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Masters Degrees (Quantum Field Theory)

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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. Read more
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. Read more
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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Our MSc Mathematics programme consists of a wide range of modules and a written project. Read more
Our MSc Mathematics programme consists of a wide range of modules and a written project. Your module choices will be mainly from the two main blocks of pure mathematics and theoretical physics but you are also able to choose certain modules from the Financial Mathematics programme and at other University of London institutions, subject to approval.

Key benefits

- An intensive course covering a wide range of basic and advanced topics.

- Intimate class environment with small class sizes (typically fewer than twenty students on a module) allowing good student-lecturer interactions.

- A full twelve-month course with a three-month supervised summer project to give a real introduction to research.

Visit the website: http://www.kcl.ac.uk/study/postgraduate/taught-courses/mathematics-msc.aspx

Course detail

- Description -

The majority of the eight modules are taken from blocks of pure mathematics and theoretical physics, with other options from the MSc Financial Mathematics and other University of London institutions available, subject to approval.

Pure Mathematics:

- Metric & Banach Spaces
- Complex Analysis
- Fourier Analysis
- Non-linear Analysis (new in 2013)
- Operator Theory
- Galois Theory
- Lie Groups & Lie Algebras
- Algebraic Number Theory
- Algebraic Geometry
- Manifolds
- Real Analysis II
- Topology
- Rings & Modules
- Representation Theory of Finite Groups

Theoretical Physics:

- Quantum Field Theory
- String Theory & Branes
- Supersymmetry
- Advanced Quantum Field Theory
- Spacetime geometry and General Relativity
- Advanced General Relativity
- Low-dimensional Quantum Field Theory

- Course purpose -

This programme is suitable for Mathematics graduates who wish to study more advanced mathematics. The programme ideally prepares students for PhD study in a mathematical discipline.

- Course format and assessment -

Eight modules assessed by written examinations; one individual project.

Career prospects

Many of our graduates take up full-time employment in various industries that require good mathematical/computer knowledge or that look for intelligent and creative people. Recent employers of our graduates include Barclays Bank, Kinetic Partners, Lloyds Banking Group and Sapient.

How to apply: http://www.kcl.ac.uk/study/postgraduate/apply/taught-courses.aspx

About Postgraduate Study at King’s College London:

To study for a postgraduate degree at King’s College London is to study at the city’s most central university and at one of the top 20 universities worldwide (2015/16 QS World Rankings). Graduates will benefit from close connections with the UK’s professional, political, legal, commercial, scientific and cultural life, while the excellent reputation of our MA and MRes programmes ensures our postgraduate alumni are highly sought after by some of the world’s most prestigious employers. We provide graduates with skills that are highly valued in business, government, academia and the professions.

Scholarships & Funding:

All current PGT offer-holders and new PGT applicants are welcome to apply for the scholarships. For more information and to learn how to apply visit: http://www.kcl.ac.uk/study/pg/funding/sources

Free language tuition with the Modern Language Centre:

If you are studying for any postgraduate taught degree at King’s you can take a module from a choice of over 25 languages without any additional cost. Visit: http://www.kcl.ac.uk/mlc

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The MSc Theoretical Physics programme will provide you with exposure to a very wide range of world-leading teaching and research skills. Read more
The MSc Theoretical Physics programme will provide you with exposure to a very wide range of world-leading teaching and research skills. As well as the wide range of modules offered by the Department of Mathematics, many optional modules are available from across the University of London, subject to approval. King's will offer you a unique module in 'General Research Techniques' which will prepare you for life as a research scientist. You will also undertake an extended research project supervised by one of our academic staff.

Key benefits

- This intensive programme covers basic topics in theoretical and mathematical physics such as general relativity and quantum field theory, and leads to advanced topics such as string theory, supersymmetry and integrable quantum field theory.

- Intimate class environment with small class sizes (typically fewer than 30 students per module) allows good student-lecturer interactions.

- A full 12-month course with a three-month supervised summer project to give a real introduction to research.

Visit the website: http://www.kcl.ac.uk/study/postgraduate/taught-courses/theoretical-physics-msc.aspx

Course detail

- Description -

The master's is organised on a module system together with an individual project. You will take eight taught modules of which at least five will be from the list: Mechanics, Relativity & Quantum Theory; Quantum Mechanics II; Quantum Field Theory; Lie Groups & Lie Algebras; Manifolds; Space-time Geometry and General Relativity; Advanced General Relativity; Supersymmetry & Gauge Theory; String Theory and Branes; Mathematical Methods for Theoretical Physics; Standard Model Physics and Beyond.

The remaining modules can be drawn from the wide range of theoretical physics or pure mathematics MSc courses available in London, the Financial Mathematics MSc in King's and at most two courses from the undergraduate programme at King's. The project is undertaken over the summer in an area of current research.

- Purpose -

The purpose of this programme is to provide a coherent and comprehensive introduction to the main building blocks of modern theoretical physics, preparing students for active research at the forefront of this discipline.

- Course format and assessment -

At least eight taught modules assessed by written examinations and one individual project.

Career prospects

Many of our very successful graduates go on to start PhD studies in theoretical physics at various universities in the United Kingdom and abroad, including with our group here at King's, for which the MSc is particularly well tailored. Our graduates also take up full-time employment in various industries that require good mathematical/computer knowledge or that look for intelligent and creative people. Recent employers of our graduates include the Algerian Space Agency, FRM Capital Advisors and Lloyds Banking Group.

How to apply: http://www.kcl.ac.uk/study/postgraduate/apply/taught-courses.aspx

About Postgraduate Study at King’s College London:

To study for a postgraduate degree at King’s College London is to study at the city’s most central university and at one of the top 20 universities worldwide (2015/16 QS World Rankings). Graduates will benefit from close connections with the UK’s professional, political, legal, commercial, scientific and cultural life, while the excellent reputation of our MA and MRes programmes ensures our postgraduate alumni are highly sought after by some of the world’s most prestigious employers. We provide graduates with skills that are highly valued in business, government, academia and the professions.

Scholarships & Funding:

All current PGT offer-holders and new PGT applicants are welcome to apply for the scholarships. For more information and to learn how to apply visit: http://www.kcl.ac.uk/study/pg/funding/sources

Free language tuition with the Modern Language Centre:

If you are studying for any postgraduate taught degree at King’s you can take a module from a choice of over 25 languages without any additional cost. Visit: http://www.kcl.ac.uk/mlc

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This is a one year advanced taught course. The aim of this course is to bring students in twelve months to the frontier of elementary particle theory. Read more
This is a one year advanced taught course. The aim of this course is to bring students in twelve months to the frontier of elementary particle theory. This course is intended for students who have already obtained a good first degree in either physics or mathematics, including in the latter case courses in quantum mechanics and relativity.

The course consists of three modules: the first two are the Michaelmas and Epiphany graduate lecture courses, which are assessed by examinations in January and March. The third module is a dissertation on a topic of current research, prepared under the guidance of a supervisor with expertise in the area. We offer a wide variety of possible dissertation topics. The dissertation must be submitted by September 15th, the end of the twelve month course period.

Course Structure

The main group of lectures are given in the first two terms of the academic year (Michaelmas and Epiphany). This part of the lecture course is assessed by examinations. In each term there are two teaching periods of four weeks, with a week's break in the middle of the term in which students will be able to revise the material. Most courses are either eight lectures or 16 lectures in length. There are 14 lectures/week in the Michaelmas term and 14 lectures/week in Epiphany term.

Core Modules

-Introductory Field Theory
-Group Theory
-Standard Model
-General Relativity
-Quantum Electrodynamics
-Quantum Field Theory
-Conformal Field Theory
-Supersymmetry
-Anomalies
-Strong Interaction Physics
-Cosmology
-Superstrings and D-branes
-Non-Perturbative Physics
-Euclidean Field Theory
-Flavour Physics and Effective Field Theory
-Neutrinos and Astroparticle Physics
-2d Quantum Field Theory

Optional Modules available in previous years included:

-Differential Geometry for Physicists
-Boundaries and Defects in Integrable Field Theory
-Computing for Physicists

Learning and Teaching

This is a full-year degree course, starting early October and finishing in the middle of the subsequent September. The aim of the course is to bring students to the frontier of research in elementary particle theory. The course consists of three modules: the first two are the Michaelmas and Epiphany graduate lecture courses. The third module is a dissertation on a topic of current research, prepared under the guidance of a supervisor with expertise in the area. We offer a wide variety of possible dissertation topics.

The lectures begin with a general survey of particle physics and introductory courses on quantum field theory and group theory. These lead on to more specialised topics, amongst others in string theory, cosmology, supersymmetry and more detailed aspects of the standard model.

The main group of lectures is given in the first two terms of the academic year (Michaelmas and Epiphany). This part of the lecture course is assessed by examinations. In each term there are two teaching periods of 4 weeks, with a week's break in the middle of the term in which students will be able to revise the material. Most courses are either 8 lectures or 16 lectures in length. There are 14 lectures/week in the Michaelmas term and 14 lectures/week in Epiphany term they are supported by weekly tutorials. In addition lecturers also set a number of homework assignments which give the student a chance to test his or her understanding of the material.

There are additional optional lectures in the third term. These introduce advanced topics and are intended as preparation for research in these areas.

The dissertation must be submitted by mid-September, the end of the twelve month course period.

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The Quantum Technologies MSc will take students to the cutting-edge of research in the emerging area of quantum technologies, giving them not only an advanced training in the relevant physics but also the chance to acquire key skills in the engineering and information sciences. Read more
The Quantum Technologies MSc will take students to the cutting-edge of research in the emerging area of quantum technologies, giving them not only an advanced training in the relevant physics but also the chance to acquire key skills in the engineering and information sciences.

Degree information

Students learn the language and techniques of advanced quantum mechanics, quantum information and quantum computation, as well as state-of-the-art implementation with condensed matter and quantum optical systems.

Students undertake modules to the value of 180 credits.

The programme consists of five core modules (75 credits), three optional modules (45 credits) and a research project with a dissertation/report (60 credits).

Core modules
-Advanced Quantum Theory
-Atom and Photon Physics
-Quantum Communication and Computation
-Research Case Studies for Quantum Technologies
-Transferable Skills in Research Case Studies for Quantum Technologies

Optional modules - students choose three of the following optional modules:
-Advanced Photonic Devices
-Introduction to Cryptography
-Nanoelectronic Devices
-Nanoscale Processing for Advanced Devices
-Optical Transmission and Networks
-Order and Excitations in Condensed Matter
-Physics and Optics of Nano-Structures
-Research Computing with C++
-Research Software Engineering with Python

Dissertation/report
All students undertake an independent research project (experimental or theoretical) related to quantum technologies, which culminates in a presentation and a dissertation of 10,000 words.

Teaching and learning
The programme is delivered through a combination of lectures and seminars, with self-study on two modules devoted to the critical assessment of current research topics and the corresponding research skills. Assessment is through a combination of problem sheets, written examinations, case study reports and presentations, as well as the MSc project dissertation.

Careers

The programme prepares graduates for careers in the emerging quantum technology industries which play an increasingly important role in: secure communication; sensing and metrology; the simulation of other quantum systems; and ultimately in general-purpose quantum computation. Graduates will also be well prepared for research at the highest level in the numerous groups now developing quantum technologies and for work in government laboratories.

Employability
Graduates will possess the skills needed to work in the emerging quantum industries as they develop in response to technological advances.

Why study this degree at UCL?

UCL offers one of the leading research programmes in quantum technologies anywhere in the world, as well as outstanding taught programmes in the subjects contributing to the field (including physics, computer science, and engineering). It also hosts the EPSRC Centre for Doctoral Training in Delivering Quantum Technologies.

The programme provides a rigorous grounding across the disciplines underlying quantum technologies, as well as the chance to work with some of the world's leading groups in research projects. The new Quantum Science and Technology Institute ('UCLQ') provides an umbrella where all those working in the field can meet and share ideas, including regular seminars, networking events and opportunities to interact with commercial and government partners.

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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. Read more
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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The course provides an introduction to the physical principles and mathematical techniques of current research in general relativity, quantum gravity, particle physics, quantum field theory, quantum information theory, cosmology and the early universe. Read more

Overview

The course provides an introduction to the physical principles and mathematical techniques of current research in general relativity, quantum gravity, particle physics, quantum field theory, quantum information theory, cosmology and the early universe.

The programme of study includes a taught component of closely-related modules in this popular area of mathematical physics. The course also includes a substantial project that will allow students to develop their interest and expertise in a specific topic at the frontier of current research, and develop their skills in writing a full scientific report.

The course will provide training in advanced methods in mathematics and physics which have applications in a wide variety of scientific careers and provide students with enhanced employability compared with undergraduate Bachelors degrees. In particular, it will provide training appropriate for students preparing to study for a PhD in the research areas listed above. For those currently in employment, the course will provide a route back to academic study.

Key facts:

- The course is taught jointly by the School of Mathematical Sciences and the School of Physics and Astronomy.

- Dissertation topics are chosen from among active research themes of the Particle Theory group, the Quantum Gravity group and the Quantum Information group.

- In addition to the lectures there are several related series of research-level seminars to which masters students are welcomed.

- The University of Nottingham is ranked in the top 1% of all universities worldwide.

Module details

Advanced Gravity

Black Holes

Differential Geometry

Gravity

Gravity, Particles and Fields Dissertation

Introduction to Quantum Information Science

Modern Cosmology

Quantum Field Theory

English language requirements for international students

IELTS: 6.0 (with no less than 5.5 in any element)

Further information



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his is a one year advanced taught course. The aim of this course is to bring students in twelve months to the frontier of elementary particle theory. Read more
his is a one year advanced taught course. The aim of this course is to bring students in twelve months to the frontier of elementary particle theory. This course is intended for students who have already obtained a good first degree in either physics or mathematics, including in the latter case courses in quantum mechanics and relativity.

The course consists of three modules: the first two are the Michaelmas and Epiphany graduate lecture courses, which are assessed by examinations in January and March. The third module is a dissertation on a topic of current research, prepared under the guidance of a supervisor with expertise in the area. We offer a wide variety of possible dissertation topics. The dissertation must be submitted by September 15th, the end of the twelve month course period.

Course Structure
The main group of lectures are given in the first two terms of the academic year (Michaelmas and Epiphany). This part of the lecture course is assessed by examinations. In each term there are two teaching periods of 4 weeks, with a week's break in the middle of the term in which students will be able to revise the material. most courses are either 8 lectures or 16 lectures in length. There are 14 lectures/week in the Michaelmas term and 14 lectures/week in Epiphany term.

Core Modules
- Introductory Field Theory
- Group Theory
- Standard Model
- General Relativity
- Quantum Electrodynamics
- Quantum Field Theory
- Conformal Field Theory
- Supersymmetry
- Anomalies
- Strong Interaction Physics
- Cosmology
- Superstrings and D-branes
- Non-Perturbative Physics
- Euclidean Field Theory
- Flavour Physics and Effective Field Theory
- Neutrinos and Astroparticle Physics
- 2d Quantum Field Theory
- Optional Modules
- Differential Geometry for Physicists
- Boundaries and Defects in Integrable Field Theory
- Computing for Physicists.

For further information on this course, please visit the Centre for Particle Theory website (http://www.cpt.dur.ac.uk/GraduateStudies)

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High-level training in statistics and the modelling of random processes for applications in science, business or health care. Read more
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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Revealing the ‘terra incognita’ between quantum mechanics and the classical world and inspiring new technologies. As a scientist, you’re a problem solver. Read more

Master's specialisation in Physics of Molecules and Materials

Revealing the ‘terra incognita’ between quantum mechanics and the classical world and inspiring new technologies.
As a scientist, you’re a problem solver. But how do you tackle a problem when there are no adequate theories and calculations become far too complicated? In the specialisation in Physics of Molecules and Materials you’ll be trained to take up this challenge in a field of physics that is still largely undiscovered: the interface between quantum and classical physics.
We focus on systems from two atoms to complete nanostructures, with time scales in the order of femtoseconds, picoseconds or nanoseconds. One of our challenges is to understand the origin of phenomena like superconductivity and magnetism. As theory and experiment reinforce each other, you’ll learn about both ‘research languages’. In this way, you’ll be able to understand complex problems by dividing them into manageable parts.

See the website http://www.ru.nl/masters/physicsandastronomy/physics

Why study Physics of Molecules and Materials at Radboud University?

- At Radboud University there’s a strong connection between theory and experiment. Theoretical and experimental physicists will teach you to become acquainted with both methods.
- In your internship(s), you’ll have the opportunity to work with unique research equipment, like free electron lasers and high magnetic fields, and with internationally known scientists.
- We collaborate with several industrial partners, such as Philips and NXP. This extensive network can help you find an internship or job that meets your interests.

If you’re successful in your internship, you have a good chance of obtaining a PhD position at the Institute for Molecules and Materials (IMM).

Admission requirements for international students

1. A completed Bachelor's degree in Physics
2. A proficiency in English
In order to take part in this programme, you need to have fluency in both written and spoken English. Non-native speakers of English* without a Dutch Bachelor's degree or VWO diploma need one of the following:
- A TOEFL score of >575 (paper based) or >232 (computer based) or >90 (internet based)
- An 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

This Master’s specialisation is an excellent preparation for a career in research, either at a university or at a company. However, many of our students end up in business as well. Whatever job you aspire, you can certainly make use of the fact that you have learned to:
- Solve complex problems
- Make accurate approximations
- Combine theory and experiments
- Work with numerical methods

Graduates have found jobs as for example:
- Consultant Billing at KPN
- Communications advisor at the Foundation for Fundamental Research on Matter (FOM)
- Systems analysis engineer at Thales
- Technical consultant at UL Transaction Security
- Business analyst at Capgemini

PhD positions

At Radboud University, we’re capable of offering many successful students in the field of Physics of Molecules and Materials a PhD position. Many of our students have already attained a PhD position, not just at Radboud University, but at universities all over the world.

Our approach to this field

In this specialisation, you’ll discover the interface between quantum mechanics and the classical world, which is still a ‘terra incognita’. We focus on two-atom systems, multi-atom systems, molecules and nanostructures. This is pioneering work, because these systems are often too complex for quantum calculations and too small for the application of classical theories.

- Theory and experiment
At Radboud University, we believe that the combination of theory and experiments is the best way to push the frontiers of our knowledge. Experiments provide new knowledge and data and sometimes also suggest a model for theoretical studies. The theoretical work leads to new theories, and creative ideas for further experiments. That’s why our leading theoretical physicists collaborate intensively with experimental material physicists at the Institute for Molecules and Materials (IMM). Together, they form the teaching staff of the Master’s specialisation in Physics of Molecules and Materials.

- Themes
This specialisation is focused on two main topics:
- Advanced spectroscopy
Spectroscopy is a technique to look at matter in many different ways. Here you’ll learn the physics behind several spectroscopic techniques, and learn how to design spectroscopic experiments. At Radboud University, you also have access to large experimental infrastructure, such as the High Magnetic field Laboratory (HFML), the FELIX facility for free electron lasers and the NMR laboratory.
- Condensed matter and molecular physics
You’ll dive into material science at the molecular level as well as the macroscopic level, on length scales from a single atom up to nanostructure and crystal. In several courses, you’ll get a solid background in both quantum mechanical and classical theories.

- Revolution
We’re not aiming at mere evolution of current techniques, we want to revolutionize them by developing fundamentally new concepts. Take data storage. The current data elements are near the limits of speed and data capacity. That’s why in the IMM we’re exploring a completely new way to store and process data, using light instead of electrical current. And this is but one example of how our research inspires future technology. As a Master’s student you can participate in this research or make breakthroughs in a field your interested in.

See the website http://www.ru.nl/masters/physicsandastronomy/physics

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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. Read more
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 >232 (computer 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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Joining the Department as a postgraduate is certainly a good move. The Department maintains strong research in both pure and applied mathematics, as well as the traditional core of a mathematics department. Read more
Joining the Department as a postgraduate is certainly a good move. The Department maintains strong research in both pure and applied mathematics, as well as the traditional core of a mathematics department. What makes our Department different is the equally strong research in fluid mechanics, scientific computation and statistics.

The quality of research at the postgraduate level is reflected in the scholarly achievements of faculty members, many of whom are recognized as leading authorities in their fields. Research programs often involve collaboration with scholars at an international level, especially in the European, North American and Chinese universities. Renowned academics also take part in the Department's regular colloquia and seminars. The faculty comprises several groups: Pure Mathematics, Applied Mathematics, Probability and Statistics.

Mathematics permeates almost every discipline of science and technology. We believe our comprehensive approach enables inspiring interaction among different faculty members and helps generate new mathematical tools to meet the scientific and technological challenges facing our fast-changing world.

The MPhil program seeks to strengthen students' general background in mathematics and mathematical sciences, and to expose students to the environment and scope of mathematical research. Submission and successful defense of a thesis based on original research are required.

Research Foci

Algebra and Number Theory
The theory of Lie groups, Lie algebras and their representations play an important role in many of the recent development in mathematics and in the interaction of mathematics with physics. Our research includes representation theory of reductive groups, Kac-Moody algebras, quantum groups, and conformal field theory. Number theory has a long and distinguished history, and the concepts and problems relating to the theory have been instrumental in the foundation of a large part of mathematics. Number theory has flourished in recent years, as made evident by the proof of Fermat's Last Theorem. Our research specializes in automorphic forms.

Analysis and Differential Equations
The analysis of real and complex functions plays a fundamental role in mathematics. This is a classical yet still vibrant subject that has a wide range of applications. Differential equations are used to describe many scientific, engineering and economic problems. The theoretical and numerical study of such equations is crucial in understanding and solving problems. Our research areas include complex analysis, exponential asymptotics, functional analysis, nonlinear equations and dynamical systems, and integrable systems.

Geometry and Topology
Geometry and topology provide an essential language describing all kinds of structures in Nature. The subject has been vastly enriched by close interaction with other mathematical fields and with fields of science such as physics, astronomy and mechanics. The result has led to great advances in the subject, as highlighted by the proof of the Poincaré conjecture. Active research areas in the Department include algebraic geometry, differential geometry, low-dimensional topology, equivariant topology, combinatorial topology, and geometrical structures in mathematical physics.

Numerical Analysis
The focus is on the development of advance algorithms and efficient computational schemes. Current research areas include: parallel algorithms, heterogeneous network computing, graph theory, image processing, computational fluid dynamics, singular problems, adaptive grid method, rarefied flow simulations.

Applied Sciences
The applications of mathematics to interdisciplinary science areas include: material science, multiscale modeling, mutliphase flows, evolutionary genetics, environmental science, numerical weather prediction, ocean and coastal modeling, astrophysics and space science.

Probability and Statistics
Statistics, the science of collecting, analyzing, interpreting, and presenting data, is an essential tool in a wide variety of academic disciplines as well as for business, government, medicine and industry. Our research is conducted in four categories. Time Series and Dependent Data: inference from nonstationarity, nonlinearity, long-memory behavior, and continuous time models. Resampling Methodology: block bootstrap, bootstrap for censored data, and Edgeworth and saddle point approximations. Stochastic Processes and Stochastic Analysis: filtering, diffusion and Markov processes, and stochastic approximation and control. Survival Analysis: survival function and errors in variables for general linear models. Probability current research includes limit theory.

Financial Mathematics
This is one of the fastest growing research fields in applied mathematics. International banking and financial firms around the globe are hiring science PhDs who can use advanced analytical and numerical techniques to price financial derivatives and manage portfolio risks. The trend has been accelerating in recent years on numerous fronts, driven both by substantial theoretical advances as well as by a practical need in the industry to develop effective methods to price and hedge increasingly complex financial instruments. Current research areas include pricing models for exotic options, the development of pricing algorithms for complex financial derivatives, credit derivatives, risk management, stochastic analysis of interest rates and related models.

Facilities

The Department enjoys a range of up-to-date facilities and equipment for teaching and research purposes. It has two computer laboratories and a Math Support Center equipped with 100 desktop computers for undergraduate and postgraduate students. The Department also provides an electronic homework system and a storage cloud system to enhance teaching and learning.

To assist computations that require a large amount of processing power in the research area of scientific computation, a High Performance Computing (HPC) laboratory equipped with more than 200 high-speed workstations and servers has been set up. With advanced parallel computing technologies, these powerful computers are capable of delivering 17.2 TFLOPS processing power to solve computationally intensive problems in our innovative research projects. Such equipment helps our faculty and postgraduate students to stay at the forefront of their fields. Research projects in areas such as astrophysics, computational fluid dynamics, financial mathematics, mathematical modeling and simulation in materials science, molecular simulation, numerical ocean modeling, numerical weather prediction and numerical methods for micromagnetics simulations all benefit from our powerful computing facilities.

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Theoretical physics is an international and highly competitive field. For several decades, Utrecht University's Institute for Theoretical Physics has been on the forefront of research in this area. Read more

Theoretical Physics

Theoretical physics is an international and highly competitive field. For several decades, Utrecht University's Institute for Theoretical Physics has been on the forefront of research in this area.

This programme serves as a gateway to understanding the fascinating world of physics, ranging from the unimaginably small scales of elementary particles to the vast dimensions of our universe.

The central goal of the Theoretical Physics programme is to obtain a detailed understanding of the collective behaviour of many particle systems from a fully microscopic point of view. In most physical systems, microscopic details determine the properties observed. Our condensed matter theorists and statistical physicists develop and apply methods for explaining and predicting these connections.

Examples include density functional theory, renormalisation-group theory and the scaling theory of critical phenomena. Dynamical properties are studied using such methods as kinetic theory and the theory of stochastic processes. These theories can be quantum mechanical, including theories of the quantum Hall effect, superconductivity, Bose-Einstein condensation, quantum magnetism and quantum computing. More classical are relationships between chaos and transport, nucleation phenomena, polymer dynamics and phase structure and dynamics of colloids.

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This renowned MSc course is designed to prepare students for PhD study in fundamental theoretical physics by bridging the gap between an undergraduate course in physics or mathematics and the research frontier. Read more
This renowned MSc course is designed to prepare students for PhD study in fundamental theoretical physics by bridging the gap between an undergraduate course in physics or mathematics and the research frontier.

The Theoretical Physics Group is internationally recognised for its contribution to our understanding of the unification of fundamental forces, the early universe, quantum gravity, supersymmetry, string theory, and quantum field theory.

The origins of the MSc course date back to the founding of the Theoretical Physics Group by Abdus Salam, one of Imperial’s Nobel Laureates.

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