Masters Degrees (Combinatorics)

This highly focused MSc explores some of the mathematics behind modern secure information and communications systems, specialising in mathematics relevant for public key cryptography, coding theory and information theory. During the course critical awareness of problems in information transmission, data compression and cryptography is raised, and the mathematical techniques which are commonly used to solve these problems are explored.

The Mathematics Department at Royal Holloway is well known for its expertise in information security and cryptography and our academic staff include several leading researchers in these areas. Students on the programme have the opportunity to carry out their dissertation projects in cutting-edge research areas and to be supervised by experts.

The transferable skills gained during the MSc will open up a range of career options as well as provide a solid foundation for advanced research at PhD level.

See the website https://www.royalholloway.ac.uk/mathematics/coursefinder/mscmathematicsofcryptographyandcommunications(msc).aspx

Why choose this course?

- You will be provided with a solid mathematical foundation and a knowledge and understanding of the subjects of cryptography and communications preparing you for research or professional employment in this area.

- The mathematical foundations needed for applications in communication theory and cryptography are covered including Algebra, Combinatorics Complexity Theory/Algorithms and Number Theory.

- You will have the opportunity to carry out your dissertation project in a cutting-edge research area; our dissertation supervisors are experts in their fields who publish regularly in internationally competitive journals and there are several joint projects with industrial partners and Royal Holloway staff.

- After completing the course former students have a good foundation for the next step of their career both inside and outside academia.

Department research and industry highlights

The members of the Mathematics Department cover a range of research areas. There are particularly strong groups in information security, number theory, quantum theory, group theory and combinatorics. The Information Security Group has particularly strong links to industry.

Course content and structure

You will study eight courses as well as complete a main project under the supervision of a member of staff.

Core courses:
Advanced Cipher Systems
Mathematical and security properties of both symmetric key cipher systems and public key cryptography are discussed as well as methods for obtaining confidentiality and authentication.

Channels
In this unit, you will investigate the problems of data compression and information transmission in both noiseless and noisy environments.

Theory of Error-Correcting Codes
The aim of this unit is to provide you with an introduction to the theory of error-correcting codes employing the methods of elementary enumeration, linear algebra and finite fields.

Public Key Cryptography
This course introduces some of the mathematical ideas essential for an understanding of public key cryptography, such as discrete logarithms, lattices and elliptic curves. Several important public key cryptosystems are studied, such as RSA, Rabin, ElGamal Encryption, Schnorr signatures; and modern notions of security and attack models for public key cryptosystems are discussed.

Main project
The main project (dissertation) accounts for 25% of the assessment of the course and you will conduct this under the supervision of a member of academic staff.

Additional courses:
Applications of Field Theory
You will be introduced to some of the basic theory of field extensions, with special emphasis on applications in the context of finite fields.

Quantum Information Theory
‘Anybody who is not shocked by quantum theory has not understood it' (Niels Bohr). The aim of this unit is to provide you with a sufficient understanding of quantum theory in the spirit of the above quote. Many applications of the novel field of quantum information theory can be studied using undergraduate mathematics.

Network Algorithms
In this unit you will be introduced to the formal idea of an algorithm, when it is a good algorithm and techniques for constructing algorithms and checking that they work; explore connectivity and colourings of graphs, from an algorithmic perspective; and study how algebraic methods such as path algebras and cycle spaces may be used to solve network problems.

Advanced Financial Mathematics
In this unit you will investigate the validity of various linear and non-linear time series occurring in finance and extend the use of stochastic calculus to interest rate movements and credit rating;

Combinatorics
The aim of this unit is to introduce some standard techniques and concepts of combinatorics, including: methods of counting including the principle of inclusion and exclusion; generating functions; probabilistic methods; and permutations, Ramsey theory.

Computational Number Theory
You will be provided with an introduction to many major methods currently used for testing/proving primality and for the factorisation of composite integers. The course will develop the mathematical theory that underlies these methods, as well as describing the methods themselves.

Complexity Theory
Several classes of computational complexity are introduced. You will discuss how to recognise when different problems have different computational hardness, and be able to deduce cryptographic properties of related algorithms and protocols.

On completion of the course graduates will have:
- a suitable mathematical foundation for undertaking research or professional employment in cryptography and/or communications

- the appropriate background in information theory and coding theory enabling them to understand and be able to apply the theory of communication through noisy channels

- the appropriate background in algebra and number theory to develop an understanding of modern public key cryptosystems

- a critical awareness of problems in information transmission and data compression, and the mathematical techniques which are commonly used to solve these problems

- a critical awareness of problems in cryptography and the mathematical techniques which are commonly used to provide solutions to these problems

- a range of transferable skills including familiarity with a computer algebra package, experience with independent research and managing the writing of a dissertation.

Assessment

Assessment is carried out by a variety of methods including coursework, examinations and a dissertation. The examinations in May/June count for 75% of the final average and the dissertation, which has to be submitted in September, counts for the remaining 25%.

Employability & career opportunities

Our students have gone on to successful careers in a variety of industries, such as information security, IT consultancy, banking and finance, higher education and telecommunication. In recent years our graduates have entered into roles including Principal Information Security Consultant at Abbey National PLC; Senior Manager at Enterprise Risk Services, Deloitte & Touche; Global IT Security Director at Reuters; and Information Security manager at London Underground.

How to apply

Applications for entry to all our full-time postgraduate degrees can be made online https://www.royalholloway.ac.uk/studyhere/postgraduate/applying/howtoapply.aspx .

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This course covers a wide range of topics from both applied and applicable mathematics and is aimed at students who want to study the field in greater depth, in areas which are relevant to real life applications.

You will explore the mathematical techniques that are commonly used to solve problems in the real world, in particular in communication theory and in physics. As part of the course you will carry out an independent research investigation under the supervision of a member of staff. Popular dissertation topics chosen by students include projects in the areas of communication theory, mathematical physics, and financial mathematics.

The transferable skills gained on this course will open you up to a range of career options as well as provide a solid foundation for advanced research at PhD level.

See the website https://www.royalholloway.ac.uk/mathematics/coursefinder/mscmathematicsforapplications.aspx

Why choose this course?

- You will be provided with a solid mathematical foundation and knowledge and understanding of the subjects of cryptography and communications, preparing you for research or professional employment in this area.

- The Mathematics Department at Royal Holloway is well known for its expertise in information security and cryptography. The academics who teach on this course include several leading researchers in these areas.

- The mathematical foundations needed for applications in communication theory and cryptography are covered including Algebra, Combinatorics Complexity Theory/Algorithms and Number Theory.

- You will have the opportunity to carry out your dissertation project in a cutting-edge research area; our dissertation supervisors are experts in their fields who publish regularly in internationally competitive journals and there are several joint projects with industrial partners and Royal Holloway staff.

- After completing the course students have a good foundation for the next step of their career both inside and outside academia.

Department research and industry highlights

The members of the Mathematics Department cover a range of research areas. There are particularly strong groups in information security, number theory, quantum theory, group theory and combinatorics. The Information Security Group has particularly strong links to industry.

Course content and structure

You will study eight courses and complete a main project under the supervision of a member of staff.

Core courses:
Theory of Error-Correcting Codes
The aim of this unit is to provide you with an introduction to the theory of error-correcting codes employing the methods of elementary enumeration, linear algebra and finite fields.

Advanced Cipher Systems
Mathematical and security properties of both symmetric key cipher systems and public key cryptography are discussed, as well as methods for obtaining confidentiality and authentication.

Main project
The main project (dissertation) accounts for 25% of the assessment of the course and you will conduct this under the supervision of a member of academic staff.

Additional courses:
Applications of Field Theory
You will be introduced to some of the basic theory of field extensions, with special emphasis on applications in the context of finite fields.

Quantum Information Theory
‘Anybody who is not shocked by quantum theory has not understood it' (Niels Bohr). The aim of this unit is to provide you with a sufficient understanding of quantum theory in the spirit of the above quote. Many applications of the novel field of quantum information theory can be studied using undergraduate mathematics.

Network Algorithms
In this unit you will be introduced to the formal idea of an algorithm, when it is a good algorithm and techniques for constructing algorithms and checking that they work; explore connectivity and colourings of graphs, from an algorithmic perspective; and study how algebraic methods such as path algebras and cycle spaces may be used to solve network problems.

Advanced Financial Mathematics
In this unit you will investigate the validity of various linear and non-linear time series occurring in finance and extend the use of stochastic calculus to interest rate movements and credit rating;

Combinatorics
The aim of this unit is to introduce some standard techniques and concepts of combinatorics, including: methods of counting including the principle of inclusion and exclusion; generating functions; probabilistic methods; and permutations, Ramsey theory.

Computational Number Theory
You will be provided with an introduction to many major methods currently used for testing/proving primality and for the factorisation of composite integers. The course will develop the mathematical theory that underlies these methods, as well as describing the methods themselves.

Complexity Theory
Several classes of computational complexity are introduced. You will discuss how to recognise when different problems have different computational hardness, and be able to deduce cryptographic properties of related algorithms and protocols.

On completion of the course graduates will have:
- knowledge and understanding of: the principles of communication through noisy channels using coding theory; the principles of cryptography as a tool for securing data; and the role and limitations of mathematics in the solution of problems arising in the real world

- a high level of ability in subject-specific skills, such as algebra and number theory

- developed the capacity to synthesise information from a number of sources with critical awareness

- critically analysed the strengths and weaknesses of solutions to problems in applications of mathematics

- the ability to clearly formulate problems and express technical content and conclusions in written form

- personal skills of time management, self-motivation, flexibility and adaptability.

Assessment

Assessment is carried out by a variety of methods including coursework, examinations and a dissertation. The examinations in May/June count for 75% of the final average and the dissertation, which has to be submitted in September, counts for the remaining 25%.

Employability & career opportunities

Our students have gone on to successful careers in a variety of industries, such as information security, IT consultancy, banking and finance, higher education and telecommunication. In recent years our graduates have entered into roles including Principal Information Security Consultant at Abbey National PLC; Senior Manager at Enterprise Risk Services, Deloitte & Touche; Global IT Security Director at Reuters; and Information Security Manager at London Underground.

How to apply

Applications for entry to all our full-time postgraduate degrees can be made online https://www.royalholloway.ac.uk/studyhere/postgraduate/applying/howtoapply.aspx .

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This programme offers you the chance to study a range of modules in pure and applicable mathematics, giving you the opportunity to increase your knowledge and abilities in these areas. Depending on your choices, you will take either 7 or 8 modules, allowing you to study several different topics in depth, and to focus on the areas that interest you most. Over 2 years you will also learn the methods of mathematical research: how to read mathematical papers and how to communicate mathematics, both in written form for your project dissertation, and orally when you give presentations about your project.

You will acquire the skills to pursue your interest in the subject, either formally with a research degree, or informally with independent reading. You will come to us as someone with a mathematics degree; you will graduate as a mathematician.

Why study this course at Birkbeck?

Offers modules in group theory, graph theory, combinatorics and applicable mathematics such as coding theory and cryptography.
Specially designed for part-time students: delivered via high-quality, face-to-face teaching in the evenings, so that you can fit study around daytime commitments.
You complete a project in your chosen area of mathematics, with guidance from an expert supervisor.
Birkbeck's mathematicians are all active researchers, mostly in the areas of algebra and combinatorics. We've developed this exciting course around those research strengths.
Birkbeck has a library and several workstation rooms. You can also use several local university libraries, including the collection of the London Mathematical Society, a 5-minute walk from Birkbeck's main building.

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In this Master's specialisation, mathematicians working in areas pertinent to (theoretical) computer science, like algebra and logic, and theoretical computer scientists, working in areas as formal methods and theorem proving, have joined forces to establish a specialisation in the Mathematical Foundations of Computer Science. The programme is unique in the Netherlands and will be built on the excellence of both research institutes and the successful collaborations therein.
The emphasis of the Master's is on a combination of a genuine theoretical and up-to-date foundation in the pertinent mathematical subjects combined with an equally genuine and up-to-date training in key aspects of theoretical computer science. For this reason, the mathematics courses in this curriculum concentrate on Algebra, Complexity Theory, Logic, Number Theory, and Combinatorics. The computer science courses concentrate on Formal Methods, Type Theory, Category Theory, Coalgebra and Theorem Proving.
Within both institutes, ICIS and WINST, there is a concentration of researchers working on mathematical logic and theoretical computer science with a collaboration that is unique in the Netherlands. The research topics range from work on algebra, logic and computability, to models of distributed, parallel and quantum computation, as well as mathematical abstractions to reason about programmes and programming languages.

See the website http://www.ru.nl/masters/mathematics/foundations

Admission requirements for international students

1. A completed Bachelor's degree in Mathematics or Computer Science
In order to get admission to this Master’s you will need a completed Bachelor's in mathematics or computer science that have a strong mathematical background and theoretical interests. We will select students based on their motivation and their background. Mathematical maturity is essential and basic knowledge of logic and discrete mathematics is expected.

2. A proficiency in English
In order to take part in the programme, you need to have fluency in English, both written and spoken. Non-native speakers of English without a Dutch Bachelor's degree or VWO diploma need one of the following:
- TOEFL score of ≥575 (paper based) or ≥90 (internet based)
- IELTS score of ≥6.5
- Cambridge Certificate of Advanced English (CAE) or Certificate of Proficiency in English (CPE), with a mark of C or higher

Career prospects

There is a serious shortage of well-trained information specialists. Often students are offered a job before they have actually finished their study. About 20% of our graduates choose to go on to do a PhD but most find jobs as systems builders, ICT specialists or ICT managers in the private sector or within government.

Our approach to this field

In this Master's specialisation, mathematicians working in areas pertinent to (theoretical) computer science, like algebra and logic, and theoretical computer scientists, working in areas as formal methods and theorem proving, have joined forces to establish a specialisation in the Mathematical Foundations of Computer Science. The programme is unique in the Netherlands and will be built on the excellence of both research institutes and the successful collaborations therein.

The emphasis of the Master's is on a combination of a genuine theoretical and up-to-date foundation in the pertinent mathematical subjects combined with an equally genuine and up-to-date training in key aspects of theoretical computer science. For this reason, the mathematics courses in this curriculum concentrate on Algebra, General Topology, Logic, Number Theory, and Combinatorics. The computer science courses concentrate on Formal Methods, Type Theory, Category Theory, Coalgebra and Theorem Proving.

Our research in this field

Within both institutes, ICIS and WINST, there is a concentration of researchers working on mathematical logic and theoretical computer science with a collaboration that is unique in the Netherlands. The research topics range from work on algebra, logic and computability, to models of distributed, parallel and quantum computation, as well as mathematical abstractions to reason about programmes and programming languages.

See the website http://www.ru.nl/masters/mathematics/foundations

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High-level training in statistics and the modelling of random processes for applications in science, business or health care.

For many complex systems in nature and society, stochastics can be used to efficiently describe the randomness present in all these systems, thereby giving the data greater explanatory and predictive power. Examples include statistical mechanics, financial markets, mobile phone networks, and operations research problems. The Master’s specialisation in Applied Stochastics will train you to become a mathematician that can help both scientists and businessmen make better decisions, conclusions and predictions. You’ll be able to bring clarity to the accumulating information overload they receive.

The members of the Applied Stochastics group have ample experience with the pure mathematical side of stochastics. This area provides powerful techniques in functional analysis, partial differential equations, geometry of metric spaces and number theory, for example. The group also often gives advice to both their academic colleagues, and organisations outside of academia. They will therefore not only be able to teach you the theoretical basis you need to solve real world stochastics problems, but also to help you develop the communications skills and professional expertise to cooperate with people from outside of mathematics.

See the website http://www.ru.nl/masters/mathematics/stochastics

Why study Applied Stochastics at Radboud University?

- This specialisation focuses both on theoretical and applied topics. It’s your choice whether you want to specialise in pure theoretical research or perform an internship in a company setting.
- Mathematicians at Radboud University are expanding their knowledge of random graphs and networks, which can be applied in the ever-growing fields of distribution systems, mobile phone networks and social networks.
- In a unique and interesting collaboration with Radboudumc, stochastics students can help researchers at the hospital with very challenging statistical questions.
- Because the Netherlands is known for its expertise in the field of stochastics, it offers a great atmosphere to study this field. And with the existence of the Mastermath programme, you can follow the best mathematics courses in the country, regardless of the university that offers them.
- Teaching takes place in a stimulating, collegial setting with small groups. This ensures that you’ll get plenty of one-on-one time with your thesis supervisor at Radboud University .
- More than 85% of our graduates find a job or a gain a PhD position within a few months of graduating.

Career prospects

Master's programme in Mathematics

Mathematicians are needed in all industries, including the banking, technology and service industries, to name a few. A Master’s in Mathematics will show prospective employers that you have perseverance, patience and an eye for detail as well as a high level of analytical and problem-solving skills.

Job positions

The skills learned during your Master’s will help you find jobs even in areas where your specialised mathematical knowledge may initially not seem very relevant. This makes your job opportunities very broad and is the reason why many graduates of a Master’s in Mathematics find work very quickly.
Possible careers for mathematicians include:
- Researcher (at research centres or within corporations)
- Teacher (at all levels from middle school to university)
- Risk model validator
- Consultant
- ICT developer / software developer
- Policy maker
- Analyst

PhD positions

Radboud University annually has a few PhD positions for graduates of a Master’s in Mathematics. A substantial part of our students attain PhD positions, not just at Radboud University, but at universities all over the world.

Our research in this field

The research of members of the Applied Stochastics Department, focuses on combinatorics, (quantum) probability and mathematical statistics. Below, a small sample of the research our members pursue.

Eric Cator’s research has two main themes, probability and statistics.
1. In probability, he works on interacting particles systems, random polymers and last passage percolation. He has also recently begun working on epidemic models on finite graphs.
2. In statistics, he works on problems arising in mathematical statistics, for example in deconvolution problems, the CAR assumption and more recently on the local minimax property of least squares estimators.

Cator also works on more applied problems, usually in collaboration with people from outside statistics, for example on case reserving for insurance companies or airplane maintenance. He has a history of changing subjects: “I like to work on any problem that takes my fancy, so this description might be outdated very quickly!”

Hans Maassen researches quantum probability or non-commutative probability, which concerns a generalisation of probability theory that is broad enough to contain quantum mechanics. He takes part in the Geometry and Quantum Theory (GQT) research cluster of connected universities in the Netherlands. In collaboration with Burkhard Kümmerer he is also developing the theory of quantum Markov chains, their asymptotic completeness and ergodic theory, with applications to quantum optics. Their focal point is shifting towards quantum information and control theory, an area which is rapidly becoming relevant to experimental physicists.

Ross Kang conducts research in probabilistic and extremal combinatorics, with emphasis on graphs (which abstractly represent networks). He works in random graph theory (the study of stochastic models of networks) and often uses the probabilistic method. This involves applying probabilistic tools to shed light on extremes of large-scale behaviour in graphs and other combinatorial structures. He has focused a lot on graph colouring, an old and popular subject made famous by the Four Colour Theorem (erstwhile Conjecture).

See the website http://www.ru.nl/masters/mathematics/stochastics

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The MSc in Mathematics and Foundations of Computer Science, run jointly by the Mathematical Institute and the Department of Computer Science, focuses on the interface between pure mathematics and theoretical computer science. 

The mathematical side concentrates on areas where computers are used, or which are relevant to computer science, namely algebra, general topology, number theory, combinatorics and logic. Examples from the computing side include computational complexity, concurrency, and quantum computing. Students take a minimum of five options and write a dissertation.

The course is suitable for those who wish to pursue research in pure mathematics (especially algebra, number theory, combinatorics, general topology and their computational aspects), mathematical logic, or theoretical computer science. It is also suitable for students wishing to enter industry with an understanding of the mathematical and logical design and concurrency.

The course will consist of examined lecture courses and a written dissertation. The lecture courses will be divided into two sections:

• Section A: Mathematical Foundations
• Section B: Applicable Theories

Each section shall be divided into schedule I (basic) and schedule II (advanced). Students will be required to satisfy the examiners in at least two courses taken from section B and in at least two courses taken from schedule II. The majority of these courses should be given in the first two terms. 

During Trinity term and over the summer students should complete a dissertation on an agreed topic. The dissertation must bear regard to course material from section A or section B, and it must demonstrate relevance to some area of science, engineering, industry or commerce.

It is intended that a major feature of this course is that candidates should show a broad knowledge and understanding over a wide range of material. Consequently, each lecture course taken will receive an assessment upon its completion by means of a test based on written work. Students will be required to pass five courses, that include two courses from section B and two at the schedule II level - these need not be distinct - and the dissertation.

The course runs from the beginning of October through to the end of September, including the dissertation.

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The MSc in Mathematics gives an in-depth training in advanced mathematics to students who have
already obtained a first degree with substantial mathematical content. Students successfully completing the MSc will acquire specialist knowledge in their chosen areas of mathematics, and the MSc is an excellent preparation for those who are considering pursuing research in mathematics.

The main areas of mathematics that may be pursued within this MSc are pure mathematics (especially algebra and combinatorics), dynamical systems, probability and statistics, and astronomy. The MSc programme is very flexible, and in consultation with your academic adviser you may choose modules in different areas or specialise in one.

Programme outline
You will normally take eight modules in total, with one module typically comprising 24 hours of lectures and 12 hours of tutorials given during a twelve-week semester. In addition to the MSc modules offered at Queen Mary, you can also choose from an extremely wide range of advanced mathematics modules offered at other Colleges of the University of London. During the summer period, supervised by an academic member of staff, you are required to complete a dissertation, working largely independently in an advanced topic in mathematics or statistics.

For details of modules typically offered, see: http://www.maths.qmul.ac.uk/postgraduate/msc-maths-stats/modules

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Studying Mathematics at postgraduate level gives you a chance to begin your own research, develop your own creativity and be part of a long tradition of people investigating analytic, geometric and algebraic ideas.

This programme allows you to further enhance your knowledge, creativity and computational skills in core mathematical subjects and their applications giving you a competitive advantage in a wide range of mathematically based careers. The modules, which are designed and taught by internationally known researchers, are accessible, relevant, interesting and challenging.

About the School of Mathematics, Statistics and Actuarial Science (SMSAS)

The School has a strong reputation for world-class research and a well-established system of support and training, with a high level of contact between staff and research students. Postgraduate students develop analytical, communication and research skills. Developing computational skills and applying them to mathematical problems forms a significant part of the postgraduate training in the School.

The Mathematics Group at Kent ranked highly in the most recent Research Assessment Exercise. With 100% of the Applied Mathematics Group submitted, all research outputs were judged to be of international quality and 12.5% was rated 4*. For the Pure Mathematics Group, a large proportion of the outputs demonstrated international excellence.

The Mathematics Group also has an excellent track record of winning research grants from the Engineering and Physical Sciences Research Council (EPSRC), the Royal Society, the EU, the London Mathematical Society and the Leverhulme Trust.

National ratings

In the Research Excellence Framework (REF) 2014, research by the School of Mathematics, Statistics and Actuarial Science was ranked 25th in the UK for research power and 100% or our research was judged to be of international quality.

An impressive 92% of our research-active staff submitted to the REF and the School’s environment was judged to be conducive to supporting the development of world-leading research.

Course structure

At least one modern application of mathematics is studied in-depth by each student. Mathematical computing and open-ended project work forms an integral part of the learning experience. There are opportunities for outreach and engagement with the public on mathematics.

You take eight modules in total: six from the list below; a short project module and a dissertation module. The modules concentrate on a specific topic from: analysis; applied mathematics; geometry; and algebra.

Modules

The following modules are indicative of those offered on this programme. This list is based on the current curriculum and may change year to year in response to new curriculum developments and innovation. Most programmes will require you to study a combination of compulsory and optional modules. You may also have the option to take modules from other programmes so that you may customise your programme and explore other subject areas that interest you.

MA961 - Mathematical Inquiry and Communication (30 credits) - https://www.kent.ac.uk/courses/modules/module/MA961
MA962 - Geometric Integration (15 credits) - https://www.kent.ac.uk/courses/modules/module/MA962
MA963 - Poisson Algebras and Combinatorics (15 credits) - https://www.kent.ac.uk/courses/modules/module/MA963
MA964 - Applied Algebraic Topology (15 credits) - https://www.kent.ac.uk/courses/modules/module/MA964
MA965 - Symmetries, Groups and Invariants (15 credits) - https://www.kent.ac.uk/courses/modules/module/MA965
MA966 - Diagram Algebras (15 credits) - https://www.kent.ac.uk/courses/modules/module/MA966
MA967 - Quantum Physics (15 credits) - https://www.kent.ac.uk/courses/modules/module/MA967
MA968 - Mathematics and Music (15 credits) - https://www.kent.ac.uk/courses/modules/module/MA968
MA969 - Applied Differential Geometry (15 credits) - https://www.kent.ac.uk/courses/modules/module/MA969
MA970 - Nonlinear Analysis and Optimisation (15 credits) - https://www.kent.ac.uk/courses/modules/module/MA970
Show more... https://www.kent.ac.uk/courses/postgraduate/146/mathematics-and-its-applications#!structure

Assessment

Assessment is by closed book examinations, take-home problem assignments and computer lab assignments (depending on the module). The project and dissertation modules are assessed mainly on the reports or work you produce, but also on workshop activities during the teaching term.

Programme aims

This programme aims to:

- provide a Master’s level mathematical education of excellent quality, informed by research and scholarship
- provide an opportunity to enhance your mathematical creativity, problem-solving skills and advanced computational skills
- provide an opportunity for you to enhance your oral communication, project design and basic research skills
- provide an opportunity for you to experience and engage with a creative, research-active professional mathematical environment
- produce graduates of value to the region and nation by offering you opportunities to learn about mathematics in the context of its application.

Careers

A postgraduate degree in Mathematics is a flexible and valuable qualification that gives you a competitive advantage in a wide range of mathematically oriented careers. Our programmes enable you to develop the skills and capabilities that employers are looking for including problem-solving, independent thought, report-writing, project management, leadership skills, teamworking and good communication.

Many of our graduates have gone on to work in international organisations, the financial sector, and business. Others have found postgraduate research places at Kent and other universities.

Learn more about Kent

Visit us - https://www.kent.ac.uk/courses/visit/openday/pgevents.html

International Students - https://www.kent.ac.uk/internationalstudent/

Why study at Kent? - https://www.kent.ac.uk/courses/postgraduate/why/

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This interdisciplinary Masters degree provides you with a broad background in some mainstream and modern aspects of mathematics and computer science. You’ll be introduced to sophisticated techniques at the forefront of both disciplines.

The programme combines teaching and research from the School of Mathematics and the School of Computing. Based on the Schools’ complementary research strengths the programme follows two main strands:

• Algorithms and complexity theory
• Numerical methods and parallel computing

You’ll have the choice to specialise in one of these strands, gaining specialist knowledge and skills that will prepare you for a wide range of careers. You’ll also develop your research skills when you complete your dissertation.

If you do not meet the full academic entry requirements then you may wish to consider the Graduate Diploma in Mathematics. This course is aimed at students who would like to study for a mathematics related MSc course but do not currently meet the entry requirements. Upon completion of the Graduate Diploma, students who meet the required performance level will be eligible for entry onto a number of related MSc courses, in the following academic year.

Course content

It is expected that you will specialise in one of two areas during the course, although this is not essential.

The two strands are:

Algorithms and complexity theory and connections to logic and combinatorics

This concerns the efficiency of algorithms for solving computational problems, and identifies hierarchies of computational difficulty. This subject has applications in many areas, such as distributed computing, algorithmic tools to manage transport infrastructure, health informatics, artificial intelligence, and computational biology.

Numerical methods and parallel computing

Many problems, in mathematics, physics, astrophysics and biology cannot be solved using analytical techniques and require the application of numerical algorithms for progress. The development and optimisation of these algorithms coupled to the recent increase in computing power via the availability of massively parallel machines has led to great advances in many fields of computational mathematics. This subject has applications in many areas, such as combustion, lubrication, atmospheric dispersion, river and harbour flows, and many more.

For information on typical modules, read Mathematics and Computer Science MSc in the course catalogue

Learning and teaching

Teaching is carried out through a mixture of lectures and smaller group activities such as workshops. Most modules are assessed by a mix of coursework and written examinations. There is also the opportunity to complete a summer project which is individually supervised by a member of staff.

Assessment

The taught course is primarily assessed by end-of-semester examinations with a small component of continuous assessment. The semester three project is assessed by a written dissertation and a short oral presentation.

Career opportunities

Each of these areas offers many career options, and the MSc will provide you with both technical and transferrable skills, for example, conducting an extended and independent research project. It will also offer you excellent preparation for doctoral research in these or related subjects. On completion of the degree you can progress onto a wide range of opportunities including:

• PhD in Mathematics, or in Computer Science
• Careers in Computing and Industries which require algorithmic tools (transport infrastructure, health informatics, computational biology, artificial intelligence, companies developing the internet (e.g. search engines).
• Many other careers (e.g. in Finance) where a mathematics background is valued.

In collaboration with both industrial and academic partners, our research has resulted in computational techniques, and software, that has been widely applied. Our industry links are extensive and include companies such as Google, Yahoo, Akamai, Microsoft, and Tracsis, as well as the NHS.

Careers support

We encourage you to prepare for your career from day one. That’s one of the reasons Leeds graduates are so sought after by employers.

The Careers Centre and staff in your faculty provide a range of help and advice to help you plan your career and make well-informed decisions along the way, even after you graduate. Find out more at the Careers website.

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This course is offered full and part-time. The full-time course lasts one calendar year, October to September; the part-time course lasts two years.

The course includes a wide range of lecture modules in analysis, geometry and topology, algebra, number theory and combinatorics.

You also undertake a written project under the direction of a supervisor who is an expert in that field.

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Our MPhil/PhD degree in Mathematics and Statistics aims to train you to conduct research of a high academic standard and to make original contributions to the subject.

The programme involves coursework (where suitable) and research training, but its major component is the preparation of a substantial research thesis. The thesis should demonstrate a sound understanding of the main issues in the area and add to existing knowledge.

Research interests in mathematics and statistics include: mathematical finance, in particular the analysis of risk and numerical computation; mathematical physics and partial differential equations; approximation theory and numerical analysis; probability and stochastic processes, pure and applied; applied statistics and multivariate analysis; covariance modelling for repeated measures and longitudinal data; medical statistics; combinatorics, algebra and designs.

Our research

Birkbeck is one of the world’s leading research-intensive institutions. Our cutting-edge scholarship informs public policy, achieves scientific advances, supports the economy, promotes culture and the arts, and makes a positive difference to society.

Birkbeck’s research excellence was confirmed in the 2014 Research Excellence Framework, which placed Birkbeck 30th in the UK for research, with 73% of our research rated world-leading or internationally excellent.

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The Department of Mathematical Sciences is committed to the idea that pure and applied mathematics are two faces of the same subject. The department offers a lively research atmosphere; faculty research areas include algebra, analysis, combinatorics, geometry/topology, number theory, and probability and statistics. Students are encouraged to take advantage of the broad range of courses and specializations available.

The department is noted for its method of graduate education. In first-year courses, the emphasis is on training the student to do mathematics in depth. Many students report that these courses are the formative experiences of their professional lives.

The MA program is intended to give the student a solid professional basis either for proceeding to the PhD program or for work in government, industry or teaching at the community college level.

The PhD degree prepares a student for university or college teaching and for higher-level employment in government and industry. Entering students having substantial graduate-level training may enter the PhD program, skipping the MA.

Graduates of the program typically find employment as teachers, those with the MA degree at community college level and those with the PhD degree at university and college level, while significant number also find employment with government and industry as statisticians, data analysts, consultants, researchers and programmers.

Recent doctoral graduate placements include: Mathematical/Statistical Engineer at Corning Incorporated, Postdoctoral Biostatistician at Janssen Research and Development, Assistant Professor at Alfred University, Risk Analyst for JP Morgan Chase, Assistant Professor at Juniata College, Assistant Professor at Ohio State University.

All applicants must submit the following:

- Online graduate degree application and application fee
- Transcripts from each college/university you have attended
- Three letters of recommendation
- Personal statement (2-3 pages) describing your reasons for pursuing graduate study, your career aspirations, your special interests within your field, and any unusual features of your background that might need explanation or be of interest to your program's admissions committee.
- Resume or Curriculum Vitae (max. 2 pages)
- Official GRE scores

And, for international applicants:
- International Student Financial Statement form
- Official bank statement/proof of support
- Official TOEFL, IELTS, or PTE Academic scores

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Operations Research

The Research Master’s specialization in Operations Research prepares students for the PhD programs in fields such as operations management or optimization techniques. This Research Master’s specialization in Operations Research is an in-depth study program focused on cutting edge research in Operations Research. A graduated student from this program gains knowledge of and insight in the tools and methodologies of operations research, including optimization, stochastic modeling, simulation, combinatorics, and game theory. Teaching is led by academics that are at the forefront of their fields.

A Research Master student is a part of a select group of finest students from all over the world. It is aimed at ambitious students that want to distinguish themselves. Students follow classes in small groups. This allows intensive interaction with professors who are top-notch researchers, both during as well as outside of classes. This program offers many different courses and students are free to choose most of the offered courses.

Students enrolled in the Research Master program qualify for research positions anywhere in the world.

Several of our Research Master students embark on a 3-year Ph.D. program and later become professors themselves. With their research, discoveries, and insights – expressed through writing, speaking, and teaching – they help shape academic world. Others opt for business practice with international companies and consulting firms.

Career Perspective MSc specialization in Operations Research

If you follow the Research Master's program in Business - be it accounting, finance, information management, marketing, operations research, or organization and strategy - you are prepared to continue in a PhD program. More than 75% of the graduates continue at a PhD position at either Tilburg University or another university. 25% of the graduates start a professional career, mostly in the consultancy or financial sector.

The Research Master's program in Business in combination with the subsequent PhD studies enable our students to find jobs at universities around the world, at research institutes or in the banking and consultancy sectors.

In the PhD phase, most accepted PhD students become university employees earning a gross salary over more than Euro 85000 over three years and are granted pension rights.

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The ALGANT Master program provides a study and research track in pure mathematics, with a strong focus on algebra, geometry and number theory. This track may be completed throughout Europe and the world, thanks to a partnership between leading research universities. The ALGANT course introduces students to the latest developments within these subjects, and provides the best possible preparation for their forthcoming doctoral studies.

The ALGANT program consists mainly of advanced courses within the field of mathematics and of a research project or internship leading to a Master thesis. Courses are offered in: algebraic geometry, algebraic and geometric topology, algebraic and analytic number theory, coding theory, combinatorics, complex function theory, cryptology, elliptic curves, manifolds. Students are encouraged to participate actively in seminars.

The university partners offer compatible basic preparation in the first year (level 1), which then leads to a complementary offer for more specialized courses in the second year (level 2). 

Program structure

Year 1 (courses in French)

Semester 1

• Modules and quadratic spaces (9 ECTS)
• Group theory (6 ECTS)
• Complex analysis (9 ECTS)
• Functional analysis (6 ECTS)

Semester 2

• Geometry (6 ECTS)
• Number theory (6 ECTS)
• Spectral theory and distributions (6 ECTS)
• Probability and statistics (6 ECTS)
• Cryptology (6 ECTS)
• Algebra and formal computations (6 ECTS)

Year 2 (courses in English)

Semester 1

• Number theory (9 ECTS)
• Algorithmic number theory (6 ECTS)
• Geometry (9 ECTS)
• Elliptic curves (6 ECTS)
• Algebraic geometry (9 ECTS)
• Analytic number theory: advanced course 1 (6 ECTS)

Semester 2

• Cohomology of groups: advanced course 2 (6 ECTS)
• The key role of certain inequalities at the interface between complex geometry (6 ECTS)

Strengths of this Master program

• Courses given by academic experts within the field of mathematics.
• Individually tailored study tracks.
• Top-quality scientific environment and facilities provided by leading global research institutes, e.g. Institut de Mathématiques de Bordeaux.
• Supported by the International Master program of the Bordeaux Initiative of Excellence.

After this Master program?

Students who successfully complete the ALGANT program will be well equipped to pursue a career in research by preparing a Ph.D.

Graduates may also directly apply for positions as highly trained mathematicians, especially in the areas of cryptography, information security and numerical communications.

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The Department of Mathematics offers graduate courses leading to M.Sc., and eventually to Ph.D., degree in Mathematics. The Master of Science program aims to provide a sound foundation for the students who wish to pursue a research career in mathematics as well as other related areas. The department emphasizes both pure and applied mathematics. Research in the department covers algebra, number theory, combinatorics, differential equations, functional analysis, abstract harmonic analysis, mathematical physics, stochastic analysis, biomathematics and topology.

Current faculty projects and research interests:

• Ring Theory and Module Theory, especially Krull dimension, torsion theories, and localization

• Algebraic Theory of Lattices, especially their dimensions (Krull, Goldie, Gabriel, etc.) with applications to Grothendieck categories and module categories equipped with torsion theories

• Field Theory, especially Galois Theory, Cogalois Theory, and Galois cohomology

• Algebraic Number Theory, especially rings of algebraic integers

• Iwasawa Theory of Galois representations and their deformations Euler and Kolyvagin systems, Equivariant Tamagawa Number
Conjecture

• Combinatorial design theory, in particular metamorphosis of designs, perfect hexagon triple systems

• Graph theory, in particular number of cycles in 2-factorizations of complete graphs

• Coding theory, especially relation of designs to codes

• Random graphs, in particular, random proximity catch graphs and digraphs

• Partial Differential Equations

• Nonlinear Problems of Mathematical Physics

• Dissipative Dynamical Systems

• Scattering of classical and quantum waves

• Wavelet analysis

• Molecular dynamics

• Banach algebras, especially the structure of the second Arens duals of Banach algebras

• Abstract Harmonic Analysis, especially the Fourier and Fourier-Stieltjes algebras associated to a locally compact group

• Geometry of Banach spaces, especially vector measures, spaces of vector valued continuous functions, fixed point theory, isomorphic properties of Banach spaces

• Differential geometric, topologic, and algebraic methods used in quantum mechanics

• Geometric phases and dynamical invariants

• Supersymmetry and its generalizations

• Pseudo-Hermitian quantum mechanics

• Quantum cosmology

• Numerical Linear Algebra

• Numerical Optimization

• Perturbation Theory of Eigenvalues

• Eigenvalue Optimization

• Mathematical finance

• Stochastic optimal control and dynamic programming

• Stochastic flows and random velocity fields

• Lyapunov exponents of flows

• Unicast and multicast data traffic in telecommunications

• Probabilistic Inference

• Inference on Random Graphs (with emphasis on modeling email and internet traffic and clustering analysis)

• Graph Theory (probabilistic investigation of graphs emerging from computational geometry)

• Statistics (analysis of spatial data and spatial point patterns with applications in epidemiology and ecology and statistical methods for medical data and image analysis)

• Classification and Pattern Recognition (with applications in mine field and face detection)

• Arithmetical Algebraic Geometry, Arakelov geometry, Mixed Tate motives

• p-adic methods in arithmetical algebraic geometry, Ramification theory of arithmetic varieties

• Topology of low-dimensional manifolds, in particular Lefschetz fibrations, symplectic and contact structures, Stein fillings

• Symplectic topology and geometry, Seiberg-Witten theory, Floer homology

• Foliation and Lamination Theory, Minimal Surfaces, and Hyperbolic Geometry

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