Mathematics MMath

Mathematics

Key information

Duration:
4 years full time
Typical A-level offer:
AAA
UCAS code:
G103
Start date:
September 2018

You'll learn from experts involved in real-world research projects in pure and applied Mathematics. So you study modules based on the latest research, ranging from analysis, mathematical biology and partial differential equations to numerical analysis and probability.

Your integrated Masters year gives you advanced skills that you apply in your individual research project, supervised by a member of faculty. This degree is accredited to meet the educational requirements of the Chartered Mathematician designation awarded by the Institute of Mathematics and its Applications.

You use industry-standard software (such as MATLAB, SAS and R), developing analytical and modelling skills, and benefit from our links to graduate employers.

The teaching is top class and you learn vital transferable skills that lay the foundation for a successful career.”Connor Osborne
Mathematics (research placement) MMath 

MMath or BSc?

We also offer this course with research placements, or as a three-year BScFind out about the benefits of an integrated Masters year.

Entry requirements

A-level

Typical offer

AAA

Subjects

A-levels must include both Mathematics and Further Mathematics, grade A.

GCSEs

You should also have a broad range of GCSEs (A*-C), including good grades in relevant subjects.

Other UK qualifications

Access to HE Diploma

Typical offer

Pass the Access to HE Diploma with 45 level 3 credits at Merit or above including 30 at Distinction.

Subjects

You will need A-level Mathematics, grade A, in addition to the Access to HE Diploma.

International Baccalaureate

Typical offer

36 points overall from full IB Diploma.

Subjects

Higher Levels must include Mathematics, with a grade of 6.

Pearson BTEC Level 3 National Extended Diploma (formerly BTEC Level 3 Extended Diploma)

Typical offer

DDD

Subjects

In addition to the BTEC Level 3 National Extended Diploma, you will need A-levels in Mathematics and Further Mathematics, both at grade A.

GCSEs

You should also have a broad range of GCSEs (A*-C), including good grades in relevant subjects.

Scottish Highers

Typical offer

AAAAA

Subjects

Highers must include Mathematics, grade A. You will also need an Advanced Higher in Mathematics (grade A).

Welsh Baccalaureate Advanced

Typical offer

Grade A and AA in two A-levels.

Subjects

A-levels must include both Mathematics and Further Mathematics, grade A.

GCSEs

You should also have a broad range of GCSEs (A*-C), including good grades in relevant subjects.

International baccalaureate

Typical offer

36 points overall from full IB Diploma.

Subjects

Higher Levels must include Mathematics, with a grade of 6.

European baccalaureate

Typical offer

Overall result of 83%

Additional requirements

Evidence of existing academic ability in Mathematics to the highest level is essential (normally with a final grade of at least 8.5).

Other international qualifications

Australia

Typical offer

Relevant state (Year 12) High School Certificate, and over 85% in the ATAR or UAI/TER/ENTER. Or a Queensland OP of 5 or below.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Austria

Typical offer

Reifeprüfung or Matura with an overall result of 2.2 or better for first-year entry. A result of 2.5 or better would be considered for Foundation Year entry.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Belgium

Typical offer

Certificat d'Enseignement Secondaire Supérieur (CESS) or Diploma van Hoger Secundair Onderwijs with a good overall average. 

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Bulgaria

Typical offer

Diploma za Sredno Obrazovanie with excellent final-year scores (normally 5.5 overall with 6 in key subjects).

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Canada

Typical offer

High School Graduation Diploma. Specific requirements vary between provinces.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

China

Typical offer

We usually do not accept Senior High School Graduation for direct entry to our undergraduate courses. However, we do consider applicants who have studied 1 or more years of Higher Education in China at a recognised degree awarding institution or who are following a recognised International Foundation Year.

If you are interested in applying for a business related course which requires an academic ability in Mathematics, you will normally also need a grade B in Mathematics from the Huikao or a score of 90 in Mathematics from the Gaokao.

Applicants who have the Senior High School Graduation may be eligible to apply to our International Foundation Year, which if you complete successfully you can progress on to a relevant undergraduate course at Sussex. You can find more information about the qualifications which are accepted by our International Study Centre at  http://isc.sussex.ac.uk/entry-requirements/international-foundation-year .

 

 

 

 

 

 

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Croatia

Typical offer

Maturatna Svjedodžba with an overall score of at least 4-5 depending on your degree choice.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Cyprus

Typical offer

Apolytirion of Lykeion with an overall average of at least 18 or 19/20 will be considered for first-year entry.

A score of 15/20 in the Apolytirion would be suitable for Foundation Year entry. Find out more about Foundation Years.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Czech Republic

Typical offer

Maturita with a good overall average.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Denmark

Typical offer

Højere Forberedelseseksamen (HF) or studentereksamen with an overall average of at least 7 on the new grading scale.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Finland

Typical offer

Finnish Ylioppilastutkinto with an overall average result in the final matriculation examinations of at least 6.5.

Additional requirements

You will need Laudatur in Mathematics.

France

Typical offer

French Baccalauréat with an overall final result of at least 15/20.

Additional requirements

You will need to be taking the science strand within the French Baccalauréat with a final result of at least 14/20 in Mathematics.

Germany

Typical offer

German Abitur with an overall result of 1.6 or better.

Additional requirements

You will need a very good final result in Mathematics (at least 14/15) at a high level.

Greece

Typical offer

Apolytirion with an overall average of at least 18 or 19/20 will be considered for first-year entry.

A score of 15/20 in the Apolytirion would be suitable for Foundation Year entry. Find out more about Foundation Years.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Hong Kong

Typical offer

Hong Kong Diploma of Secondary Education (HKDSE) with grades of 5, 4, 4 from three subjects including two electives. 

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Hungary

Typical offer

Erettsegi/Matura with a good average.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

India

Typical offer

Standard XII results from Central and Metro Boards with an overall average of 75-80%. 

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Iran

Typical offer

High School Diploma and Pre-University Certificate.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Ireland

Typical offer

Irish Leaving Certificate (Higher Level) at H1 H1 H2 H2 H2.

Additional requirements

Highers must include Mathematics, grade H1.

Israel

Typical offer

Bagrut, with at least 8/10 in at least six subjects, including one five-unit subject.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Italy

Typical offer

Italian Diploma di Maturità or Diploma Pass di Esame di Stato with a Final Diploma mark of at least 90/100.

Additional requirements

Evidence of existing academic ability to the highest level in Mathematics is essential.

Japan

Typical offer

Upper Secondary Leaving Certificate is suitable for entry to our Foundation Years. Find out more about Foundation Years.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Latvia

Typical offer

Atestats par Visparejo videjo Izglitibu with very good grades in state exams.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Lithuania

Typical offer

Brandos Atestatas including scores of 80-90% in at least three state examinations (other than English).

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Luxembourg

Typical offer

Diplôme de Fin d'Etudes Secondaires.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Malaysia

Typical offer

Sijil Tinggi Persekolahan Malaysia (STPM). As well as various two or three-year college or polytechnic certificates and diplomas.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Netherlands

Typical offer

Voorereidend Wetenschappelijk Onderwijs (VWO), normally with an average of at least 7.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Nigeria

Typical offer

You are expected to have one of the following:

  • Higher National Diploma
  • One year at a recognised Nigerian University
  • Professional Diploma (Part IV) from the Institute of Medical Laboratory Technology of Nigeria
  • Advanced Diploma

You must also have a score of C6 or above in WAEC/SSC English.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Norway

Typical offer

Norwegian Vitnemal Fra Den Videregaende Skole- Pass with an overall average of 5.

Additional requirements

Evidence of existing academic ability at a high level in Mathematics is essential.

Pakistan

Typical offer

Bachelor (Pass) degree in arts, commerce or science.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Poland

Typical offer

Matura with three extended-level written examinations, normally scored within the 7th stanine.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Portugal

Typical offer

Diploma de Ensino Secundario normally with an overall mark of at least 16/20. 

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Romania

Typical offer

Diploma de Bacalaureat with an overall average of 8.5-9.5 depending on your degree choice.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Singapore

Typical offer

A-levels, as well as certain certificates and diplomas.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Slovakia

Typical offer

Maturitna Skuska or Maturita with honours, normally including scores of 1 in at least three subjects.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Slovenia

Typical offer

Secondary School Leaving Diploma or Matura with at least 23 points overall.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

South Africa

Typical offer

National Senior Certificate with very good grades. 

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Spain

Typical offer

Spanish Título de Bachillerato (LOGSE) with an overall average result of at least 8.5.

Additional requirements

Evidence of a high level of existing academic ability in Mathematics is essential.

Sri Lanka

Typical offer

Sri Lankan A-levels.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Sweden

Typical offer

Fullstandigt Slutbetyg with good grades.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Switzerland

Typical offer

Federal Maturity Certificate.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

Turkey

Typical offer

Devlet Lise Diplomasi or Lise Bitirme is normally only suitable for Foundation Years, but very strong applicants may be considered for first year entry. Find out more about Foundation Years.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

USA

Typical offer

We look at your full profile taking into account everything you are studying. You must have your high school graduation diploma and we will be interested in your Grade 12 GPA. However, we will also want to see evidence of the external tests you have taken. Each application is looked at individually, but you should normally have one or two of the following:

  • APs (where we would expect at least three subject with 4/5 in each)
  • SAT Reasoning Tests (normally with a combined score of 1300) or ACT grades
  • and/or SAT Subject Tests (where generally we expect you to have scores of 600 or higher). 

We would normally require APs or SAT Subject Tests in areas relevant to your chosen degree course.

Subject-specific knowledge

Evidence of existing academic ability at a high level in Mathematics is essential.

Please note

Our entry requirements are guidelines and we assess all applications on a case-by-case basis.

My country is not listed

If your qualifications aren’t listed or you have a question about entry requirements, email ug.enquiries@sussex.ac.uk.

English language requirements

IELTS (Academic)

6.5 overall, including at least 6.0 in each component

IELTS scores are valid for two years from the test date. Your score must be valid when you begin your Sussex course. You cannot combine scores from more than one sitting of the test.

If you are applying for degree-level study we can consider your IELTS test from any test centre, but if you require a Confirmation of Acceptance for Studies (CAS) for an English language or pre-sessional English course (not combined with a degree) the test must be taken at a UK Visas and Immigration (UKVI)-approved IELTS test centre.

Find out more about IELTS.

Other English language requirements

Proficiency tests

Cambridge Advanced Certificate in English (CAE)

For tests taken before January 2015: Grade B or above

For tests taken after January 2015: 176 overall, including at least 169 in each skill

We would normally expect the CAE test to have been taken within two years before the start of your course.

You cannot combine scores from more than one sitting of the test. Find out more about Cambridge English: Advanced.

Cambridge Certificate of Proficiency in English (CPE)

For tests taken before January 2015: grade C or above

For tests taken after January 2015: 176 overall, including at least 169 in each skill

We would normally expect the CPE test to have been taken within two years before the start of your course.

You cannot combine scores from more than one sitting of the test. Find out more about Cambridge English: Proficiency.

Pearson (PTE Academic)

62 overall, including at least 56 in all four skills.

PTE (Academic) scores are valid for two years from the test date. Your score must be valid when you begin your Sussex course. You cannot combine scores from more than one sitting of the test. Find out more about Pearson (PTE Academic).

TOEFL (iBT)

88 overall, including at least 20 in Listening, 19 in Reading, 21 in Speaking, 23 in Writing.

TOEFL (iBT) scores are valid for two years from the test date. Your score must be valid when you begin your Sussex course. You cannot combine scores from more than one sitting of the test. Find out more about TOEFL (iBT).

The TOEFL Institution Code for the University of Sussex is 9166.

English language qualifications

AS/A-level (GCE)

Grade C or above in English Language.

Hong Kong Advanced Level Examination (HKALE)/ AS or A Level: grade C or above in Use of English

French Baccalaureat

A score of 12 or above in English.

GCE O-level

Grade C or above in English.

Brunei/Cambridge GCE O-level in English: grades 1-6.

Singapore/Cambridge GCE O-level in English: grades 1-6.

GCSE or IGCSE

Grade C or above in English as a First Language.

Grade B or above in English as a Second Language

German Abitur

A score of 12 or above in English.

Ghana Senior Secondary School Certificate

If awarded before 1993: grades 1-6 in English language.

If awarded between 1993 and 2005: grades A-D in English language.

Hong Kong Diploma of Secondary Education (HKDSE)

 Level 4, including at least 3 in each component in English Language.

Indian School Certificate (Standard XII)

The Indian School Certificate is accepted at the grades below when awarded by the following examination boards:

Central Board of Secondary Education (CBSE) – English Core only: 70%

Council for Indian School Certificate Examinations (CISCE) - English: 70% 

International Baccalaureate Diploma (IB)

English A or English B at grade 5 or above.

Malaysian Certificate of Education (SPM) 119/GCE O-level

If taken before the end of 2008: grades 1-5 in English Language.

If taken from 2009 onwards: grade C or above in English Language.

The qualification must be jointly awarded by the University of Cambridge Local Examinations Syndicate (UCLES).

West African Senior School Certificate

Grades 1-6 in English language when awarded by the West African Examinations Council (WAEC) or the National Examinations Council (NECO).

Country exceptions

Select to see the list of exempt English-speaking countries

If you are a national of one of the countries below, or if you have recently completed a qualification equivalent to a UK Bachelors degree or higher in one of these countries, you will normally meet our English requirements. Note that qualifications obtained by distance learning or awarded by studying outside these countries cannot be accepted for English language purposes.

You will normally be expected to have completed the qualification within two years before starting your course at Sussex. If the qualification was obtained earlier than this we would expect you to be able to demonstrate that you have maintained a good level of English, for example by living in an English-speaking country or working in an occupation that required you to use English regularly and to a high level.

Please note that this list is determined by the UK’s Home Office, not by the University of Sussex.

List of exempt countries

  • Antigua and Barbuda
  • Australia
  • Bahamas
  • Barbados
  • Belize
  • Canada**
  • Dominica
  • Grenada
  • Guyana
  • Ireland
  • Jamaica
  • New Zealand
  • St Kitts and Nevis
  • St Lucia
  • St Vincent and the Grenadines
  • Trinidad and Tobago
  • United Kingdom
  • USA

** Canada: you must be a national of Canada; other nationals not on this list who have a degree from a Canadian institution will not normally be exempt from needing to provide evidence of English.

Admissions information for applicants

Transfers into Year 2

Yes. Find out more about transferring into Year 2 of this course. We don’t accept transfers into the third or final year.

If your qualifications aren’t listed or you have a question about entry requirements, email ug.enquiries@sussex.ac.uk.

Why choose this course?

  • 1st in the UK for career prospects (The Guardian University Guide 2018).
  • 97% of our mathematics research is rated world leading, internationally excellent or internationally recognised (Research Excellence Framework 2014).
  • 91% for overall satisfaction  enjoy a stimulating and supportive learning environment (National Student Survey 2016).

Course information

How will I study?

You attend a number of lectures each week, supported by regular workshops. Some make use of IT/computer laboratories. Lecture notes are available online.

You have set assignments, mainly problem-solving. In workshops, you work in small groups. Postgraduate students are on hand to help you solve mathematical problems. Assessment includes problem sheets, short exams and presentations, as well as an unseen written examination.

You learn the foundations of analysis, algebra, numerics, modelling and applications, and develop presentation skills.

Modules

These are the modules running in the academic year 2017. Modules running in 2018 may be subject to change.

Core modules


Customise your course

Our courses are designed to broaden your horizons and give you the skills and experience necessary to have the sort of career that has an impact.

Gain programming skills and apply them to areas such as digital media, business and interactive design. Find out about our Year in Computing

How will I study?

You broaden your mathematical skills by studying probability and statistics, and take more advanced lectures in analysis, algebra and numerics. You also take a careers module, which includes talks by potential employers.

You further develop numerical, analytical and presentation skills, and also learn to make informed decisions about your career so you can submit strong and competitive job applications.

Modules

These are the modules running in the academic year 2017. Modules running in 2018 may be subject to change.

Core modules


Customise your course

Our courses are designed to broaden your horizons and give you the skills and experience necessary to have the sort of career that has an impact.

Gain programming skills and apply them to areas such as digital media, business and interactive design. Find out about our Year in Computing

Study abroad (optional)

Apply to study abroad – you’ll develop an international perspective and gain an edge when it comes to your career. Find out where your course could take you.

Placement (optional)

Spending a year or term with a company is a great way to network and gain practical skills. When you leave Sussex, you'll benefit from having the experience employers are looking for.

Recent students have gone on placements at:

  • Rolls-Royce Motor Cars Ltd
  • NATS Holdings
  • HEFCE.

Find out more about placements and internships.

“It was a brilliant opportunity to create a network of contacts and develop an understanding of the day-to-day tasks of an accountant.” Ollie WellsMathematics BSc
Corporation Tax Assistant

Please note

If you’re receiving – or applying for – USA federal Direct Loan funds, you can’t transfer to the version of this program with an optional study abroad period in any country or optional placement in the USA. Find out more about American Student Loans and Federal Student Aid

How will I study?

You specialise in your chosen area of mathematics by selecting from a wide range of mathematics options as well as some from its applications.

You will develop independent study and specialised problem-solving skills. 

Modules

These are the modules running in the academic year 2017. Modules running in 2018 may be subject to change.

Options

How will I study?

You specialise further in your chosen area of mathematics by selecting from a wide range of options both within and beyond mathematics, building on your choices in Year 3.

You choose an individual research topic and supervisor for your final-year project.

You'll develop independent study, advanced problem-solving and presentation skills – essential for graduate-level jobs or further study.

Modules

These are the modules running in the academic year 2017. Modules running in 2018 may be subject to change.

Core modules

Options

Students and staff share their experiences of Mathematics at Sussex

Fees

Fees are not yet set for entry in the academic year 2018. Note that your fees, once they’re set, may be subject to an increase on an annual basis.

The UK Government has confirmed that if you’re an EU student applying for entry in September 2018, you'll pay the same fee rate as UK students for the duration of your course, even if the UK leaves the EU before the end of your course. You'll also continue to have access to student loans and grants. Find out more on the UK Government website.

Find out about typical living costs for studying at Sussex

Scholarships

Our focus is personal development and social mobility. To help you meet your ambitions to study at Sussex, we deliver one of the most generous scholarship programmes of any UK university.

Careers

Graduate destinations

100% of Department of Mathematics students were in work or further study six months after graduating. Recent students have started jobs as:

  • statistician, Cabinet Office
  • strategic accounts intern, Google
  • analyst, Taylor Curcio

(HESA EPI, Destinations of Leavers from Higher Education Survey 2015)

Your future career

You can be confident that our courses will provide you with the necessary skills and experience to secure employment or further studies with some of the world’s leading institutions. We run a careers course – including talks by potential employers, advice on choosing a career path, and CV and application writing – and you’ll have plenty of opportunity to engage with potential employers.

Our graduates go onto a range of careers in finance and accounting, digital media, teaching, engineering and healthcare.

Working while you study

Our Careers and Employability Centre can help you find part-time work while you study. Find out more about career development and part-time work

 

Calculus

  • 15 credits
  • Autumn Teaching, Year 1

Topics covered on this module include:

  1. Functions of one real variable: graphical representation, inverse functions, composition of functions, polynomial, trigonometric, exponential and hyperbolic functions.
  2. Limits, continuity and differentiation: one-sided limits, infinite limits, algebra of limits, continuity and the intermediate value theorem, differentiation from first principles, product rule and chain rule, Rolle's Theorem, the mean value theorem and Taylor's theorem, stationary points of a function.
  3. Integration: indefinite and definite integrals, fundamental theorem of calculus, integration by parts and integration by substitution.
  4. Solutions to first order ODEs.
  5. Solutions to second order ODEs.

Geometry

  • 15 credits
  • Autumn Teaching, Year 1

Topics include: vectors in two and three dimensions. Vector algebra: addition, scalar product, vector product, including triple products. Applications in two- and three-dimensional geometry: points, lines, planes, geometrical theorems. Area and volume. Linear dependence and determinants. Polar co-ordinates in two and three dimensions. Definitions of a group and a field. Polynomials. Complex numbers, Argand plane, De Moivre's theorem. Matrices: addition, multiplication, inverses. Transformations in R^2 and R^3: isometries. Analytical geometry: classification and properties of conics.

Introduction to Pure Mathematics

  • 15 credits
  • Autumn Teaching, Year 1

In this module, the topics you will cover will include:

  • numbers; introduction of mathematical symbols, natural numbers, integers, rationals, real numbers, basic number algebra
  • ordering, inequalities, absolute value (modulus), homogeneity, triangle inequality
  • concept of algebraic structure, groups
  • sequences, induction principle, well-ordering principle, sums, products, factorials, Fibonacci numbers, fractions
  • irrational roots of integers, divisibility, prime numbers, Euclidean Division, highest common factor, Euclidean algorithm, number theory, atomic property of primes, coprime factorisation, fundamental theorem of arithmetic, square-free numbers
  • logic; concept of proof, logical argument, direct proof, propositional manipulation, basic logic, and, or, not, implication, contraposition, contradiction, logical equivalence, quantifiers
  • operations with sets; equality, intersection, difference, union, empty set, ordered pairs, cartesian products, power set
  • counting; maps and functions, distinguished functions, injections, surjections, bijections, one-to-one correspondences, pigeonhole principle, counting the power set, counting subsets of the power set, cherry picking, binomial coefficients, binomial formula, combinatorics, inclusion-exclusion formula, permutations, counting maps
  • functions and maps; formal definition, finite and infinite sets, pigeonhole principle revisited, counterimage, inverse functions, partial inverses
  • relations; relations, equivalence relations, modular arithmetic and quotient sets, order relations, partial ordering, total ordering, linear ordering
  • rigorous extension of N to Z and Q
  • rings, fields.

Mathematics in Everyday Life

  • 15 credits
  • Autumn Teaching, Year 1

This module covers:

  • Money: Rule of 72, repayments, annuities, APR, compounding, present value, tax system, Student Loans.
  • Differential equations: how they arise, how they can be solved. Applications include radiocarbon dating, cooling of liquids, evaporation of mothballs, escape of water down plugholes, war, epidemics, predator-prey models.
  • Applications in sports and games -- tennis, rugby, snooker, darts, athletics, soccer, ranking methods.
  • Business applications: stock control, linear programming, pound-cost average, hierarchies and promotions policies, the Kelly strategy.
  • Voting methods and paradoxes: Arrow's Impossibility Theorem. 
  • Simpson's Paradox, disease testing (false positives etc), gambling, TV game shows.
  • Mathematical essay: eg book reviews, topic descriptions.

Analysis 1

  • 15 credits
  • Spring Teaching, Year 1

In this module, the topics you will cover will include:

  • Sequences; convergence, Cauchy sequences, subsequences
  • Series; proof and application of convergence/divergence criteria
  • Limits of functions; definitions, examples and properties
  • Continuity; intermediate value theorem, uniform continuity
  • Differentiability; definition, proofs of mean value theorems.

Discrete Mathematics

  • 15 credits
  • Spring Teaching, Year 1

This module gives the fundamental properties of the natural numbers, continuing from IPM, counting and the first part of group theory. It includes the principles of counting, divisibility and congruences, arithmetics functions, primitive roots of an integer, quadratic residues and group theory.

Linear Algebra

  • 15 credits
  • Spring Teaching, Year 1

In this module, you will cover:

  • matrices
  • elementary row and column operations
  • vector spaces
  • linear independence
  • basis and dimension
  • inner products and orthogonality
  • Gram-Schmidt orthonormalisation process
  • linear transformations
  • determinants
  • eigen-values and eigen-vectors
  • polynomials
  • Cayley-Hamilton theorem
  • quadratic forms.

Numerical Analysis 1

  • 15 credits
  • Spring Teaching, Year 1

In this module, you will cover topics such as:

Part I: Introduction to Computing with MATLAB 

  • basic arithmetic and vectors
  • m-file functions
  • for-loops
  • if and else
  • while statements.

Part II: Introduction to Numerical Analysis

  • operating with floating point numbers, round-off error, cancellation error
  • polynomial interpolation, basic idea of interpolation, order of approximation, Lagrangian interpolation, Runge's example, piecewise linear interpolation
  • numerical differentiation, finite difference quotients, order of approximation
  • numerical integration, derive standard numerical integration scheme's and analyse, using polynomial interpolation (midpoint formula, trapezoidal rule, Simpson's formula)
  • nonlinear equations, bisection method, fixed point iteration method, Newton's method/secant method.

Algebra

  • 15 credits
  • Autumn Teaching, Year 2

Topics covered on this module include:

  1. Definition of a group, examples, abelian groups, subgroups, cosets, Lagrange's theorem, cyclic groups.
  2. The symmetric group S_n and conjugacy.
  3. Homomorphisms, normal subgroups and quotient groups.
  4. Definition of a ring, examples, polynomial rings, ideals and homomorphisms.
  5. Definition of a field and construction of extensions.

Analysis 2

  • 15 credits
  • Autumn Teaching, Year 2

Topics covered: power series, radius of convergence; Taylor series and Taylor's formula; applications and examples; upper and lower sums; the Riemann integral; basic properties of the Riemann integral; primitive; fundamental theorem of calculus; integration by parts and change of variable; applications and examples. Pointwise and uniform convergence of sequences and series of functions: interchange of differentiation or integration and limit for sequences and series; differentiation and integration of power series term by term; applications and examples. Metric spaces and normed linear spaces: inner products; Cauchy sequences, convergence and completeness; the Euclidean space R^n; introduction to general topology; applications and examples.

Calculus of Several Variables

  • 15 credits
  • Autumn Teaching, Year 2

This module covers topics including:

  • Partial derivatives, the gradient vector, higher order partial derivatives and Hessian matrix
  • Multiple integration, cylindrical polar coordinates and spherical polar coordinates
  • Implicit Function Theorem and Taylor's Theorem
  • Line integrals
  • Green's theorem, Gauss's theorem and Stokes' theorem
  • CV and career guidance

Introduction to Probability

  • 15 credits
  • Autumn Teaching, Year 2

This module will cover topics including:

  • Elementary probability theory: axioms, probability measure, conditional probability, independence, Bayes formula and permutations and combinations
  • Discrete distributions: expectation, variance, standard distributions, probability generating functions and sums of random variables and random walk
  • Continuous distributions: densities, expectation, variance, standard distributions, transformations, linear function of independent normal random variables, normal approximations, central limit theorem, moment generating functions and law of large numbers
  • Joint distributions: discrete and continuous, conditional distributions, covariance and correlation

Complex Analysis

  • 15 credits
  • Spring Teaching, Year 2

In this module, the topics you will cover will include:

  • holomorphic functions, Cauchy's theorem and its consequences
  • power series, integration, differentiation and analysis of convergence
  • Taylor expansions and circle of convergence
  • Laurent expansions and classification of isolated singularities
  • residue theorem and evaluation of integrals
  • Rouche's theorem and the fundamental theorem of algebra.

Differential Equations

  • 15 credits
  • Spring Teaching, Year 2

In this module, the topics you will cover will include:

Ordinary differential equations:

  • solution methods; variation of the constant formula, separation of variables
  • solution of linear ODE with constant coefficients
  • Lipschitz continuity
  • existence and uniqueness (Picard-Lindeloef), maximal solutions, Gronwall
  • higher order equations into system of first order
  • boundary value problems.

Partial differential equations:

  • partial and total derivatives
  • first order PDEs; method of characteristics for semilinear and quasilinear equations, initial boundary value problems.

Numerical Analysis 2

  • 15 credits
  • Spring Teaching, Year 2

Topics covered include:

  • Linear systems (conditioning, LU factorization, basic iterative methods, convergence analysis)
  • Nonlinear systems: fixed-point methods, Newton’s method
  • Applications:
    • iterative methods for eigen-problems,
    • least squares,
    • optimisation,
    • linear programming.

Probability and Statistics

  • 15 credits
  • Spring Teaching, Year 2

Topics covered on this module include:

  1. Multivariate discrete and continuous distributions; Topic includes expectations, covariance, lower dimensional marginals, distributions of sums of independent random variables and central limit theorem (CLT), transformations of random variables. Introduction to statistical linear regression of uniform random samples and point estimators for mean and variance.
  2. Conditional expectations.
  3. Multivariate normal distributions, multivariate CLT and introduction of basic statistics and statistical distributions: Sample mean and variance, t; 2 and F distributions.
  4. Parameter estimation: Exponential family of distributions, Unbiased estimators, sucient estimators and maximum likelihood estimators. The german tank problem. Condence intervals.
  5. Hypothesis testing: z- and t-tests, 2 tests, including Pearson's test and test for proportions, contingency tables.

Advanced Numerical Analysis

  • 15 credits
  • Autumn Teaching, Year 3

This module will cover topics including:

  • Iterative methods for linear systems
  • Iterative methods for nonlinear systems
  • Optimisation
  • Eigenvalue problems
  • Numerical methods for ordinary differential equations
  • Runge-Kutta methods
  • Linear multistep methods

Communicating STEM

  • 15 credits
  • Autumn Teaching, Year 3

This module aims to develop the skills and understanding required for explaining scientific concepts to a range of audiences, with a particular focus on school aged pupils (5-18). This module has been designed to encourage more Physics and Maths Undergraduates to think about a career in teaching following their University studies. It has received funding from the NCTL and is viewed by the NCTL as a good way to bridge the gap between University and a profession in teaching.

The module begins with an exploration of the features, and types of scientific explanations. This is then applied to the types of learner from primary and secondary school and to those who have special needs. The skills of good scientific explanations are then developed through engaging with common misconceptions, progression of scientific ideas and those concepts that are particularly troublesome to grasp. This will be underpinned by engaging with two key learning theories. 

The final sessions will bring these ideas together and develop knowledge and understanding of questioning, models and modelling of scientific concepts and processes to aid effective explanations.

Cryptography

  • 15 credits
  • Autumn Teaching, Year 3

Topics covered include:

  • Symmetric-key cryptosystems.
  • Hash functions and message authentication codes.
  • Public-key cryptosystems.
  • Complexity theory and one-way functions.
  • Random number generation.
  • Attacks on cryptosystems.
  • Cryptographic standards.

E-Business and E-Commerce Systems

  • 15 credits
  • Autumn Teaching, Year 3

Topics for this module include:

  • elementary economic theory and its interaction with e-business
  • alternative e-business strategies (as theories and as case studies)
  • legal and behavioural issues
  • marketing, branding, and customer relationship issues
  • software systems for e-business and e-commerce
  • commercial website management.

Financial Mathematics

  • 15 credits
  • Autumn Teaching, Year 3

You will study generalized cash flows, time value of money, real and money interest rates, compound interest functions, equations of value, loan repayment schemes, investment project evaluation and comparison, and bonds.

Functional Analysis

  • 15 credits
  • Autumn Teaching, Year 3

In this module, you cover:

  • Banach spaces, Banach fixed-point theorem, Baire's theorem
  • bounded linear operators and on Banach spaces, continuous linear functionals, Banach-Steinhaus uniform boundedness principle
  • open mapping and closed graph theorems, Hahn-Banach theorem
  • Hilbert spaces, orthogonal expansions, Riesz representation theorem.

Galois Theory

  • 15 credits
  • Autumn Teaching, Year 3

A quadratic equation in one variable has a formula for its solutions. So do cubic and quartic equations, whereas a general quintic has no such formula. The theory of field equations and its connection to the theory of groups explains this.

The syllabus will include:

  • Consideration of the historic problems
  • Quadratic equations, complex roots of 1, cubic equations, quartic equations
  • Insolvability of the quintic
  • Ruler and compass constructions, squaring the circle, duplicating the cube
  • Field extensions
  • Applications to ruler-and-compass constructions
  • Normal extensions
  • Application to finite fields, splitting fields
  • Galois group of polynomials
  • Application to x5 - 1 = 0
  • Fundamental Theorem of Galois Theory
  • Galois group for cubic polynomial
  • Solutions of equations in radicals and soluble groups

Introduction to Mathematical Biology

  • 15 credits
  • Autumn Teaching, Year 3

The module will introduce you to the concepts of mathematical modelling with applications to biological, ecological and medical phenomena. The main topics will include:

  • Continuous populations models for single species;
  • Discrete population models for single species;
  • Phase plane analysis;
  • Interacting populations (continuous models);
  • Enzyme kinetics;
  • Dynamics of infectious diseases and epidemics.

Linear Statistical Models

  • 15 credits
  • Autumn Teaching, Year 3
Topics include: full-rank model (multiple and polynomial regression), estimation of parameters, analysis of variance and covariance; model checking; comparing models, model selection; transformation of response and regressor variables; models of less than full rank (experimental design), analysis of variance, hypothesis testing, contrasts; simple examples of experimental designs, introduction to factorial experiments; and use of a computer statistical package to analyse real data sets.

Partial Differential Equations

  • 15 credits
  • Autumn Teaching, Year 3

Topics include: Second-order Partial Differential Equations: wave equation, heat equation, Laplace equation. D'Alembert's solution, separation of variables, Duhamel's principle, energy method, Maximum principle, Green's identities.

Probability Models

  • 15 credits
  • Autumn Teaching, Year 3

You cover topics including:

  • short revision of probability theory
  • expectation and conditional expectation
  • convergence of random variables, in particular laws of large numbers, moment generating functions, and central limit theorem
  • stochastic processes in discrete time in particular Markov chains, including random walk, martingales in discrete time, Doob's optional stopping theorem, and martingale convergence theorem.

Topology and Advanced Analysis

  • 15 credits
  • Autumn Teaching, Year 3

This module will introduce you to some of the basic concepts and properties of topological spaces. The subject of topology has a central role in all of Mathematics and having a proper understanding of its concepts and main theorem is essential as part of an undergraduate mathematics curriculum.

Topics that will be covered in this module include:

  • Topological spaces
  • Base and sub-base
  • Separation axioms
  • Continuity
  • Metrisability
  • Completeness
  • Compactness and Coverings
  • Total Boundedness
  • Lebesgue numbers and Epsilon-nets
  • Sequential Compactness
  • Arzela-Ascoli Theorem
  • Montel's theorem
  • Infinite Products
  • Box and Product Topologies
  • Tychonov Theorem. 

Coding Theory

  • 15 credits
  • Spring Teaching, Year 3

Topics include: 

  • Introduction to error-correcting codes. The main coding theory problem. Finite fields.
  • Vector spaces over finite fields. Linear codes. Encoding and decoding with a linear code.
  • The dual code and the parity check matrix. Hamming codes. Constructions of codes.
  • Weight enumerators. Cyclic codes.

Continuum Mechanics

  • 15 credits
  • Spring Teaching, Year 3

Topics include: 

  • Kinematics: Eulerian and Lagrangian descriptions, velocity, acceleration, rate of change of physical quantities, material derivatives, streamlines.
  • Deformation: stress and strain tensors, Hooke's law, equilibrium equations.
  • Conservation laws for mass, momentum and energy.
  • Phase/group velocities of travelling wave solutions.
  • Models of fluid and solid mechanics.

Dynamical Systems

  • 15 credits
  • Spring Teaching, Year 3
  • General dynamical systems: semiflow, stability and attraction, omega-limit set, global attractor
  • Ordinary Differential Equations: Linear systems, Lyapunov function, linearised systems around fixed points, two-dimensional
    systems, periodic orbit
  • Discrete systems (iterations): Linear systems, linearised systems around fixed points, chaos

Measure and Integration

  • 15 credits
  • Spring Teaching, Year 3

In this module, you cover:

  • countably additive measures, sigma-algebras, Borel sets, measure spaces
  • outer measures and Caratheodory's construction of measures
  • construction and properties of Lebesgue measure in Euclidean spaces
  • measurable and integrable functions, Lebesgue integration theory on measure spaces, L^p spaces and their properties
  • convergence theorems: monotone convergence, dominated convergence, Fatou's lemma
  • application of limit theorems to continuity and differentiability of integrals depending on a parameter
  • properties of finite measure spaces and probability theory.

Medical Statistics

  • 15 credits
  • Spring Teaching, Year 3
Topics include: logistic regression, fitting and interpretation. Survival times; Kaplan-Meier estimate, log-rank test, Cox proportional hazard model. Designing medical research. Clinical trials; phases I-IV, randomised double-blind controlled trial, ethical issues, sample size, early stopping. Observational studies: prospective/retrospective, longitudinal/cross-sectional. Analysis of categorical data; relative risk, odds ratio; McNemar's test, meta-analysis (Mantel-Haenszel method). Diagnostic tests; sensitivity and specificity; receiver operating characteristic. Standardised mortality rates.

Multimedia Design and Applications

  • 15 credits
  • Spring Teaching, Year 3

In this module, you develop a basic understanding of human perception and how this relates to the capture, display, storage and transmission of digital media.

Your studies in this module cover the theory and the software and hardware required for the capture, display, storage and transmission of:

  • audio
  • video
  • image
  • graphical-based digital media.

Optimal Control of Partial Differential Equations

  • 15 credits
  • Spring Teaching, Year 3

You will be introduced to optimal control problems for partial differential equations. Starting from basic concepts in finite dimensions (existence, optimality conditions, adjoint, Lagrange functional and KKT system) you will study the theory of linear-quadratic elliptic optimal control problems (weak solutions, existence of optimal controls, adjoint operators, necessary optimality conditions, Langrange functional and adjoint as Langrangian multiplier) as well as basic numerical methods for your solution (gradient method, projected gradient method and active set strategy). The extension to semi-linear elliptic control problems will also be considered.

Perturbation theory and calculus of variations

  • 15 credits
  • Spring Teaching, Year 3

The aim of this module is to introduce you to a variety of techniques primarily involving ordinary differential equations, that have applications in various branches of applied mathematics. No particular application is emphasised.

Topics covered include

  • Dimensional analysis and scaling:
  • physical quantities and their measurement;
  • dimensions;
  • change of units;
  • physical laws;
  • Buckingham Pi Theorem;
  • scaling.
  • Regular perturbation methods:
  • direct method applied to algebraic equations and initial value problems (IVP);
  • Poincar method for periodic solutions;
  • validity of approximations.
  • Singular perturbation methods:
  • finding approximate solutions to algebraic solutions;
  • finding approximate solutions to boundary value problems (BVP) including boundary layers and matching.
  • Calculus of Variations:
  • necessary conditions for a function to be an extremal of a fixed or free end point problem involving a functional of integral form;
  • isoperimetric problems.

Random processes

  • 15 credits
  • Spring Teaching, Year 3

Topics covered on this module include:

Rationalisation:
After the introduction of the Poisson process, birth and death processes as well as epidemics models can be presented in full generality as applications of the pooled Poisson process. At the same time, you will be introduced to the Kolmogorov equations and to the techniques for solving them. Renewal theory is needed to better understand queues, and, for this reason, it is discussed before queues.
Modernisation:
A modern introductory course on stochastic processes must include at least a section on compound renewal processes (with a focus on the compound Poisson process) as well as a chapter on the Wiener process and on Ito stochastic calculus. This is necessary given the importance this process has in several applications from finance to physics. Modernisation is achieved by including a new introductory chapter divided into three parts.
  1. Poisson processes:
    1. Density and distribution of into-event time.
    2. Pooled Poisson process.
    3. Breaking down a Poisson process.
    4. Applications of the Poisson process, eg birth-and-death processes, the Kolmogorov equations.
  1. Renewal processes:
    1. The ordinary renewal process.
    2. The equilibrium renewal process.
    3. The compound renewal process.
    4. Applications of renewal processes, queues.
  1. Wiener processes:
    1. Definition and properties
    2. Introduction to stochastic integrals
    3. Introduction to stochastic differential equations.

Researching STEM

  • 15 credits
  • Spring Teaching, Year 3

This module aims to develop the skills and understanding required for explaining scientific concepts to a range of audiences, with a particular focus on school aged pupils (11-18). At the start of the module there will be a lecture and a seminar to introduce the project. Tutorials will follow to aid the student with their research project.

The module will involve time in an educational institution carrying out a minor research project based on an aspect of science communication. You will consider an area of interest, relating to communicating STEM, and carry out some literature-based research. You will then design and carry out a small research project. This is likely to be based on a small number of class observations in a chosen school, possibly with some intervention activities that you would evaluate based on your literature review.

Assessment is in the form of a written assignment, to include a literature review and evaluations of observations (from the educational institution) based on theory.

MMath Project

  • 30 credits
  • Autumn & Spring Teaching, Year 4

The work for the project and the writing of the project report will have a major role in bringing together material that you have mastered up to Year 3 and is mastering in Year 4. It will consist of a sustained investigation of a mathematical topic at Masters' level. The project report will be typeset using TeX/LaTeX (mathematical document preparation system). The use of mathematical typesetting, (mathematics-specific) information technology and databases and general research skills such as presentation of mathematical material to an audience, gathering information, usage of (electronic) scientific libraries will be taught and acquired during the project.

Advanced Numerical Analysis

  • 15 credits
  • Autumn Teaching, Year 4

This module will cover topics including:

  • Iterative methods for linear systems: Jacobi and Gauss-Seidel, conjugate gradient, GMRES and Krylov methods
  • Iterative methods for nonlinear systems: fixed point iteration, Newton's method and Inexact Newton
  • Optimisation: simplex methods, descent methods, convex optimisation and non-convenx optimisation
  • Eigenvalue problems: power method, Von Mises method, Jacobi iteration and special matrices
  • Numerical methods for ordinary differential equations: existence of solutions for ODE's, Euler's method, Lindelöf-Picard method, continuous dependence and stability of ODE's
  • Basic methods: forward and backward Euler, stability, convergence, midpoint and trapezoidal methods (order of convergence, truncation error, stability convergence, absolute stability and A-stability)
  • Runge-Kutta methods: one step methods, predictor-corrector methods, explicit RK2 and RK4 as basic examples, and general theory of RK methods such as truncation, consitency, stability and convergence 
  • Linear multistep methods: multistep methods, truncation, consistency, stability, convergence, difference equaitons, Dahlquist's barriers, Adams family and backward difference formulas
  • Boundary value problems in 1d, shooting methods, finite difference methods, convergence analysis, Galerkin methods and convergence analysis

Cryptography

  • 15 credits
  • Autumn Teaching, Year 4

You will cover the following areas: 

  • symmetric-key cryptosystems
  • hash functions and message authentication codes
  • public-key cryptosystems
  • complexity theory and one-way functions
  • primality and randomised algorithms
  • random number generation
  • elliptic curve cryptography
  • attacks on cryptosystems
  • quantum cryptography
  • cryptographic standards.

E-Business and E-Commerce Systems

  • 15 credits
  • Autumn Teaching, Year 4

The module provides a theoretical and technical understanding of the major issues for existing large-scale E-Business and E-Commerce systems. Theoretical aspects include alternative E-Business strategies, marketing, branding, customer relationship issues and commercial website management. The technical part covers the standard methods for large-scale data storage, data movement, transformation, and application integration, together with the fundamentals of application architecture. Examples focus on the most recent developments in E-Business and E-Commerce distributed systems. Critical analysis of current and emerging technologies for E-Business and E-Commerce is carried out during seminars.

Financial Mathematics

  • 15 credits
  • Autumn Teaching, Year 4

You will study generalized cash flows, time value of money, real and money interest rates, compound interest functions, quations of value, loan repayment schemes, investment project evaluation and comparison, bonds, term structure of interest rates, some simple stochastic interest rate models and project writing.

Functional Analysis

  • 15 credits
  • Autumn Teaching, Year 4

In this module, you cover:

  • Banach spaces, Banach fixed-point theorem, Baire's theorem
  • bounded linear operators and on Banach spaces, continuous linear functionals, Banach-Steinhaus uniform boundedness principle
  • open mapping and closed graph theorems, Hahn-Banach theorem
  • Hilbert spaces, orthogonal expansions, Riesz representation theorem.

Introduction to Cosmology

  • 15 credits
  • Autumn Teaching, Year 4

This module covers:

  • Observational Overview: in visible light and other wavebands; the cosmological principle; the expansion of the universe; particles in the universe.
  • Newtonian Gravity: the Friedmann equation; the fluid equation; the aceleration equation.
  • Geometry: flat, spherical and hyperbolic; infinite vs. observable universes; introduction to topology
  • Cosmological Models: solving equations for matter and radiation dominated expansions and for mixtures (assuming flat geometry and zero cosmological constant); variation of particle number density with scale factor; variation of scale factor with time and geometry.
  • Observational Parameters: Hubble, density, deceleration.
  • Cosmological Constant: fluid description; models with a cosmological constant.
  • The Age of the Universe: tests; model dependence; consequences
  • Dark Matter: observational evidence; properties; potential candidates (including MACHOS, neutrinos and WIMPS)
  • The Cosmic Microwave Background: properties; derivation of photo to baryon ratio; origin of CMB (including decoupling and recombination).
  • The Early Universe: the epoch of matter-radiation equality; the relation between temperature and time; an overview of physical properties and particle behaviour.
  • Nucleosynthesis: basics of light element formation; derivation of percentage, by mass, of Helium; introduction to observational tests; contrasting decoupling and nucleosynthesis.
  • Inflation: definition; three problems (what they are and how they can be solved); estimation of expansion during Inflation; contrasting early time and current inflationary epochs; introduction to cosmological constant problem and quintessence.
  • Initial Singularity: definition and implications.
  • Connection to General Relativity: brief introduction to Einstein equations and their relation to the Friedmann equation.
  • Cosmological Distance Scales: proper, luminosity, angular distances; connection to observables.
  • Structures in the Universe: CMB anisotropies; galaxy clustering
  • Constraining Cosmology: connection to CMB, large scale structure (inc BAO and weak lensing) and supernovae.

Mathematical Fluid Mech

  • 15 credits
  • Autumn Teaching, Year 4

The aim of this module is to provide an introduction to fluid mechanics, regarded from the perspective of the mathematical analysis of underlying PDE models. As such the course is at the interface between pure and applied mathematics.

The mdoule focuses on the basic equations of fluid dynamics, namely the Navier-Stokes and Euler equations. These are the equations governing the motion of fluids, such as water or air.

The module starts with the derivation of the basic conservation laws. Some simple cases of solutions are analyzed in detail and then a general existence theory in bounded and unbounded domain is obtained, based on energy methods.

Object Oriented Programming

  • 15 credits
  • Autumn Teaching, Year 4

You will be introduced to object-oriented programming, and in particular to understanding, writing, modifying, debugging and assessing the design quality of simple Java applications.

You do not need any previous programming experience to take this module, as it is suitable for absolute beginners.

Programming in C++

  • 15 credits
  • Autumn Teaching, Year 4

After a review of the basic concepts of the C++ language, you are introduced to object oriented programming in C++ and its application to scientific computing. This includes writing and using classes and templates, operator overloading, inheritance, exceptions and error handling. In addition, Eigen, a powerful library for linear algebra is introduced. The results of programs are displayed using the graphics interface dislin.

Topology and Advanced Analysis

  • 15 credits
  • Autumn Teaching, Year 4

Topics that will be covered in this module include:

  • Topological spaces
  • Base and sub-base
  • Separation axioms
  • Continuity
  • Metrisability
  • Completeness
  • Compactness and Coverings
  • Total Boundedness
  • Lebesgue numbers and Epsilon-nets
  • Sequential Compactness
  • Arzela-Ascoli Theorem
  • Montel's theorem
  • Infinite Products
  • Box and Product Topologies
  • Tychonov Theorem
  • Banach-Alaoglu theorem.

Advanced Partial Differential Equations

  • 15 credits
  • Spring Teaching, Year 4

You will be introduced to modern theory of linear and nonlinear Partial Differential Equations. Starting from the theory of Sobolev spaces and relevant concepts in linear operator theory, which provides the functional analytic framework, you will treat the linear second-order elliptic, parabolic, and hyperbolic equations (Lax-Milgram theorem, existence of weak solutions, regularity, maximum principles), e.g., the potential, diffusion, and wave equations that arise in inhomogeneous media.

The emphasis will be on the solvability of equations with different initial/boundary conditions, as well as the general qualitative properties of their solutions. They then turn to the study of nonlinear PDE, focusing on calculus of variation.

Coding Theory

  • 15 credits
  • Spring Teaching, Year 4

Topics covered include: 

  • Introduction to error-correcting codes. The main coding theory problem. Finite fields.
  • Vector spaces over finite fields. Linear codes. Encoding and decoding with a linear code.
  • The dual code and the parity check matrix. Hamming codes. Constructions of codes.
  • Weight enumerators. Cyclic codes. MDS codes.

Differential Geometry

  • 15 credits
  • Spring Teaching, Year 4

On this module, we will cover:

  • Manifolds and differentiable structures
  • Lie derivatives
  • Parallel transport
  • Riemannian metrics and affine connections
  • Curvature tensor
  • Sectional curvature
  • Scalar curvature
  • Ricci curvature
  • Bianchi identities
  • Schur's lemma
  • Complete manifolds
  • Hopf-Rinow theorem
  • Hadmard's theorem
  • Geodescis and Jacobi fields
  • Bonnet-Meyer and Synge theorems
  • Laplace-Beltrami operator
  • Heat kernels and index theorem.

Financial Portfolio Analysis

  • 15 credits
  • Spring Teaching, Year 4

You will study valuation, options, asset pricing models, the Black-Scholes model, Hedging and related MatLab programming. These topics form the most essential knowledge for you if you intend to start working in the financial fields. They are complex application problems. Your understanding of mathematics should be good enough to understand the modelling and reasoning skills required. The programming element of this module makes complicated computations manageable and presentable.

Mathematical Models in Finance and Industry

  • 15 credits
  • Spring Teaching, Year 4

Topics include: partial differential equations (and methods for their solution) and how they arise in real-world problems in industry and finance. For example: advection/diffusion of pollutants, pricing of financial options.

Measure and Integration

  • 15 credits
  • Spring Teaching, Year 4

In this module, you cover:

  • countably additive measures, sigma-algebras, Borel sets, measure spaces
  • outer measures and Caratheodory's construction of measures
  • construction and properties of Lebesgue measure in Euclidean spaces
  • measurable and integrable functions, Lebesgue integration theory on measure spaces, L^p spaces and their properties
  • convergence theorems: monotone convergence, dominated convergence, Fatou's lemma
  • application of limit theorems to continuity and differentiability of integrals depending on a parameter
  • properties of finite measure spaces and probability theory.

Monte Carlo Simulations

  • 15 credits
  • Spring Teaching, Year 4

The module will cover topics including:

  • Introduction to R 
  • Pseudo-random number generation 
  • Generation of random variates 
  • Variance reduction 
  • Markov-chain Monte Carlo and its foundations 
  • How to analyse Monte Carlo simulations 
  • Application to physics: the Ising model 
  • Application to statistics: goodness-of-fit tests

Random processes

  • 15 credits
  • Spring Teaching, Year 4

Topics covered include:

Rationalisation:
After the introduction of the Poisson process, birth and death processes as well as epidemics models can be presented in full generality as applications of the pooled Poisson process. At the same time, the students will be introduced to the Kolmogorov equations and to the techniques for solving them. Renewal theory is needed to better understand queues, and, for this reason, it is discussed before queues.
Modernisation:
A modern introductory course on stochastic processes must include at least a section on compound renewal processes (with a focus on the compound Poisson process) as well as a chapter on the Wiener process and on Ito stochastic calculus. This is necessary given the importance this process has in several applications from finance to physics. Modernisation is achieved by including a new introductory chapter divided into three parts.
  1. Poisson processes:
    1. Density and distribution of into-event time.
    2. Pooled Poisson process.
    3. Breaking down a Poisson process.
    4. Applications of the Poisson process, eg birth-and-death processes, the Kolmogorov equations.
  1. Renewal processes
    1. The ordinary renewal process.
    2. The equilibrium renewal process.
    3. The compound renewal process.
    4. Applications of renewal processes, queues.
  1. Wiener process
    1. Definition and properties
    2. Introduction to stochastic integrals
    3. Introduction to stochastic differential equations.
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