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Masters Degrees (Geometry)

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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. … Read more

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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Working at a frontier of mathematics that intersects with cutting edge research in physics. Mathematicians can benefit from discoveries in physics and conversely mathematics is essential to further excel in the field of physics. Read more

Working at a frontier of mathematics that intersects with cutting edge research in physics.

Mathematicians can benefit from discoveries in physics and conversely mathematics is essential to further excel in the field of physics. History shows us as much. Mathematical physics began with Christiaan Huygens, who is honoured at Radboud University by naming the main building of the Faculty of Science after him. By combining Euclidean geometry and preliminary versions of calculus, he brought major advances to these areas of mathematics as well as to mechanics and optics. The second and greatest mathematical physicist in history, Isaac Newton, invented both the calculus and what we now call Newtonian mechanics and, from his law of gravity, was the first to understand planetary motion on a mathematical basis.

Of course, in the Master’s specialisation in Mathematical Physics we look at modern mathematical physics. The specialisation combines expertise in areas like functional analysis, geometry, and representation theory with research in, for example, quantum physics and integrable systems. You’ll learn how the field is far more than creating mathematics in the service of physicists. It’s also about being inspired by physical phenomena and delving into pure mathematics.

At Radboud University, we have such faith in a multidisciplinary approach between these fields that we created a joint research institute: Institute for Mathematics, Astrophysics and Particle Physics (IMAPP). This unique collaboration has lead to exciting new insights into, for example, quantum gravity and noncommutative geometry. Students thinking of enrolling in this specialisation should be excellent mathematicians as well as have a true passion for physics.

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

Why study Mathematical Physics at Radboud University?

- This specialisation is one of the few Master’s in the world that lies in the heart of where mathematics and physics intersect and that examines their cross-fertilization.

- You’ll benefit from the closely related Mathematics Master’s specialisations at Radboud University in Algebra and Topology (and, if you like, also from the one in Applied Stochastics).

- Teaching takes place in a stimulating, collegial setting with small groups. This ensures that at Radboud University you’ll get plenty of one-on-one time with your thesis supervisor.

- You partake in the Mastermath programme, meaning you can follow the best mathematics courses, regardless of the university in the Netherlands that offers them. It also allows you to interact with fellow mathematic students all over the country.

- As a Master’s student you’ll get the opportunity to work closely with the mathematicians and physicists of the entire IMAPP research institute.

- More than 85% of our graduates find a job or a gain a PhD position within a few months of graduating. About half of our PhD’s continue their academic careers.

Career prospects

Mathematicians are needed in all industries, including the industrial, banking, technology and service industry and also within management, consultancy and education. 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 indeed and is 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 Mathematical Physics Department, emphasise operator algebras and noncommutative geometry, Lie theory and representation theory, integrable systems, and quantum field theory. Below, a small sample of the research our members pursue.

Gert Heckman's research concerns algebraic geometry, group theory and symplectic geometry. His work in algebraic geometry and group theory concerns the study of particular ball quotients for complex hyperbolic reflection groups. Basic questions are an interpretation of these ball quotients as images of period maps on certain algebraic geometric moduli spaces. Partial steps have been taken towards a conjecture of Daniel Allcock, linking these ball quotients to certain finite almost simple groups, some even sporadic like the bimonster group.

Erik Koelink's research is focused on the theory of quantum groups, especially at the level of operator algebras, its representation theory and its connections with special functions and integrable systems. Many aspects of the representation theory of quantum groups are motivated by related questions and problems of a group representation theoretical nature.

Klaas Landsman's previous research programme in noncommutative geometry, groupoids, quantisation theory, and the foundations of quantum mechanics (supported from 2002-2008 by a Pioneer grant from NWO), led to two major new research lines:

1. The use of topos theory in clarifying the logical structure of quantum theory, with potential applications to quantum computation as well as to foundational questions.

2. Emergence with applications to the Higgs mechanism and to Schroedinger's Cat (aka as the measurement problem). A first paper in this direction with third year Honours student Robin Reuvers (2013) generated worldwide attention and led to a new collaboration with experimental physicists Andrew Briggs and Andrew Steane at Oxford and philosopher Hans Halvorson at Princeton.

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

Radboud University Master's Open Day 10 March 2018



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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 master’s programme Mathematics focuses on analysis and number theory. From applied to fundamental research, and from algebra to data science, our master’s programme spans these fields entirely. Read more

The master’s programme Mathematics focuses on analysis and number theory. From applied to fundamental research, and from algebra to data science, our master’s programme spans these fields entirely.

What does this master’s programme entail?

The two-year master's programme Mathematics has two components: an analysis-oriented component with topics such as dynamical systems, differential equations, probability theory and stochastics, percolation and mathematics in the life sciences, and an algebra/geometry-oriented component with topics such as algebraic number theory, algebraic geometry, algebraic topology and cryptology. The goal of each programme is to train the student as an independent researcher, and to develop the necessary skills and proficiency to advance your career.

Read more about our Mathematics programme.

Why study Mathematics at Leiden University?

  • Your study programme can be fine-tuned to your own mathematical interests, both pure and applied.
  • You will be educated by renowned researchers like Spinoza prize winner Aad van der Vaart and Hendrik Lenstra and receive a top level education in Mathematics.
  • The institute has an extensive international network which allows you to broaden your horizon and provide you with ample opportunities to join interdisciplinary seminars and pursue interdisciplinary research projects.

Find more reasons to choose Mathematics at Leiden University.

Mathematics: the right master’s programme for you?

The master’s programme in Mathematics in Leiden focuses on analysis, probability and statistics, number theory and (arithmetic) geometry. If you are looking for an opportunity to specialize in one of these areas, Leiden is an excellent possibility. Students who have obtained a Master of Science degree in Mathematics possess a thorough theoretical basis, know how to work in a multinational environment, and are able to operate well on the international market.

Read more about the entry requirements for Mathematics.

Specialisations



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EXPLORE THE INTERSECTION OF FUNDAMENTAL AND APPLIED RESEARCH. In our Master’s programme in. Mathematical Sciences. , you’ll engage in the core activities of mathematics while discovering new patterns and relationships and learning to construct models with predictive power. Read more

EXPLORE THE INTERSECTION OF FUNDAMENTAL AND APPLIED RESEARCH

In our Master’s programme in Mathematical Sciences, you’ll engage in the core activities of mathematics while discovering new patterns and relationships and learning to construct models with predictive power. You’ll also study the science of structures within the different branches of the physical sciences. Our institute of Mathematical Sciences offers you a versatile, internationalised learning environment at the intersection of fundamental and applied research

PERSONALIZING YOUR MASTER'S

You can tailor your programme by selecting one of the following 8 tracks:

  • Algebraic Geometry and Number Theory
  • Applied Analysis
  • Complex Systems
  • Differential Geometry and Topology
  • Logic
  • Probability, Statistics, and Stochastic Modelling
  • Pure Analysis
  • Scientific Computing

You can also choose to do a Research project in History of Mathematics. Choosing a track will allow you to pursue your Master's degree through either a theoretical or more applied focus.

Honours programmes

You have the opportunity to combine the Mathematical Sciences Master's programme with other Master's programmes within the Graduate School of Natural Sciences, such as Theoretical Physics. If you have an interest in geometry and plan to do a PhD in this field, you can apply for the selective honours programma of the Utrecht Geometry Centre.

PROGRAMME OBJECTIVE

Our Mathematical Sciences MSc degree programme will prepare you for a challenging career in academic research or as a professional mathematician.



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The fields of graphics, vision and imaging increasingly rely on one another. Read more

The fields of graphics, vision and imaging increasingly rely on one another. This unique and timely MSc provides training in computer graphics, geometry processing, virtual reality, machine vision and imaging technology from world-leading experts, enabling students to specialise in any of these areas and gain a grounding in the others.

About this degree

Graduates will understand the basic mathematical principles underlying the development and application of new techniques in computer graphics and computer vision and will be aware of the range of algorithms and approaches available, and be able to design, develop and evaluate algorithms and methods for new problems, emerging technologies and applications.

Students undertake modules to the value of 180 credits.

The programme consists of four core modules (60 credits), four optional modules (60 credits) and a research project (60 credits).

Core modules

  • Computer Graphics (15 credits)
  • Image Processing (15 credits)
  • Mathematical Methods, Algorithmics and Implementations (15 credits)
  • Research Methods and Reading (15 credits)

Optional modules

Students must choose a minimum of 15 and a maximum of 30 credits from Group One options. Students must choose a minimum of 30 and a maximum of 45 credits from Group Two options.

Group One Options (15 to 30 credits)

  • Machine Vision (15 credits)
  • Virtual Environments (15 credits)

Group Two Options (30 to 45 credits)

  • Acquisition and Processing of 3D Geometry (15 credits)
  • Computational Modelling for Biomedical Imaging (15 credits)
  • Computational Photography and Capture (15 credits)
  • Geometry of Images (15 credits)
  • Graphical Models (15 credits)
  • Information Processing in Medical Imaging (15 credits)
  • Introduction to Machine Learning (15 credits)
  • Inverse Problems in Imaging (15 credits)
  • Numerical Optimisation (15 credits)
  • Robotic Sensing, Manipulation and Interaction (15 credits)
  • Robotic Vision and Navigation (15 credits)

Please note: the availability and delivery of optional modules may vary, depending on your selection.

Dissertation/report

All students undertake an independent research project related to a problem of industrial interest or on a topic near the leading edge of research, which culminates in a 60–80 page dissertation.

Teaching and learning

The programme is delivered through a combination of lectures and tutorials. Lectures are often supported by laboratory work with help from demonstrators. Student performance is assessed by unseen written examinations, coursework and a substantial individual project.

Further information on modules and degree structure is available on the department website: Computer Graphics, Vision and Imaging MSc

Careers

Graduates are ready for employment in a wide range of high-technology companies and will be able to contribute to maintaining and enhancing the UK's position in these important and expanding areas. The MSc provides graduates with the up-to-date technical skills required to support a wealth of research and development opportunities in broad areas of computer science and engineering, such as multimedia applications, medicine, architecture, film animation and computer games. Our market research shows that the leading companies in these areas demand the deep technical knowledge that this programme provides. Graduates have found positions at global companies such as Disney, Sony and Siemens. Others have gone on to PhD programmes at leading universities worldwide.

Recent career destinations for this degree

  • Business Analyst, Adobe
  • Software Engineer, FactSet Research Systems
  • MRes in Engineering, Imperial College London
  • Software Engineer, Sengtian Software
  • PhD in Computer Graphics, UCL

Employability

UCL received the highest percentage (96%) for quality of research in Computer Science and Informatics in the UK's most recent Research Excellence Framework (REF2014).

Our graduates have some of the highest employment rates of any university in the UK. This degree programme also provides a foundation for further PhD study or industrial research.

Careers data is taken from the ‘Destinations of Leavers from Higher Education’ survey undertaken by HESA looking at the destinations of UK and EU students in the 2013–2015 graduating cohorts six months after graduation.

Why study this degree at UCL?

UCL Computer Science contains some of the world's leading researchers in computer graphics, geometry processing, computer vision and virtual environments.

Research activities include geometric acquisition and 3D fabrication, real-time photo-realistic rendering, mixed and augmented reality, face recognition, content-based image-database search, video-texture modelling, depth perception in stereo vision, colour imaging for industrial inspection, mapping brain function and connectivity and tracking for SLAM (simultaneous localisation and mapping).

Research Excellence Framework (REF)

The Research Excellence Framework, or REF, is the system for assessing the quality of research in UK higher education institutions. The 2014 REF was carried out by the UK's higher education funding bodies, and the results used to allocate research funding from 2015/16.

The following REF score was awarded to the department: Computer Science

96% rated 4* (‘world-leading’) or 3* (‘internationally excellent’)

Learn more about the scope of UCL's research, and browse case studies, on our Research Impact website.



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The Master of Science in Mathematics (120 ECTS) is a research-based master’s programme in which you can specialize in the following fields of mathematics. Read more

The Master of Science in Mathematics (120 ECTS) is a research-based master’s programme in which you can specialize in the following fields of mathematics: Pure Mathematics: Algebra, Analysis and Geometry; and Applied Mathematics: Statistics, Financial Mathematics, Computational Mathematics, Plasma-Astrophysics. 

What is the Master of Mathematics all about?

Besides a solid, all-round education in mathematics, the programme offers you the possibility to focus on either pure or applied mathematics. This allows you to acquire both breadth of knowledge and depth in your own areas of interest. Pure and applied mathematics courses are firmly grounded in the core research activities of the Department of Mathematics. Gradually, you will gain experience and autonomy in learning how to cope with new concepts, higher levels of abstraction, new techniques, new applications, and new results. This culminates in the Master’s thesis, where you become actively involved in the research performed in the various mathematical research groups of the Departments of Mathematics, Physics, Astronomy and Computer Sciences.

 This is an initial Master's programme and can be followed on a full-time or part-time basis.

Structure

The programme of the Master of Science in Mathematics consists of 120 ECTS. You choose one of the two profiles – Pure Mathematics or Applied Mathematics (54 ECTS) – and one of the two options – Research Option or Professional Option (30 ECTS). The profile allows you to specialize either in pure mathematics (algebra, geometry, analysis), or in applied mathematics (statistics, computational mathematics, fluid dynamics).

There is one common course: ‘Mathematics of the 21st Century’ (6 ECTS). To complete the programme, you carry out a research project that results in a master’s thesis (30 ECTS).

Department

All staff members of the Department of Mathematics are actively involved in the two-year Master of Science in Mathematics programme. The academic staff at the Department of Mathematics consists of leading experts in their fields. Researchers in pure mathematics focus on algebraic geometry, group theory, differential geometry, functional analysis, and complex analysis. Researchers in mathematical statistics deal with extreme values, robust statistics, non-parametric statistics, and financial mathematics. Research in the applied mathematics group is in computational fluid dynamics and plasma-astrophysics.

Career perspectives

Mathematicians find employment in industry and in the banking, insurance, and IT sectors. Many graduates from the research option pursue a career in research and start a PhD in mathematics, mathematical physics, astrophysics, engineering, or related fields.



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Mathematics is at the heart of advances in science, engineering and technology, as well as being an indispensable problem-solving and decision-making tool in many other areas of life. Read more
Mathematics is at the heart of advances in science, engineering and technology, as well as being an indispensable problem-solving and decision-making tool in many other areas of life. This MSc course enables you to delve deeply into particular aspects of pure and applied mathematics, through a wide choice of modules in fascinating areas such as fractal geometry, coding theory and analytic theory. You’ll complete your MSc with a piece of independent study, exploring the history of modern geometry, advances in approximation theory, variational methods applied to eigenvalue problems, or algebraic graph theory and culminating in a dissertation on the topic of your choice.

Key features of the course

•Ideal for mathematically inclined scientists and engineers as well as mathematicians
•Extends your knowledge and refines your abilities to process information accurately, and critically analyse and communicate complex ideas
•Develops an enhanced skill set that will put you at an advantage in careers as diverse as mathematics, education, computer science, economics, engineering and finance.
•The most popular MSc in mathematics in the UK.
This qualification is eligible for a Postgraduate Loan available from Student Finance England. For more information, see Fees and funding

Course details

You can take a number of different routes towards your qualification - see the full module list for all options.

Modules

The modules in this qualification are categorised as entry, intermediate and dissertation. Check our website for start dates as some modules are not available for study every year.

Entry:

• Calculus of variations and advanced calculus (M820)
• Analytic number theory I (M823)

Intermediate:

• Nonlinear ordinary differential equations (M821)
• Applied complex variables (M828) - next available in October 2017 and following alternate years
• Analytic number theory II (M829) - next available in October 2018 and following alternate years
• Approximation theory (M832) - next available in October 2018 and following alternate years
• Advanced mathematical methods (M833) - next available in October 2017 and following alternate years
• Fractal geometry (M835) - next available in October 2017 and following alternate years
• Coding theory (M836) - next available in October 2018 and following alternate years
• Dissertation: Dissertation in mathematics (M840)

Module study order:

•You must normally pass at least one entry level module before studying an intermediate module.
•You must pass Analytic number theory I (M823) before studying Analytic number theory II (M829).
•You must normally pass four modules before studying the Dissertation in mathematics (M840).
•Some topics for the dissertation have prerequisite modules

Otherwise within each category modules may be studied in any order, and you may register for a module while studying a pre-requisite for that module (i.e. before you know whether you have passed the pre-requisite module or not).

To gain this qualification, you need 180 credits as follows:

150 credits from this list:

Optional modules

• Advanced mathematical methods (M833)
• Analytic number theory I (M823)
• Analytic number theory II (M829)
• Applied complex variables (M828)
• Approximation theory (M832)
• Calculus of variations and advanced calculus (M820)
• Coding theory (M836)
• Fractal geometry (M835)
• Nonlinear ordinary differential equations (M821)

Plus

Compulsory module

Dissertation in mathematics (M840)

The modules quoted in this description are currently available for study. However, as we review the curriculum on a regular basis, the exact selection may change over time.

Credit transfer

For this qualification, we do not allow you to count credit for study you have already done elsewhere.

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The Masters in Mathematics/Applied Mathematics offers courses, taught by experts, across a wide range. Mathematics is highly developed yet continually growing, providing new insights and applications. Read more

The Masters in Mathematics/Applied Mathematics offers courses, taught by experts, across a wide range. Mathematics is highly developed yet continually growing, providing new insights and applications. It is the medium for expressing knowledge about many physical phenomena and is concerned with patterns, systems, and structures unrestricted by any specific application, but also allows for applications across many disciplines.

Why this programme

  • Mathematics at the University of Glasgow is ranked 3rd in Scotland (Complete University Guide 2017).
  • The School has a strong international reputation in pure and applied mathematics research and our PGT programmes in Mathematics offer a large range of courses ranging from pure algebra and analysis to courses on mathematical biology and fluids.
  • You will be taught by experts across a wide range of pure and applied mathematics and you will develop a mature understanding of fundamental theories and analytical skills applicable to many situations.
  • You will participate in an extensive and varied seminar programme, are taught by internationally renowned lecturers and experience a wide variety of projects.
  • Our students graduate with a varied skill set, including core professional skills, and a portfolio of substantive applied and practical work.

Programme structure

Modes of delivery of the Masters in Mathematics/Applied Mathematics include lectures, laboratory classes, seminars and tutorials and allow students the opportunity to take part in project work.

If you are studying for the MSc you will take a total of 120 credits from a mixture of Level-4 Honours courses, Level-M courses and courses delivered by the Scottish Mathematical Sciences Training Centre (SMSTC).

You will take courses worth a minimum of 90 credits from Level-M courses and those delivered by the SMSTC. The remaining 30 credits may be chosen from final-year Level-H courses. The Level-M courses offered in a particular session will depend on student demand. Below are courses currently offered at these levels, but the options may vary from year to year.

Level-H courses (10 or 20 credits)

  • Algebraic & geometric topology
  • Continuum mechanics & elasticity
  • Differential geometry
  • Fluid mechanics
  • Functional analysis
  • Further complex analysis
  • Galois theory
  • Mathematical biology
  • Mathematical physics
  • Numerical methods
  • Number theory
  • Partial differential equations
  • Topics in algebra.

Level-M courses (20 credits)

  • Advanced algebraic & geometric topology
  • Advanced differential geometry & topology
  • Advanced functional analysis
  • Advanced methods in differential equations
  • Advanced numerical methods
  • Biological & physiological fluid mechanics
  • Commutative algebra & algebraic geometry
  • Elasticity
  • Further topics in group theory
  • Lie groups, lie algebras & their representations
  • Magnetohydrodynamics
  • Operator algebras 
  • Solitons
  • Special relativity & classical field theory.

SMSTC courses (20 credits)

  • Advanced Functional Analysis
  • Advanced Mathematical Methods

The project titles are offered each year by academic staff and so change annually.

Career prospects

Career opportunities are diverse and varied and include academia, teaching, industry and finance.

Graduates of this programme have gone on to positions such as:

Maths Tutor at a university.



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This course aims to bring you, in 12 months, to a position where you can embark with confidence on a wide range of careers, including taking a PhD in Mathematics or related disciplines. Read more

This course aims to bring you, in 12 months, to a position where you can embark with confidence on a wide range of careers, including taking a PhD in Mathematics or related disciplines. There is a wide range of taught modules on offer, and you will also produce a dissertation on a topic of current research interest taken from your choice of a wide range of subjects offered.

Course structure and overview

  • Six taught modules in October-May
  • A dissertation in June-September.

Modules: Six of available options

In previous years, optional modules available included:

Modules in Pure Mathematics:

  •  Algebraic Topology IV
  •  Analysis III and IV
  •  Codes and Cryptography III
  •  Differential Geometry III
  •  Galois Theory III
  •  Representation Theory III and IV
  •  Riemannian Geometry IV
  •  Topology III
  •  Topics in Algebra and Geometry IV

Modules in Probability and Statistics:

  •  Bayesian Statistics III and IV 
  •  Mathematical Finance III and IV
  •  Decision Theory III
  •  Operations Research III
  •  Statistical Methods III
  • Stochastic Processes III and IV

Modules in Applications of Mathematics:

  •  Advanced Quantum Theory IV
  •  Continuum Mechanics III and IV
  •  Dynamical Systems III
  •  General Relativity IV
  •  Mathematical Biology III 
  •  Partial Differential Equations III and IV
  •  Quantum Information III
  •  Quantum Mechanics III
  •  Solitons III and IV

Course 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 give the students a wide mathematical background allowing them to either proceed to PhD or to apply the gained knowledge in industry.

The course consists of three modules: the first two are the Michaelmas and Epiphany lecture courses covering variety of topics in pure and applied mathematics and statistics. 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 main group of lectures is given in the first two terms of the academic year (Michaelmas and Epiphany), there are also two revision lectures in the third term (Easter). This part of the course is assessed by examinations. Students choose 6 modules, each module has 2 lectures per week and one fortnightly problems class. There are 10 teaching weeks in the Michaelmas term and 9 teaching weeks in Epiphany term. In addition lecturers also set a number of homework assignments which give the student a chance to test their understanding of the material.

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



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This course is for people who need to strengthen their maths knowledge fairly quickly before starting their training to become a secondary mathematics teacher. Read more
This course is for people who need to strengthen their maths knowledge fairly quickly before starting their training to become a secondary mathematics teacher.

Course overview

This 24-week fast track course will boost your maths knowledge so that you are able to teach maths up to A Level. To join the Subject Knowledge Enhancement (SKE) course, you should already have received a conditional offer for initial teacher training – or you should at least be considering an application for it.

Typically, you should have already received a conditional offer for initial teacher training (ITT) – or you should at least be considering an application for it. Our course ends in July so that you are ready for ITT in September.

For Home/EU students, there is no tuition fee and you will be paid a bursary while you are studying on the course.

You will cover areas such as geometry, calculus, statistics, mechanics, patterns and equations as well as common errors and misconceptions in mathematics. At the initial interview, we will advise you whether this 24-week fast track course is suitable for you or whether an alternative course would be better for your circumstances.

Once you have successfully completed the SKE course and your initial teacher training, your career prospects will be very good. There is a national shortage of mathematics teachers and there are excellent job prospects, both in the North East and throughout the country.

Reports on our provision by external examiners have been very positive. Comments include: “It is clear that the course remains very successful in preparing candidates appropriately for their subsequent teacher training course”.

Course content

This course enhances your skills and knowledge in mathematics, preparing you for teacher training. You will study the following units:
-Patterns and Equations
-Structure and Pattern
-Development of Geometric Thinking
-Nature of Mathematics
-Investigating Calculus
-Enriching and Strengthening Mathematics
-Elementary Geometry
-Development of Mathematics
-Statistics
-Mechanics
-Using LOGO and 3D Geometry
-Errors and Misconceptions in Mathematics
-Matrices
-Investigating Sequences and Series

Teaching and assessment

We use a wide variety of teaching and learning methods that include lectures, independent work, directed tasks, written assignments and use of ICT. In addition to attending taught sessions, we encourage you to undertake some voluntary work experience in a school. Assessment methods include written work, exams and presentations. We assess all units.

Facilities & location

This course is based on the banks of the River Wear at The Sir Tom Cowie Campus at St Peter’s. Interactive whiteboards are available in our classrooms and we encourage you to use mathematical software, such as Autograph graph-plotting software, to support your learning.

When it comes to IT provision you can take your pick from over a hundred PCs in the St Peter’s Library, a dedicated computer classroom, and wireless access zones. If you have any problems, just ask the friendly helpdesk team.

The University of Sunderland is a vibrant learning environment with a strong international dimension thanks to the presence of students from around the world.

Employment & careers

This course enhances your maths subject knowledge, allowing you to progress on to a teacher training programme such as the University of Sunderland’s PGCE Mathematics Secondary Education and then achieve Qualified Teacher Status.

There is a shortage of mathematics teachers in England so there are excellent career opportunities both in the North East and nationwide.

The starting salary of a Newly Qualified Teacher is over £22,000, with extra if you work in London. Teachers can expect to see their salaries rise by an average of 30 per cent after their first four years in the job.

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About the programme. In addition to being a science in its own right, mathematics plays a fundamental role in the quantitative areas of practically all other academic disciplines, particularly in the natural sciences, engineering, business administration, economics, medicine and psychology. Read more

About the programme

In addition to being a science in its own right, mathematics plays a fundamental role in the quantitative areas of practically all other academic disciplines, particularly in the natural sciences, engineering, business administration, economics, medicine and psychology. Mathematical results permeate nearly all facets of life and are a necessary prerequisite for the vast majority of modern technologies – and as our IT systems become increasingly powerful, we are able to mathematically handle enormous amounts of data and solve ever more complex problems.

Special emphasis is placed on developing students' ability to formalise given problems in a way that facilitates algorithmic processing as well as enabling them to choose or develop, and subsequently apply, suitable algorithms to solve problems in an appropriate manner. The degree programme is theoretical in its orientation, with strongly application-oriented components. Studying this programme, you can gain advanced knowledge in the mathematical areas of Cryptography, Computer Algebra, Algorithmic Algebra and Geometry, Image and Signals Processing, Statistics and Stochastic Simulation, Dynamical Systems and Control Theory as well as expert knowledge in Computer Science fields such as Data Management, Machine Learning and Data Mining.

Furthermore, you will have the chance to learn how to apply your knowledge to tackle problems in areas as diverse as Marketing, Predictive Analytics, Computational Finance, Digital Humanities, IT Security and Robotics.

Programme syllabus

The core modules consist of two mathematics seminars and the presentation of your master's thesis.The compulsory elective modules are divided into eight module groups:

1)   Algebra, Geometry and Cryptography

This module group imparts advanced results in the areas of algebra and geometry, which constitute the fundament for algorithmic calculations, particularly in cryptography but also in many other mathematical areas.

2)   Mathematical Logic and Discrete Mathematics

The theoretical possibilities and limitations of algorithm-based solutions are treated in this module group.

3)   Analysis, Numerics and Approximation Theory

Methods from the fields of mathematical analysis, applied harmonic analysis and approximation theory for modelling and approximating continuous and discrete data and systems as well as efficient numerical implementation and evaluation of these methods are the scope of this module group.

4) Dynamical Systems and Optimisation

Dynamical systems theory deals with the description of change over time. This module group is concerned with methods used for the modelling, analysis, optimisation and design of dynamical systems, as well as the numerical implementation of such techniques.

5) Stochastics, Statistics

This module group deals with methods for modelling and analysing complex random phenomena as well as the construction, analysis and optimisation of stochastic algorithms and techniques used in statistical data analysis.

6) Data Analysis and Data Management and Programming

This module group examines the core methods used in computer science for the analysis of data of heterogeneous modalities (e.g. multimedia data, social networks and sensor data) and for the realisation of data analysis systems.

7) Applications

In this module group, you will practise applying the mathematical methods learned in module groups 1 to 6 to real-world applications such as Marketing, Predictive Analytics and Computational Finance.

8) Key Competencies and Language Training

In this module group, you will choose seminars that develop your non-subject-specific skills, such as public speaking and academic writing and other soft skills; you may also undertake internships. This serves to complement your technical expertise gained during your degree studies and helps to prepare you for your professional life after university.



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Master's specialisation in Algebra and Topology. The Algebra and Topology section is an active research group consisting of renowned experts covering a remarkably broad range of topics. Read more

Master's specialisation in Algebra and Topology

The Algebra and Topology section is an active research group consisting of renowned experts covering a remarkably broad range of topics. The group consists of two full professors (I. Moerdijk, Spinoza Laureate 2012, and B. Moonen), four permanent members, and a large number of post-docs and PhD students. More information about the research activities of the group can be found at http://www.math.ru.nl/topology.

The section offers a Master's specialisation in Algebra in Topology, which is a 2-year programme aimed at students with an interest in pure mathematics and its applications.

The Master's programme has a strong focus on current research developments. It introduces students to a broad range of techniques and concepts that play a central role in modern mathematics. In addition to providing a strong theoretical basis, the programme offers excellent opportunities for a further specialisation focusing on applications of pure mathematics or on interactions with other fields.

The programme offers courses in Algebra, Topology, Geometry, Number Theory, and Logic and Computation. There are strong interactions with other Master's specialisations at Radboud University, notably the ones in Mathematical Physics and in Mathematical Foundations of Computer Science.

In addition, the programme offers a variety of seminars from beginning Master's level to research level. Moreover, students have the possibility to incorporate courses from related programmes (e.g. Mathematical Physics and Mathematical Foundations of Computer Science into their programme, as well as individual reading courses. Each student concludes his programme by studying a special topic and writing a Master's thesis about it.

Excellent students having completed this Master's programme or a similar programme elsewhere can in principle continue and enrol in the PhD Programme, but admission for this is limited and highly selective.

See the website http://www.ru.nl/masters/algebratopology

Admission requirements for international students

1. A completed Bachelor's degree in Mathematics or related area

Entering the Master’s programme in Mathematics requires a Bachelor’s degree in Mathematics that is the equivalent to a Dutch university diploma (this does not include a Bachelor’s from a university of applied science, in Dutch hbo; in German Fachhochschule). That means we expect you to have a solid background in the core areas groups, rings, fields and topology. We expect students to have passed core mathematics courses during their Bachelor’s in:

The Examination Board will determine if an international student has the required mathematical knowledge to be admitted. The Examination Board will also indicate if the student is required to follow specific courses from the Bachelor's programme to eliminate possible deficiencies.

- Basic notions in Mathematics

- Linear Algebra

- Algebra

- Analysis

- Topology

- Geometry

- Differential Equations

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:

- TOEFL score of ≥575 (paper 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

Mathematicians are needed in all industries, including the banking, technology and service industries, amongst many others. 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 specialised 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 not seem initially relevant. This makes your job opportunities very broad and is 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.

See the website http://www.ru.nl/masters/algebratopology

Radboud University Master's Open Day 10 March 2018



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Applied Mathematical Sciences offers a clear and relevant gateway into a successful career in business, education or scientific research. Read more
Applied Mathematical Sciences offers a clear and relevant gateway into a successful career in business, education or scientific research. The programme arms students with the essential knowledge required by all professional mathematicians working across many disciplines. You will learn to communicate their ideas effectively to peers and others, as well as the importance of research, planning and self-motivation.

Students will take a total of 8 courses, 4 in each of the 1st and 2nd Semesters followed by a 3-month Project in the summer. A typical distribution for this programme is as follows:

Core courses

:

Modelling and Tools;
Optimization;
Dynamical Systems;
Applied Mathematics (recommended);
Applied Linear Algebra (recommended).

Optional Courses

:

Mathematical Ecology;
Functional Analysis;
Numerical Analysis of ODEs;
Pure Mathematics;
Statistical Methods;
Stochastic Simulation;
Software Engineering Foundations;
Mathematical Biology and Medicine;
Partial Differential Equations;
Numerical Analysis;
Geometry.

Typical project subjects

:

Pattern Formation of Whole Ecosystems;
Climate Change Impact;
Modelling Invasive Tumour Growth;
Simulation of Granular Flow and Growing Sandpiles;
Finite Element Discretisation of ODEs and PDEs;
Domain Decomposition;
Mathematical Modelling of Crime;
The Geometry of Point Particles;
Can we Trust Eigenvalues on a Computer?

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You can study this Mathematical Sciences MSc programme full-time or part-time. It offers students the opportunity to specialise in a broad range of areas across pure and applied mathematics, statistics and probability, and theoretical physics. Read more
You can study this Mathematical Sciences MSc programme full-time or part-time. It offers students the opportunity to specialise in a broad range of areas across pure and applied mathematics, statistics and probability, and theoretical physics.

The topics we cover include:

- advanced probability theory
- algebra
- asymptotic methods
- geometry
- mathematical biology
- partial differential equations
- quantum field theory
- singularity theory
- stochastic analysis
- standard model/string theory.

By completing the first semester you qualify for the PG certificate. By completing the second, you qualify for the PG Diploma. Then, by completing your dissertation, you qualify for the MSc.

Key Facts

REF 2014
92% of our research impact judged at outstanding and very considerable, 28% improvement in overall research at 4* and 3*.

Facilities
A dedicated student resource suite is available in the Department, with computer and reading rooms and a social area.

Why Department of Mathematical Sciences?

Range and depth of study options

We offer a very wide range of modules, from advanced algebra and geometry, to partial differential equations, probability theory, stochastic analysis, and mathematical physics. With these you can tailor your programme to specialise in one of these areas, or gain a broad understanding of several. This allows you to build up the required background for the project and dissertation modules, which offer the opportunity to undertake an in-depth study of a topic of your choice, supervised by a leading expert in the field.

Exceptional employability

At Liverpool, we listen to employers’ needs. Alongside key problem solving skills, employers require strong communication skills. These are integral to this programme. Graduates go on to research degrees, or become business and finance professionals, or to work in management training, information technology, further education or training (including teacher training) and scientific research and development.

Teaching quality

We are proud of our record on teaching quality, with five members of the Department having received the prestigious Sir Alastair Pilkington Award for Teaching. We care about each student and you will find the staff friendly and approachable.

Accessibility

We take students from a wide variety of educational backgrounds and we work hard to give everyone the opportunity to shine.

Supportive atmosphere

We provide high quality supervision and teaching, computer labs, and and you will benefit from the friendly and supportive atmosphere in the Department, as evidenced by student feedback available on our university website. A common room and kitchen for the exclusive use of the Department’s students, and a lively maths society help to foster a friendly and supportive environment.

Career prospects

The excellent University Careers Service is open to all postgraduates. Graduates of the MSc and PhD programmes move on to many different careers. Recent graduates have moved into fast track teacher programmes, jobs in finance (actuarial, banking, insurance), software development, drugs testing and defence work, as well as University postdoctoral or lecturing posts. The MSc programme is of course a natural route into doctoral study in Mathematics and related fields, both at Liverpool and elsewhere. Some of our PhD students move on to postdoctoral positions and to academic teaching jobs and jobs in research institutes, both in the UK and elsewhere.

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