. Learning a language takes time, so it would be very useful to learn or review the various number systems (integer, rational, real, complex), modular arithmetic, set theory notation, and the basic techniques of proofs (direct, by induction, by contradiction, by contrapositive, by cases, by construction).

Synergies: This module is core; it is relevant to, and required for, all other maths modules at Warwick. Most later pure mathematics modules specifically:

Leads to: The following modules have this module listed as assumed knowledge or useful background:
MA139  

Assumed knowledge: MA141 Analysis 1

Useful background: 

Synergies: Analysis is one of the two most fundamental parts of pure mathematics, the other being Algebra. This module and the first term module MA141 Analysis 1 but analysis also has close connections to applied mathematics, probability theory and physics. The most natural synergies are therefore:

Leads to: The following modules have this module listed as assumed knowledge or useful background:
MA140  

Assumed knowledge: Grade A in A-level Maths or equivalent

Synergies:

Leads to: The following modules have this module listed as assumed knowledge or useful background:
MA141  

Assumed knowledge: Grade A in A-level Further Maths or equivalent

Useful background: None

Analysis and Algebra are the two fundamental areas of pure mathematics. This module forms the foundations for all the following Analysis-based modules. Along with MA144 Methods of Mathematical Modelling 2.

Leads to: The following modules have this module listed as assumed knowledge or useful background:

All future Analysis or Analysis-related modules,

including the following Year 2 modules:

and familiarity with the results (if not the proofs) will be needed in:
MA142  

Recommended prerequisites: MA1K2 Refresher Mathematics

Leads to: The following modules have this module listed as assumed knowledge or useful background:

Synergies: specifically:
MA143  

Assumed knowledge: MA142 Calculus 1 but analysis also has close connections to applied mathematics, probability theory and physics. The most natural synergies are therefore:


    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA144  

    Assumed knowledge: Basic knowledge of vectors and matrices (e.g. dot and cross products, determinant of a 3x3 matrix). Equations of ellipses and hyperbolae.

    Synergies: 

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA145  

    Assumed knowledge: Grade A in A-level Further Maths or equivalent.

    Useful background: This module focuses on multivariable calculus, and so refreshing your memory on differential and integral calculus covered in school and in Term 1 modules will be useful. Reviewing techniques such as integration by substitution and the chain rule for differentiation, as well as content on vectors and matrices will be useful.

    Synergies: Many other first-year mathematical modules, including specifically:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA146  

    Assumed knowledge: None (standard entry criteria for Maths students suffice)

    Useful background: Modelling with differential equations, solution techniques for linear differential equations of first and second order, eigenvalues and -vectors of 2x2 matrices, python and jupyter notebooks.

    Synergies: MA124 Maths by Computer (python programming, problem solving on the computer)

    Leads to: The following modules have this module listed as assumed knowledge or useful background:

    Learning Outcomes: By the end of the module students should be able:

    • To understand the modelling cycle in science and engineering, to formulate mathematical models and problems using differential equations, and to use a variety of methods to reveal their main underlying dynamics.
    • To apply a range of techniques to solve simple ordinary differential equations (first order, second order, first order systems), and to gain insight into the qualitative behaviour of solutions.
    • To confidently deploy computational methods and software to validate results, to approximate solutions of more challenging problems, and to further investigate them.
    MA147  

    Assumed knowledge: None (standard entry criteria for Maths-related subjects suffice)

    Useful background: Modelling with differential equations, solution techniques for linear differential equations of first and second order, eigenvalues and eigenvectors of 2x2 matrices

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA148  

    Assumed knowledge: Grade A in A-level Further Maths or equivalent

    Useful background: This module is most closely related to:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA149  

    Assumed knowledge: Grade A in A-level Further Maths or equivalent

    Useful background:

    This module is most closely related to:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA150  

    Assumed knowledge: Grade A in A-level Further Maths or equivalent.

    Useful background: 

    Synergies: this module is most closely related to:

    Leads to: Most modules rely on the ideas of Linear Algebra, and it is either assumed knowledge or useful background for all future study in mathematics. The most direct following module is:

    though the scientific content will continue in:
    MA151  

    Assumed knowledge: Grade A in A-level Further Maths or equivalent

    Useful background: Knowledge of modular arithmetic, manipulation of polynomials (including long division), language of functions, terms ‘commutative’ and ‘associative’

    Synergies: Specifically:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA152  

    Assumed knowledge: MA141 Analysis 1 but analysis also has close connections to applied mathematics, probability theory and physics. The most natural synergies are therefore:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA1K2  
    Frequently Asked Questions
    • When does the module start/end? The module runs from 8th September 2025 until Friday 10th October 2025
    • Where can I find the website? https://warwick.ac.uk/ma1k2
    • I cannot access the website, what should I do?
      First note that you need to use your Warwick IT account to access the website. If you haven't already, ensure that you set up your IT account, which is part of the enrolment process. Once you have done this, allow 24 hours for the system to update, and if you still cannot access the page, then please complete the following webform: https://forms.office.com/e/LBkDA1P5hH
    • Is the module core? Yes, for all maths and joint-degree first-year undergraduates, i.e. those on the following courses:
      • Mathematics
      • Mathematics and X where X is Statistics, Physics, Economics, Business, Philosophy
      • Discrete Mathematics
      • MORSE, Data Science
    • Is there academic support available? Yes - see the Academic Support page for more information
    • Is the module assessed? Yes, there will be two computer-marked tests (each worth 25% of the module grade, pass mark 70%, can retake) and one self-marked exam (worth 50% of the module grade, cannot retake). For more information, see How this module is assessed.
    • Will the mark count towards my degree? No
    • Will the mark appear on my transcript? Yes
    • Is this a prerequisite for continuation of my degree, or for any future modules? No, the grade you achieve has no effect on your continuation/progression/classification of your degree. However, A-level maths will be needed so use this opportunity to refresh your skills and knowledge.
    • What happens if I fail MA1K2? There is no requirement to pass MA1K2 and no consequences should you fail the module. You will not be offered resits for MA1K2, as it is exempt from the right-to-remedy-failure policy.
    • Is there more information about the exam? Yes, there is a section all about the exam on the Moodle page, including a video, and sample questions.
    • When should I start? As soon as you are able. There are 16 examinable chapters, so we recommend spreading the work over a few weeks leading up to the start of term
    • How will I learn? There will be pre-recorded lectures, some self-study chapters and exercise sheets on each chapter for you to try. The module will be asynchronous (i.e. no live events) but there will be support on hand from a team of academics.
    • I have started late and don't have time to do everything. How should I prioritise? Prioritise the areas where you feel you could do with a little extra revision. You could try the two tests to check which areas you already feel confident in and which you need some revision in. You can always re-take the tests later to check if you have improved.
    • Do I need to buy any books? No, but you might find it useful to consult the books you already have from your A-levels (or equivalent)
    • I have a question not on this list. Where can I ask it? If you have a question about the module which isn't covered in this list, please ask on the MA1K2 forum  or contact the mathematics Taught Programmes Office via UGmathematics@warwick.ac.uk
    MA222  

    Assumed knowledge:

    MA142 Calculus 1:

    • Sequences
    • Convergence
    • Cauchy sequences
    • Series
    • Continuous functions
    • Differentiable functions

    MA138 Sets and Numbers:

    • Set theory
    • Proofs
    • Cardinality

    MA271 Mathematical Analysis 3:

    • Pointwise and uniform convergence of sequences of functions
    • Open and closed sets in ${\mathbb R}^n$

    Synergies: The following module goes well together with Metric Spaces:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA240  
    • Assumed knowledge:
      MA146 Methods of Mathematical Modelling 1: First order linear equations: first order linear equations, examples of existence and unique-ness, integration techniques (integrating factors, ..). Second order equations: general homogeneous equations and linear second order equations with constant coefficients, reduction to 2x2 systems, sketching the flow under a vector. Nonlinear equations and 2x2 systems: linear stability such as predator and prey models. Difference equation: discrete population models, stability and instability of solutions.

      MA151 Algebra I: Group theory: groups, abelian groups, cyclic groups, subgroups, key examples (numbers, dihedral group, isometry groups, symmetric group, alternating group), order of an element, Lagrange's Theorem
    MA241  

    Assumed knowledge: The module starts from first principles and requires only interest in Mathematics and some level of mathematical maturity

    Useful background:

    Synergies:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA243  

    Assumed knowledge: Basics of linear algebra:

    MA150 Algebra 2 or MA149 Linear Algebra or MA148 Vectors and Matrices

    • Vector spaces
    • Bases and dimension
    • Linear maps
    • Rank and nullity
    • Represent linear maps by matrices
    • Euclidean inner product
    • Eigenvalues and eigenvectors

    Useful background: Familiarity with the basic language of geometry:

     MA144 Methods of Mathematical Modelling 2 or MA145 Mathematical Methods and Modelling 2 

    • Euclidean distance and norm
    • Parametrised curves

    Synergies: The following modules go well together with Geometry:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA250  

    Assumed knowledge: We will use facts and techniques of the following topics, knowledge of which may have been obtained in the modules stated below or otherwise: Solutions to linear first and second order ODEs, separation of variables techniques - MA146 Methods of Mathematical Modelling 1 or MA147 Mathematical Methods and Modelling 1 or MA133 Differential Equations.

    Differentiation along a curve, Leibniz' rule and the divergence theorem (MA144 Methods of Mathematical Modelling 2 or MA145 Mathematical Methods and Modelling 2). Diagonalisation of symmetric matrices (MA150 Algebra 2 or MA149 Linear Algebra or MA148 Vectors and Matrices).

    Synergies: Sometimes, solving partial differential equations comes down to having to solve ordinary differential equations, and then computational techniques as discussed in MA2K4 Numerical Methods and Computing can prove useful. Eigenvalue problems for second order ordinary differential equations are investigated in MA254 Theory of ODEs in more depth. There are many applications of partial differential equations, for example, those discussed in PX263 Electromagnetic Theory and Optics and PX264 Physics of Fluids.

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA252  

    Assumed knowledge: None

    Useful background:

    Synergies: This module complements the following:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA254  

    Assumed knowledge:

    Synergies:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA256  

    Assumed knowledge: Students should have a good knowledge of differential equations and matrix-vector manipulation. Some knowledge of stochastic modelling would be a plus. The following modules will provide a good background to this module:

    Useful background: A good understanding of mathematical models of biological systems will help students to follow the material in this course. The book listed below by Murray "Mathematical Biology, An Introduction" provides a guide to modelling biological systems with differential equations.

    Synergies: The following year 2 modules will go well with this module:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA257  

    Assumed knowledge:

    MA151 Algebra 1 or MA267 Groups and Rings - rings, subrings, fields

    Useful background: Interest in Number Theory is essential

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA259  
    MA260  

    Assumed knowledge:

    • MA139 Analysis 2: sequences, convergence, Cauchy sequences, series, continuity, differentiability
    • MA132 Foundations: sets, proofs, cardinality
    • MA270 Analysis 3: pointwise and uniform convergence of sequences of functions, open and closed sets in R^n.

    Synergies: The following module goes well together with Norms, Metrics & Topologies:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA262  

    Assumed knowledge: MA124 Maths by Computer
    MA263  

    Assumed knowledge:

    • MA139 Analysis 2: epsilon-delta definition of continuity and continuous limits, properties of continuous functions, definition of derivative, Mean Value Theorem, Taylor's theorem with remainder, supremum and infimum.
    • MA144 Methods of Mathematical Modelling 2:partial derivatives, multiple integrals, parameterisation of curves and surfaces, arclength and area, line and surface integrals, vector fields.
    • MA150 Algebra 2: Rank-Nullity Theorem and its geometric interpretation, dependence of matrix representation of a linear map with respect to a choice of bases, determinant.
    • MA270 Analysis 3 : Differentiable Functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ and uniform convergence.

    Useful background: Plotting graphs and contour plots of simple functions of two variables; the use of appropriate mathematical software for this purpose is encouraged.

    Synergies: 

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA265  

    Assumed knowledge: We will use facts and techniques of the following topics, knowledge of which may have been obtained in the modules stated below or otherwise: Solutions to linear first and second order ODEs, separation of variables techniques - MA146 Methods of Mathematical Modelling 1 or MA147 Mathematical Methods and Modelling 1 or MA133 Differential Equations.

    Differentiation along a curve, Leibniz' rule and the divergence theorem (MA144 Methods of Mathematical Modelling 2 or MA145 Mathematical Methods and Modelling 2). Diagonalisation of symmetric matrices (MA150 Algebra 2 or MA149 Linear Algebra or MA148 Vectors and Matrices).

    Synergies: Sometimes, solving partial differential equations comes down to having to solve ordinary differential equations, and then computational techniques as discussed in MA2K4 Numerical Methods and Computing can prove useful. Eigenvalue problems for second order ordinary differential equations are investigated in MA254 Theory of ODEs in more depth. There are many applications of partial differential equations, for example, those discussed in PX263 Electromagnetic Theory and Optics and PX264 Physics of Fluids.

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA266  

    Assumed knowledge: Knowledge of vector spaces and matrices from MA150 Algebra 2 or MA149 Linear Algebra or MA148 Vectors and Matrices. In particular, understanding change of basis matrices, eigenvalues and eigenvectors, elementary row and column operations and diagonalisation of matrices

    Useful background: Group theory from MA151 Algebra 1 or MA267 Groups and Rings especially abelian groups

    Synergies: The following modules go well with this module:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA267  

    Assumed knowledge:

    MA138 Sets and Numbers

    • Number theory: congruence modulo-n, prime factorisation, Euclidean algorithm, greatest common divisors (gcd) and least common multiples (lcm).
    • Sets and functions: basic set theory, injective and surjective functions, equivalence relations.
    • Polynomials: multiplication and division, Euclidean algorithm, Remainder Theorem.

    Synergies: 

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA268  

    Assumed knowledge:

    MA132 Foundations

    • Number theory: prime factorisation, Euclidean algorithm, gcd and lcm, Chinese Remainder Theorem
    • Sets and functions: basic set theory, injective and surjective functions, bijections and their inverses, relations

    MA151 Algebra 1:

    • Binary operations
    • Fermat's Little Theorem, Euler's Theorem
    • Polynomials: multiplication and division, Euclidean algorithm, Remainder Theorem
    • Permutations: multiplication, decomposing into disjoint cycles, transpositions, even and odd permutations
    • Group theory: groups, abelian groups, cyclic groups, subgroups, key examples (numbers, dihedral group, isometry groups, symmetric group, alternating group), order of an element, Lagrange's Theorem
    • Ring theory: definitions of rings, subrings, ideals, cosets of ideals, quotient rings, fields. Key examples ($\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}/n\mathbb{Z}$, matrix rings, polynomial rings).
    • Matrices, determinants, echelon form, elementary matrices
    • Vector spaces, subspaces, bases, span, linear independence
    • Linear transformations

    Synergies: 

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA269  

    Assumed knowledge:

    Useful background:

    Synergies: The following modules go well together with this module:

    • MA250 Partial Differential Equations: Fourier transforms can be used to solve many linear PDEs on infinite domains, including all the ones covered in MA250
    • MA254 Theory of ODEs: Asymptotics can be seen as a generalization of the concept of linearization near critical points covered in this course
    • MA2K4 Numercial Methods and Computing: Asymptotics are used to simplify complex mathematical models, and numerical Fourier transforms are extensively used to speed up numerics

    Leads to:

    • MA3B8 Complex Analysis: Experience of using complex contour integration will help prepare for the more rigorous treatment in Complex Analysis
    • MA3D1 Fluid Dynamics: Many of the techniques and concepts covered in this course are used in fluid dynamics, including both asymptotics and Fourier transforms, and several examples in this course originate in fluid dynamics
    • MA3G1 Theory of Partial Differential Equations: Many of the techniques covered here can be used to solve, or understand the solutions of, many forms of PDEs
    MA270  

    Assumed knowledge: Notions of convergence, and basic results for sequences, series, differentiation and integration from introductory analysis modules like MA141 Analysis 1 and MA139 Analysis 2; knowledge of vector spaces from MA150 Algebra 2

    Useful background: Basic results about curves, surfaces and vector fields from MA144 Methods of Mathematical Modelling 2

    Synergies: MA250 Partial Differential Equations

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA271  

    Assumed knowledge: Notions of convergence, and basic results for sequences, series, differentiation and integration from introductory analysis modules like MA140 Mathematical Analysis 1 or MA142 Calculus 1 and MA152 Mathematical Analysis 2 or MA143 Calculus 2; knowledge of vector spaces from MA149 Linear Algebra or MA148 Vectors and Matrices

    Useful background: Basic results about curves, surfaces and vector fields from MA145 Mathematical Methods and Modelling 2 or MA133 Differential Equations

    Synergies: MA250 Partial Differential Equations

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA2K4  

    Assumed knowledge:

    Useful background: Good working knowledge in linear algebra and analysis

    Synergies: 

    Leads to: The following modules will have this module listed as assumed knowledge or useful background:

    MA301

    		  
    Assumed knowledge
    We will assume knowledge of the linear wave equation, including its well posedness and solution via separation of variables, from MA265 Methods of Mathematical Modelling 3 or MA250 Partial Differential Equations. Basic notions of asymptotic approximation from MA269 Asymptotics and Integral Transforms will be recalled as needed.
    Synergies
    The scalar wave theory explored here will complement the study of Maxwell’s equations in the MA302 Electromagnetism.
    MA3J4 Mathematical Modelling and PDEs has some discussion of linear dispersive waves in the final part, this module will pick up some of these ideas (group velocity and dispersion relations) and extend them to heterogeneous systems.
    The existence and uniqueness of solutions to PDEs is explored in detail in MA3G1 Theory of Partial Differential Equations. Conversely, this course will instead focus on explicit characterisation and approximation of solutions for the purpose of applications.
    Leads to
    This would lead nicely to MA4L0 Advanced Topics in Fluids, which has a brief discussion of gravity and capillary waves. will cover the theory of elliptic eigenvalue problems in more detail.

    MA302

    		  
    Assumed knowledge:
    Synergies:

     

    MA341

    		  

    Assumed knowledge: The module starts from first principles and requires only interest in Mathematics and some level of mathematical maturity

    Useful background:

    Synergies:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA343  

    Assumed knowledge: Basics of linear algebra:

    MA150 Algebra 2 or MA149 Linear Algebra or MA148 Vectors and Matrices

    • Vector spaces
    • Bases and dimension
    • Linear maps
    • Rank and nullity
    • Represent linear maps by matrices
    • Euclidean inner product
    • Eigenvalues and eigenvectors

    Useful background: Familiarity with the basic language of geometry:

     MA144 Methods of Mathematical Modelling 2 or MA145 Mathematical Methods and Modelling 2 

    • Euclidean distance and norm
    • Parametrised curves

    Synergies: The following modules go well together with Geometry:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA352  

    Assumed knowledge: None

    Useful background:

    Synergies: This module complements the following:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA354  

    Assumed knowledge:

    Synergies:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA356  

    Assumed knowledge: Students should have a good knowledge of differential equations and matrix-vector manipulation. Some knowledge of stochastic modelling would be a plus. The following modules will provide a good background to this module:

    Useful background: A good understanding of mathematical models of biological systems will help students to follow the material in this course. The book listed below by Murray "Mathematical Biology, An Introduction" provides a guide to modelling biological systems with differential equations.

    Synergies: The following year 2 modules will go well with this module:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA357  

    Assumed knowledge:

    MA151 Algebra 1 or MA267 Groups and Rings - rings, subrings, fields

    Useful background: Interest in Number Theory is essential

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA359  

    Assumed knowledge:

    MA270 Analysis 3 or MA271 Mathematical Analysis 3:

    • Uniform Convergence

    MA139 Analysis 2 or MA152 Mathematical Analysis 2:

    • Riemann Integration
    • Uniform Continuity

    MA260 Norms, Metrics and Topologies or MA222 Metric Spaces:

    • Definition of a topology
    • Open and closed sets
    • Norms

    Useful background: Good knowledge of core analysis courses including material about continuity and convergence.

    Synergies: The following modules go well together with Measure Theory:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA369  

    Assumed knowledge:

    Useful background:

    Synergies: The following modules go well together with this module:

    • MA250 Partial Differential Equations: Fourier transforms can be used to solve many linear PDEs on infinite domains, including all the ones covered in MA250
    • MA254 Theory of ODEs: Asymptotics can be seen as a generalization of the concept of linearization near critical points covered in this course
    • MA2K4 Numercial Methods and Computing: Asymptotics are used to simplify complex mathematical models, and numerical Fourier transforms are extensively used to speed up numerics

    Leads to:

    • MA3B8 Complex Analysis: Experience of using complex contour integration will help prepare for the more rigorous treatment in Complex Analysis
    • MA3D1 Fluid Dynamics: Many of the techniques and concepts covered in this course are used in fluid dynamics, including both asymptotics and Fourier transforms, and several examples in this course originate in fluid dynamics
    • MA3G1 Theory of Partial Differential Equations: Many of the techniques covered here can be used to solve, or understand the solutions of, many forms of PDEs
    MA377  

    Assumed knowledge:

    MA266 Multilinear Algebra: Jordan normal forms.

    MA268 Algebra 3 or MA267 Groups and Rings: Rings, Integral domains, UFD, PID, ED, Smith normal forms over integers, classification of finitely generated abelian groups.

    Useful background: Interest in Algebra and good working knowledge of Linear Algebra

    Synergies: The following modules go well together with Rings and Modules:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA390  

    Assumed knowledge:

    Useful background:

    Synergies:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA397  

    Assumed knowledge: None

    Useful background: None

    Synergies: Covers second year material useful for third year core modules.
    MA398  

    Assumed knowledge: The following core first and second year modules are particularly important:

    Useful background: Some knowledge of numerical concepts such as accuracy, iteration and stability as provided in MA2K4 Numerical Methods and Computing will become important in the context of this module. General interdisciplinary curiosity will also be supported through interactions with areas such as medical imaging and data science.

    Synergies: The following modules link up well with Matrix Analysis and Algorithms, either through methodology, computational or application-oriented content:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3A6  

    Assumed knowledge:

    It is an extremely good idea to revise finitely generated abelian groups and ideals before starting this course.

    Useful background:

    Synergies:

    • MA3D5 Galois Theory: Like this course, Galois Theory studies algebraic numbers; if you're interested in one you will probably enjoy the other as well. Some results will be stated without proof in this course, and proved in Galois Theory. The two courses have a lot of overlap in the pre-requisites, especially around polynomial rings and factorising integer polynomials.

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3B8  

    Assumed knowledge:

    Please note that MA270 Analysis 3 for the purposes of this course.

    Useful knowledge: The "assumed knowledge" (and their prerequisites) will be enough.

    Synergies: This course connects with virtually every other domain in both pure and applied mathematics.

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3D1  

    Assumed knowledge:

    MA259 Multivariable Calculus:

    • Multivariate scalar and vector functions
    • Differential identities
    • Integral theorems
    • Ability to perform line, surface and volumetric integrals

    MA250 Partial Differential Equations:

    • Derivation and solution of various differential equations as applied to fluid dynamics, most notably Laplace equations
    • Heat equation
    • Understanding of appropriate boundary conditions that accompany the equations
    • Methods of solution, including separation of variables and fundamental solution

    Useful background:

    MA3B8 Complex Analysis:

    • Cauchy-Riemann conditions
    • Holonomic functions
    • Complex integration
    • Conformal maps

    Synergies: Those who enjoy fluid dynamics may be interested in the following modules:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3D4  

    Assumed knowledge:

    Synergies:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3D5  

    Assumed knowledge:

    • MA150 Algebra 2: Linear independences, bases of vector spaces, dimension, linear maps, rank-nullity formula
    • MA132 Foundations: Factorisation of polynomials, long division and the Euclidean algorithm
    • MA268 Algebra 3: Fields, rings and ideals, quotient rings and the first isomorphism theorem - especially polynomial rings and their quotient rings by ideals. Groups and homomorphisms, normal subgroups and quotient groups and the isomorphism theorems.

    Useful background: Whilst we do use the technical machinery described as assumed knowledge, at some level this module is extremely hands on, and you will benefit by practising the Euclidean algorithm for polynomials, working with permutations (in permutation groups), and manipulating complex numbers (inverses, solving quadratic polynomials, and especially the roots of unity).

    Synergies:

    Although each subject is more general in differing respects, the fundamental objects of study in each module include fields that contain the rational numbers and are finite dimensional as a vector space over them - the set of all complex numbers you can write using rationals and the square root of 2 is an example. In Galois theory we study the symmetries of such fields, while in Algebraic Number Theory the focus is on number-theoretic questions, such as questions about factorisation.

    Galois Theory uses groups of permutations and their subgroups as fundamental objects that capture the symmetry of field extensions and of solutions of polynomials. Any familiarity with permutations and groups is good, and in particular soluble groups appear in both modules: in Galois Theory they capture the symmetries that arise when you repeatedly extract square roots, cube roots and higher.

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3D9  

    Assumed knowledge:

    Useful background: Some familiarity with MA254 Theory of ODEs may be useful.

    Synergies: This module goes well with the following modules:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3E1  

    Assumed knowledge:

    Useful background: MA266 Multilinear Algebra (orthonormal bases, dual vector spaces, tensor products)

    Synergies: This module goes well with other third year algebra modules, particularly Introduction to Group Theory

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3F1  

    Assumed knowledge:  

    MA222 Metric Spaces:

    • Topological spaces
    • Continuous functions
    • Homeomorphisms
    • Compactness
    • Connectedness

    MA151 Algebra 1:

    • Groups
    • Subgroups
    • Homomorphisms and Isomorphisms

    Useful background:  

    Synergies: The following modules go well together with Introduction to Topology:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3G1  

    Assumed knowledge:

    Useful background:

    Synergies:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3G6  

    Assumed knowledge: The ring theory part of the second year Maths core. In particular, the definitions of rings, ideals, homomorphisms and quotients of rings by ideals

    Useful background: The rest of the Algebra modules from the 1st and 2nd year core

    Synergies: The following modules go well together with Commutative Algebra:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3G7  

    Assumed knowledge:

    Useful background: 

    Synergies:

    Leads To: The following modules have this module listed as assumed knowledge or useful background:
    MA3G8  

    Assumed knowledge:

    MA3G7 Functional Analysis I:

    • Normed spaces
    • Banach spaces
    • Lebesgue spaces
    • Hilbert spaces
    • Dual spaces
    • Linear operators

    MA260 Norms, Metrics and Topologies or MA222 Metric Spaces:

    • Normed spaces
    • Metric spaces
    • Continuity
    • Topological spaces
    • Compactness
    • Completeness

    Useful background: 

    MA260 Norms, Metrics and Topologies or MA222 Metric Spaces:

    • Nowhere dense sets
    • Baire category theorem

    MA359 Measure Theory:

    • Lebesgue measure
    • Measurable functions
    • Integral with respect to a measure

    Synergies: The following modules go well together with Functional Analysis II:

    Leads To: The following modules have this module listed as assumed knowledge or useful background:
    MA3H0  

    Assumed knowledge:

    Useful background:

    Synergies: The following year 3 modules link up well with Numerical Analysis and PDEs, either through the use of numerical analysis, or by covering various aspects of partial differential equations:
    MA3H2  

    Assumed knowledge: MA359 Measure Theory and  ST350 Measure Theory for Probability. Alternatively, the students need to know the following basic facts: probability measure and expectation (including conditional expectation); convergence of random variables; the law of large numbers and central limit theorems; basic theory of Markov chains and random walks; relevant theorems of analysis such the Fubini's theorem, the dominated and the monotone convergence theorems. Most of the above facts are summarised on the course's Moodle page and covered by Chapter 1 and the Appendix of Rick Durret's book 'Probability: theory and examples'.

    Useful background: This module provides an introduction to phase transitions for Markov processes and Bernoulli percolation models. Phase transitions are ubiquitous in Nature: freezing and evaporation of water and spontaneous magnetisation of a ferromagnet are some of the most familiar examples. However the rigorous mathematical theory of phase transition is both exciting and hard. One source of difficulty is the non-analytical dependence of the observables detecting the phase transition (e. g. magnetisation) on the parameters controlling the phase (e. g. temperature). In the course we will treat rigorously two of the simplest models exhibiting phase transition: firstly, we will investigate the extinction phase transition for the well know Galton-Watson branching process from population dynamics, secondly - the percolation transition for Bernoulli percolation model on tree graphs and Z to the d.

    Galton-Watson branching process was introduced in the 19th century to investigate the chance of the perpetual survival of aristocratic families in Victorian Britain and has since became both a useful model for population dynamics and an interesting probabilistic model in its own right. Bernoulli percolations were introduced in the late 1950's to model the propagation of fluid through porous media and gained. Probabilistically it is the simplest model of spatial disorder. Each of the models is very easy to define, yet there are still many open research questions concerning both the branching process and the percolation model. For example, there have been already two Fields medals awarded in the 21st century for studying percolations (Smirnov and Duminil-Copin). Yet there are still some fundamental unresolved questions (e, g. the continuity of the percolation function), which will certainly bring you the Fields medal if you can answer them before you are 40!

    The beauty of the models we are studying in the course is in the possibility to understand them using elementary probabilistic methods. This is in part due to simplified proofs due to Duminil-Copin, Hofstadt, Heydenreich and many others which appeared only in the last decade. Thus the course will equip you with modern tools for studying probabilistic models of phase transitions. The acquired knowledge will allow you understand research papers on branching processes and percolations and will be applicable to the study phase transitions in applications such as biological and physical systems, communication networks and financial markets.

    Synergies: The following modules go well together with Markov Processes:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3H3  

    Assumed knowledge: Elementary set theory and Proof by induction, as covered in:

    Useful background: Set theoretic reasoning in topology, as in:

    Synergies:

    MA3H5  

    Assumed knowledge:

    MA263 Multivariable Analysis or MA259 Multivariable Calculus

    • Basic theory of differentiation, including statements (though not proofs) of Inverse and Implicit Function Theorems

    MA260 Norms, Metrics and Topologies or MA222 Metric Spaces

    • Integrations in several variables and familiarity with basic notions of point-set topology and metric spaces
    • Topologies
    • Continuity
    • Compactness
    • Connectedness
    • Completeness

    Useful background: MA254 Theory of ODEs - We will use some results about the existence, uniqueness and dependence on parameters of solutions to ODEs in one or two places of the course. These results will be stated and could be taken on faith, but some prior acquaintance might be useful.

    Synergies: The theory of manifolds is fundamental in many areas of modern mathematics. Modules that go well with this Module are (of course some choice should be made depending on whether your tastes are more analytic, geometric or topological):

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3H6  

    Assumed knowledge: Introductory topology and second year abstract algebra:

    Synergies: MA3H5 Manifolds

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3H7  

    Assumed knowledge:

    Useful background:

    Synergies:

    MA3J2  

    Assumed knowledge:

    MA241 Combinatorics I:

    • Graph theory
    • Hall's Theorem
    • Graph colouring

    MA222 Metric Spaces:

    • Norms on Euclidean space
    • Open and closed sets
    • Compactness

    MA268 Algebra 3:

    • Basic examples of finite fields

    ST120 Introduction to Probability:

    • Events
    • Probabilities
    • Random variables

    Useful background:

    ST120 Introduction to Probability:

    • Poisson distribution
    • Chebyshev's inequality
    • The Central Limit Theorem

    MA243 Geometry:

    • Projective geometry

    Synergies: The following module goes well together with Combinatorics II:

    Additional Resources
    MA3J3  

    Assumed knowledge: This module will assume knowledge from core mathematics modules, in particular, first-year knowledge of differential equations, and bits of both MA268 Algebra 3 and MA259 Multivariable Calculus. In more detail:

    MA268 Algebra 3: Groups, in particular permutation groups and groups and groups of non-singular matrices (Dihedral groups especially). Quotient Groups, isomorphism theorems and orbit-stabilizer.

    MA259 Multivariable Calculus: Differentiable functions, Inverse Function Theorem and Implicit Function Theorem.

    Useful background: Having taken MA254 Theory of ODEs (or taking in parallel) will be beneficial, but not essential.

    Synergies: Although containing a fair amount of "pure" maths, this is largely applying those theories and so this sits well beside modules such as MA256 Introduction to Mathematical Biology, MA390 Topics in Mathematical Biology, MA4E7 Population Dynamics, MA3J4 Mathematical Modelling with PDEs and MA4M1 Epidemiology by Example. It will also add extra context for modules such as MA3H5 Manifolds, MA3E1 Groups and Representations and MA4E0 Lie Groups, although here we only intersect with the basics of those.

    Leads to: This module doesn't formally lead to any further modules, but sits alongside, and gives useful context to, many of the modules in "Synergies" above.
    MA3J4  

    Assumed knowledge:

    Useful background:

    Synergies: The following modules go well together with Mathematical Modelling:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3J8  

    Prerequisites: There are no formal prerequisites beyond the core module MA260 Norms, Metrics and Topology, but any programming module and any of the following modules would be useful complements: MA3H0 Numerical Analysis and PDE.
    MA3K0  

    Assumed knowledge: Basic probability theory: random variables, law of large numbers, Chebycheff inequality, distribution functions, expectation and variance, Bernoulli distribution, normal distribution, Poisson distribution, exponential distribution, de Moivre Laplace theorem e.g. ST120 Introduction to Probability

    Some basic skills in analysis: MA270 Analysis 3. The module works in Euclidean vector space ${R}^n $ , so norm, basic inequalities, scalar product, linear mappings and matrix algebra (eigenvalues, eigenvectors, singular values etc) are relevant.

    Useful background: Know what a a probability measure/distribution is. Earlier probability modules will be of some use but not necessary. The framework is some mild probability theory (e.g. ST202 Stochastic Processes). Know what the Central Limit Theorem is (de Moirvre Laplace for general random variables).

    Synergies: In general the module is a mathematical basis for machine learning, data science and random matrix theory. The following modules provide some synergies and connections:

    There are also strong links and thus suitable combinations to the following modules:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3K1  

    Assumed knowledge: You are expected to have a solid background in the core mathematics modules taught in Year 1 and 2 of the Maths degree: MA141 Analysis 1, MA139 Analysis 2 and MA270 Analysis 3 and MA149 Linear Algebra are essential. MA263 Multivariable Analysis is useful. ST120 Introduction to Probability and statistics is required.
    MA3K4  

    Assumed knowledge:

    Useful background: Interest in Algebra

    Synergies: This module will go well with:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3K6  

    Assumed knowledge:

    Useful background: 
    MA3K7  

    Assumed knowledge:

    MA3K8  

    Assumed knowledge:

    Useful background: Newton's laws of motion, scalar potential of electrostatic field or gravitational field. However, this is a mathematics module and a physics background will not be required.

    Synergies: Variational problems arise whenever some quantity is to be optimised. This quantity can come from geometry (length, area), physics (energy), biology, economics. So all modules in which optimisation is considered are related to this module. Examples of such modules include:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA3K9  

    Assumed knowledge:

    • MA150 Algebra 2: A foundational understanding of vectors, matrices, and linear transformations is essential. These concepts are fundamental to many DSP techniques and algorithms

    Useful background: 

    • MA124 Mathematics by Computer: Our tutorials will leverage Python to explore DSP concepts. While we will offer coding guidance, a basic knowledge of Python will enhance your comprehension of the implementations and aid in evaluations
    • MA266 Multilinear Algebra: Grasping advanced linear algebra is crucial for understanding multi-dimensional DSP techniques. Many DSP algorithms and transformations originate from these linear algebra concepts
    • MA269 Asymptotics and Integral Transforms: This module provides insights into integral transforms, vital for signal analysis in the frequency domain. Asymptotic analysis is key for assessing signal behaviour at boundary conditions

    Synergies: This module pairs well with:

    • MA2K4 Numerical Methods and Computing: Digital Signal Processing often necessitates the use of numerical methods for signal analysis and filtering. This module introduces computational techniques directly applicable to DSP, enhancing both the efficiency and precision of signal processing algorithms
    • MA398 Matrix Analysis and Algorithms: Given that many DSP techniques revolve around matrix operations and transformations, a profound understanding of matrix analysis can significantly enhance the optimization and execution of DSP algorithms. This module delves into advanced matrix computations, essential for multi-dimensional signal processing
    MA3P2  
    Assumed knowledge
    MA151 Algebra 1 - basics of group theory
    MA150 Algebra 2 - matrices, row operations, determinants
    Not required but thematically linked: MA260 Norms, Metrics and Topologies
    Synergies

     
    MA424  

    Assumed knowledge:

    MA260 Norms, Metrics and Topologies or MA222 Metric Spaces:

    • Metric and topological spaces
    • Continuous functions
    • Homeomorphisms
    • Compactness
    • The Cantor set

    MA259 Multivariable Calculus:

    • Differentiable functions
    • Diffeomorphisms

    Useful background:

    Synergies:

    MA426  

    Assumed knowledge:

    MA268 Algebra 3:

    • Basic theory of groups, rings, fields

    MA257 Introduction to Number Theory:

    • Primes
    • Divisibility
    • Congruences

    MA3B8 Complex Analysis:

    • Holomorphic and meromorphic functions
    • Laurent series
    • Poles
    • Residues

    Useful background:

    MA4L7 Algebraic Curves:

    • Affine and projective plane curves
    • Bezout's theorem
    • Singular points
    • Change of coordinates

    MA4A5 Algebraic Geometry:

    • Algebraic varieties
    • Rational maps
    • Morphisms

    Synergies: The following modules go well together with Elliptic Curves:
    MA427  

    Assumed knowledge:

    MA150 Algebra 2:

    • Eigenvalues and eigenvectors

    MA222 Metric Spaces / MA260 Norms, Metrics and Topologies:

    • Metric spaces
    • Continuity
    • Compactness

    MA359 Measure Theory:

    • Abstract measures
    • Lebesgue measure
    • Convergence Theorems
    • L^1 and L^2 spaces

    Useful background:

    ST120 Introduction to Probability:

    • Probability spaces
    • Notion of random variable
    • Law of large numbers

    MA3G7 Functional Analysis I:

    • Hilbert Spaces
    • Orthonormal basis
    • Dual spaces

    MA3G8 Functional Analysis II:

    • Banach spaces
    • Hanh-Banach theorem
    • Convex sets

    MA433 Fourier Analysis:

    • Fourier series and their properties

    Synergies:

    MA433  

    Assumed knowledge: Familiarity with measure theory at the level of MA359 Measure Theory especially Fubini's Theorem, Dominated and Monotone Convergence Theorems.

    Useful background: Further knowledge of Functional Analysis such as: MA3G8 Functional Analysis II is helpful but not necessary. Topics such as norms of bounded linear operators will be reviewed in the module. Some basics about Hilbert spaces will also be reviewed in the module. The uniform boundedness principle will be stated without proof, but the other major results from functional analysis are not used.

    Synergies: The following modules go well together with Fourier Analysis:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA442  
    This module is a continuation of MA3K4 Introduction to Group Theory. The material in MA3K4 will be reviewed as we go through, but it will be assumed that students are already familiar with it. One significant difference in notation is that we shall be using right rather than left actions.
    In addition to the material below, the topic of groups defined by presentations using generators and defining relations (which are used in algebraic topology for example) will be covered more formally than in earlier group theory courses, with plenty of examples. These groups are defined as quotient groups of free groups, so we shall start with the definition and basic properties of free groups. We shall also introduce the Todd-Coxeter coset enumeration procedure, again with lots of examples. This method can often be used to prove that a given presentation defines a finite group.
    MA448  

    Assumed knowledge:

    Useful background: 

    Synergies:
    MA453  

    Assumed knowledge: Linear algebra and ring theory from Year 2
    MA473  

    Assumed knowledge: The groups theory and some geometric ideas from the second year Maths core:

    MA266 Multilinear Algebra:

    • Euclidean spaces
    • Abelian groups

    MA268 Algebra 3:

    • Groups, generators and relations.

    Useful background: Interest in Group Theory:

    MA3K4 Introduction to Group Theory:

    • Semidirect products

    Synergies: The following modules go well together with Reflection Groups:
    MA482  

    Assumed knowledge:

    Useful background: There will be links with material from several other modules: Solutions to elliptic and parabolic linear PDEs are very closely related, and students who have taken Additional Resources
    MA4A2  

    Assumed knowledge:

    MA359 Measure Theory:

    • Lebesgue integration
    • Fubini’s Theorem
    • Dominated Convergence Theorem
    • Divergence Theorem
    • Riesz Representation Theorem

    MA259 Multivariable Calculus:

    • Differentiable functions
    • Partial derivatives
    • Chain rule
    • Implicit and Inverse Function Theorem

    Useful background: 

    Synergies: This module fits well with MA4M2 Mathematics of Inverse Problems. Essential for research in much of geometry, analysis, probability and applied mathematics etc.

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA4A5  

    Assumed knowledge: MA3H5 Manifolds is useful background, but will be fully recalled.

    Synergies: The following modules go well together with Algebraic Geometry:

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA4C0  

    Assumed knowledge:

    MA266 Multilinear Algebra:

    • Bilinear forms
    • Eigenvalues and Eigenvectors

    MA259 Multivariable Calculus:

    • Differentiation of functions of several variables, including the Chain Rule
    • Inverse and Implicit Function theorems

    MA222 Metric Spaces:

    • Basic point set topology

    MA254 Theory of ODEs:

    • Existence and uniqueness of solutions to ODEs and their smooth dependence on parameters and initial conditions

    Useful background:

    Synergies:

    MA4E0  

    Assumed knowledge:

    • MA222 Metric Spaces : topological spaces
    • MA259 Multivariable Calculus : calculus of several variables including the Implicit Function and Inverse Function Theorems
    • MA3H5 Manifolds : knowledge of manifolds, tangent spaces and vector fields will help, although all necessary results from Manifolds will be reviewed in this course

    Useful background: A knowledge of calculus of several variables including the Implicit Function and Inverse Function Theorems, as well as the existence theorem for ODEs. A basic knowledge of manifolds, tangent spaces and vector fields will help. Results needed from the theory of manifolds and vector fields will be stated but not proved in the course.

    Synergies: Lie groups have both algebraic and geometric sides. These sides are studied deeply in the following two modules:
    MA4E7  

    Assumed knowledge: No formal prerequisites. Some practical knowledge of how to analyse and solve differential equations will be assumed. In particular, finding fixed points and their stability in systems of ODEs, drawing phase plane diagrams, solving difference equations, a very small amount of PDEs etc.

    Useful background: Additional Resources
    ST403  

    Assumed knowledge: At least one of:

    Useful background:

    Synergies: The following module goes well together with Brownian Motion:
    MA4H0  

    Assumed knowledge:

    Useful background:

    Synergies: This module provides a complementary view of dynamical systems theory to others offered by the department. It concentrates on continuous time and aspects relevant to physics and biology. If you want a well rounded training in dynamical systems theory you are recommended to take one of the other modules:

    Books on the subject: You may find the following three books useful:

    • MW Hirsch, S Smale & RL Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos
    • JD Meiss, Differential Dynamical Systems
    • RC Robinson, An Introduction to Dynamical Systems
    MA4H4  

    Assumed knowledge: Group theory, Euclidean and hyperbolic geometry, Fundamental group and covering spaces. These subjects are covered by Warwick courses MA3F1 Introduction to Topology.

    Useful background: Any course in algebra, geometry or topology, in particular,

    Synergies:

    The following modules go well together with Geometric Group Theory:
    MA4J0  

    Assumed knowledge:

    Useful background:

    Synergies:

    MA4J1  

    Assumed knowledge: This module assumes knowledge of various aspects of first and second year core maths material. Modules from other departments may also cover the necessary background. We list where the relevant material can be found for Maths and joint degree students.

    Useful background: MA3G1 Theory of PDEs

    Synergies: The third year modules listed under "Useful Background" would go well alongside this module. Fourth year modules which would also synergise well are:

    Some background on the theory of ODEs and PDEs would also be useful, as covered in Additional Resources
    MA4J3  

    Assumed knowledge: None

    Useful background:

    Synergies:

    MA4J5  

    Assumed knowledge:

    MA398 Matrix Analysis and Algorithms:

    • Methodological foundations in linear algebra and matrix algorithms as well as hands-on experience in programming

    ST120 Introduction to Probability:

    • Basic probability theory
    • Random variables

    Useful background:

    ST202 Stochastic Processes:

    • Markov processes and Markov chains

    MA241 Combinatorics:

    • Foundations of graph theory

    MA252 Combinatorial Optimisation:

    • Algorithms in graph theory and NP-hard problems

    Synergies: The following modules go well together with Structures of Complex Systems:
    MA4J7  

    Assumed knowledge:

    Useful knowledge: A certain level of mathematical maturity (comfort with proofs and routine computations). Some category theory (categories, functors, natural transformations) will make learning the material much easier.

    Synergies: This is a capstone of the undergraduate mathematics programme. It is a valuable course for anybody with an interest in "modern topology" (bringing us up to the 1960's). It is essential background for postgraduate study in geometry or topology, as well as many areas of algebra, number theory, and applied mathematics.

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA4J8  

    Assumed knowledge: Divisibility and ideals from MA3G6 Commutative Algebra module or my book "Undergraduate Commutative Algebra".

    Useful background: Material from, or an interest in:

    Synergies: The material of commutative algebra has many basic and more advanced links with Algebraic Geometry and Algebraic Number Theory. The module follows Additional Resources
    MA4L0  

    Assumed knowledge:

    Familiarity with the mathematical description of fluid dynamics is necessary.

    Experience with partial differential equations and methods of their solutions.

    Useful background:

    Some experience of modelling with PDEs.

    Synergies: The following modules would go well with Advanced Topics in Fluids:
    MA4L2  

    Assumed knowledge: Basic probability theory and some combinatorics:

    ST120 Introduction to Probability:

    • Notions of events and their probability
    • Conditional probabilities
    • Law of large numbers
    • Random variables
    • Joint distributions
    • Independence of random variables
    • Moment generating functions
    • Law of large numbers

    MA241 Combinatorics:

    • Basic counting
    • Binomial and multinomial theorems
    • Generating functions
    • Basics of graph theory

    MA3H2 Markov Processes and Percolation Theory:

    • Notion of Markov process

    Useful background: In addition, some notions of measure theory can be useful indeed:

    MA359 Measure Theory:

    • Fatou's lemma
    • Monotone and dominated convergence theorems
    • Fubini's theorem
    • Riesz representation theory

    Synergies: The models of statistical mechanics provide excellent illustrations for the following module:
    MA4L6  

    Assumed knowledge: Some basic real and complex analysis, including: uniform convergence, the Identity Theorem from complex analysis and especially Cauchy's Residue Theorem. This is covered in the modules MA270 Analysis 3 and (ideally) MA3B8 Complex Analysis. Although the module will not assume much specific content or results, it will have a serious "analytic" flavour of estimating objects and handling error terms. The most important thing is to be comfortable with this style of mathematics, which might be familiar from previous courses in analysis, measure theory or probability.

    Useful background: The module will assume very little from number theory, but in a few places it will be useful to know things like: the Chinese Remainder Theorem, the structure of the multiplicative group mod q. These are covered in e.g. MA268 Algebra 3 and MA257 Introduction to Number Theory.

    Synergies: This is fundamentally an analysis module, and would go well (in terms of the general style of the mathematics) with modules like: MA433 Fourier Analysis, MA4J0 Advanced Real Analysis and possibly MA427 Ergodic Theory. Those interested in number theory will probably also enjoy MA426 Elliptic Curves.
    MA4L7  

    Assumed knowledge: Some familiarity with basic ideas of commutative algebra is a prerequisite. As a rough guide, the lectures need the first half of MA3B8 Complex Analysis will be mentioned in explaining the purely algebraic discussion of zeros and poles of a rational function. In a similar way, the Cauchy integral theorem is good motivation for the full statement of the Riemann-Roch theorem, although it is not needed for the proof.

    Synergies: The course is a basic introduction to the study of algebraic varieties (and schemes) and their cohomology. The Riemann-Roch theorem for curves is a first major step towards the classification of algebraic curves, surfaces and higher dimensional varieties, that makes up a large component of modern algebraic geometry, and has applications across the mathematical sciences and theoretical physics.

    Leads to: The following modules have this module listed as assumed knowledge or useful background:
    MA4M1  

    Assumed knowledge:

    • Knowledge of general behaviour/steady states of ODEs. To revise or check your background knowledge on ODEs we recommend reading the following textbook which is available online through Warwick’s library: An Introduction to Ordinary Differential Equations (James Robinson)”
    • Basic programming skills (ODE solvers, functions, for/if/while loops, plotting, vectors and matrices)
    • The module is taught in R but the first 2 weeks is aimed at getting everyone to a basic level either by refreshing their knowledge or by learning R given prior programming experience in another language
    • Basic knowledge of key probability distributions and their properties (including gamma, Erlang, Poisson, binomial, beta, uniform, exponential, normal).

    Useful background: There are no strict prerequisites, but other modules that could provide a useful background include those on modelling such as:

    For programming:

    Synergies: The following module goes well together with Epidemiology by Example:

    • MA4E7 Population Dynamics - this is a complementary course which is recommended to be taken before or simultaneously with Epidemiology by Example. Population Dynamics provides a lot more detail on development of different ecological and epidemiological models and their behaviour (especially qualitative) and is assessed 100% through exam. Epidemiology by Example focuses on programming to tackle a series of real-world-inspired epidemiological problems, bringing in components like model fitting and health economics, which would not be able to be assessed through traditional written examination.
    MA4M2  

    Assumed knowledge:

    MA3G7 Functional Analysis I:

    • Linear operators
    • Compact operators
    • Dual spaces

    Useful background:

    MA3G8 Functional Analysis II:

    • Compactness and weak convergence

    Synergies:
    MA4M6  

    Assumed knowledge:

    Useful background: The module will explore examples coming from various areas of mathematics. Some level of comfort with basic definitions from the following modules will be helpful to better appreciate the module content:

    Synergies: There are set theoretic subtleties at the foundations of category theory, as well as beautiful connections with logic, that we will not dive into. MA4J7 Cohomology and Poincaré Duality.

    Leads to: This module provides essential background for postgraduate studies in modern algebraic topology and algebraic geometry.
    MA4N1  

    Assumed knowledge: You probably already know enough Maths for this module but some experience with programming and debugging will be essential

    Useful background:
    MA4N3  

    Assumed knowledge: 

    MA150 Algebra 2

    • Eigenvalues and eigenvectors

    MA260 Norms, Metrics and Topologies:

    • Metric spaces
    • Continuity
    • Compactness
    • Connectedness
    • Cantor sets

    MA263 Multivariable Analysis:

    • Differentiable functions, diffeomorphisms

    MA359 Measure Theory:

    • Abstract measures
    • Lebesgue measure
    • Convergence Theorems
    • L^1 and L^2 spaces

    Useful background:

    MA3G7 Functional Analysis I:

    • Hilbert Spaces
    • Dual spaces

    MA3H5 Manifolds:

    • Definition of manifold
    • Tangent bundle

    MA424 Dynamical Systems:

    • Topological dynamics

    Synergies: 
    MA4N5  

    Assumed knowledge: Background in Algebra and Probability

    Useful background: The following modules expand your knowledge of Algebra and Probability. If you find them interesting, you will find this module interesting too.

    Synergies:
    MA4N6  

    Assumed knowledge:

    MA359 Measure Theory:

    • Lebesgue integration
    • Riesz Representation Theorem

    Useful background:


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