Mathematics MSc

Program Overview

This one-year MSc program provides a rigorous foundation in mathematics research, offering advanced courses in pure and applied mathematics, culminating in a research dissertation. The program emphasizes research skills, presentation abilities, and critical thinking, preparing students for doctoral studies or specialized careers in various fields utilizing mathematical expertise. Graduates have pursued research at top global universities or secured employment in industries like computing, finance, and data science.

Program Outline

Degree Overview:

This one-year MSc program is designed to provide students with a strong foundation for a successful career in mathematics research. The program offers a variety of advanced lecture modules across pure and applied mathematics, allowing students to tailor their studies to their interests. Possible topics include algebra, geometry, topology, financial mathematics, and mathematical biology. Students will gain valuable research experience by working with an experienced mathematician on a dissertation topic of their choice. The program also emphasizes the development of problem-solving, presentation, and argumentation skills.

Outline:

The program consists of core modules and optional modules.

Core Modules:

• Research Skills in Mathematics (15 credits): This module provides training in research-level skills used in mathematics and related technical disciplines, with a focus on mathematical/scientific writing and presentation skills.
Students will gain experience in using computer packages for presenting mathematical and statistical material and learn how to prepare coherent, structured, and accurate reports. Topics covered include conducting literature searches, summarizing information, accessing papers through preprint servers, and managing reference lists. Students will apply these skills to write a literature review in an area of interest and prepare and deliver a talk on this topic.
• Dissertation (60 credits): The dissertation is an extensive piece of work (10-20,000 words) that allows students to synthesize theoretical knowledge on a subject of interest.
Students will gain experience in the phases of a large project, including planning to a deadline, researching background information, problem specification, carrying out relevant analyses, and reporting both at length through the dissertation and in summary through an oral presentation.

Optional Modules (Students take 105 credits from this group):

• Machine Learning (15 credits): This module explores the interface between computer science and statistics, focusing on developing tools for modeling and understanding complex data sets.
• Special Topics (15 credits): This module is available to students who have already covered the material in some of the courses offered on the MSc and who would otherwise have difficulty finding sufficient credits to complete the program.
A course of study would be proposed by a member of staff, which may consist of material from a book or from a graduate lecture course taken by PhD students. The student would be expected to write a short report as part of a portfolio of notes and exercises.
• Financial Mathematics (15 credits): This module describes the mathematical ideas behind developments in modern finance, including the Capital Asset Pricing Model and the Black-Scholes option pricing formula.
It includes a project.
• Functional Analysis (15 credits): This module studies infinite-dimensional vector spaces equipped with extra structure, focusing on Banach spaces and especially Hilbert spaces.
Applications of the theory examined include Fourier series and the Fourier transform, and differential equations.
• Further Topics in Mathematical Biology (15 credits): This module focuses on the mathematical modeling of biological phenomena, emphasizing deterministic models based on systems of differential equations.
Examples are drawn from a range of biological topics, including the spread of epidemics, predator-prey dynamics, cell biology, medicine, and other biological phenomena that require a mathematical approach to understand.
• Topics in Mathematical Physics (15 credits): This module introduces students to advanced concepts and techniques in modern mathematical physics, preparing them for research-level activities.
It examines how key physical ideas are precisely formulated in the language of mathematics, such as the idea that fundamental particles arise as excitations of relativistic quantum fields.
• Special Topics 2 (15 credits): This module is available to students who have already covered the material in some of the courses offered on the MSc and who would otherwise have difficulty finding sufficient credits to complete the program.
A course of study would be proposed by a member of staff, which may consist of material from a book or from a graduate lecture course taken by PhD students. The student would be expected to write a short report as part of a portfolio of notes and exercises.
• Mathematical Modelling of Natural Systems (15 credits): This module provides a practical introduction to techniques for modeling natural systems.
Students will learn how to construct, analyze, and interpret mathematical models using a combination of differential equations, scientific computing, and mathematical reasoning.
• Further Topics in Number Theory (15 credits): This module treats examples of further topics in number theory, accessible with the aid of advanced mathematical background.
• Probability with Measure Theory (15 credits): This module introduces the modern basis for probability theory, drawing upon real analysis.
It allows students to study random objects with complex or surprising properties.
• Directed Reading in Mathematics (15 credits): This module serves as a preparation for the MSc dissertation and will cover material deemed useful by the dissertation supervisor, consisting of specialist background material for a student.
The precise content and subject matter will depend on the interests of the student, as well as the level and background of the individual student. The module will be assessed with a portfolio consisting of a report and exercises set by the supervisor.
• Advanced Topics in Algebra A (30 credits): This module studies algebraic structures involved in fields, groups, Galois Theory, rings, and commutative algebra, giving applications to number theory, roots of polynomials, and algebraic geometry.
• Advanced Topics in Waves and Fluid Dynamics A (30 credits): This module covers waves and fluid dynamics, focusing on the flow of fluids, including gas, liquid, and more complex media.
It explores wave motions, viscous fluid flow, and advanced topics such as three-dimensional flows, boundary layers, vortex dynamics, or magnetohydrodynamics.
• Algebraic Topology (30 credits): This module covers algebraic topology, following on from metric spaces.
It studies the shape of spaces and builds a bridge between shapes and algebra.
• Analytical Dynamics and Classical Field Theory (30 credits): This module discusses Newton's laws of mechanics, Lagrange, Hamilton, and Jacobi's work, and Noether's theorem relating symmetries and conservation laws.
It also introduces Einstein's theory of gravity, General Relativity.
• Stochastic Processes and Finance (30 credits): This module studies stochastic processes and applies this knowledge to mathematical finance, particularly arbitrage-free pricing and the Black-Scholes model.

Assessment:

Assessment methods include coursework, exams, and a dissertation. Regular feedback is provided to support student development throughout the course.

Teaching:

Teaching is delivered through a variety of lectures and small group seminars.

Careers:

The advanced topics covered and the extensive research training make this course excellent preparation for a PhD. Sheffield maths graduates have secured postgraduate research positions at many of the world's top 100 universities. Mathematics graduates also develop numerical, problem-solving, and data analysis skills that are useful in many careers, including computing, banking, and data science. Employers that have hired Sheffield maths graduates include Amazon, Barclays, Dell, Goldman Sachs, IBM, PwC, Sky, the NHS, and the Civil Service.

Other:

The School of Mathematical and Physical Sciences is leading the way with groundbreaking research and innovative teaching. They provide students with the skills and knowledge to support them in a wide range of careers. Mathematicians at the University of Sheffield have expertise across many areas of pure mathematics, applied mathematics, probability, and statistics. The school strives to be an inclusive and welcoming environment for all who enjoy mathematics. Their mathematics and statistics researchers work on a wide variety of topics, from the most abstract questions in algebraic geometry and number theory to the calculations behind infectious disease, black holes, and climate change. In the Research Excellence Framework 2021, 96% of their mathematical sciences research was rated in the highest two categories as world-leading or internationally excellent. Mathematics and statistics staff have received honors from the Royal Society, the Society for Mathematical Biology, and the Royal Statistical Society, who also provide professional accreditation for their statistics courses. They are an organizer of the Transpennine Topology Triangle, a key focal point for topology research in the UK. They also have strong links with the Society for Industrial and Applied Mathematics, the Institute of Mathematics and its Applications, the European Physical Society, and the International Society on General Relativity and Gravitation.