糖心Vlog

At university, emphasis is placed on understanding general mathematical theorems. they apply in many different cases, and understanding why a result is true enables us to creatively use the underlying ideas to tackle new problems.

Study the language and structure of mathematical proofs, illustrated by results from number theory. You will see the concept of congruence of integers, which is a simplified form of arithmetic where seemingly impossible problems become solvable. In relation, you’ll encounter the abstract idea of an equivalence relation.

Sets and functions form the basic language of mathematics. You will study functions of a real variable and abstract functions between arbitrary sets, and you will explore how to count sets, both finite combinatorial arrangements and infinite sets.

Interested in how mathematicians build theories from basic concepts to complex ideas, like eigenvalues and integration? Journey from polynomial operations to matrices and calculus through this module.

Starting with polynomials and mathematical induction, you will learn fundamental proof techniques. You will explore matrices, arrays of numbers encoding simultaneous linear equations, and their geometric transformations, which are essential in linear algebra. Eigenvalues and eigenvectors, which characterise these transformations, will be introduced, highlighting their role in applications including population growth and Google's page rankings.

Next, we will reintroduce you to calculus, from its invention by Newton and Leibniz, to its formalisation by Cauchy and Weierstrass. You will explore sequence convergence, techniques for evaluating limits, and key continuity tools like the intermediate value theorem. Differentiation techniques develop a geometric understanding of function graphs, leading to mastering integration methods for solving differential equations and calculating areas under curves. We conclude with a first look at vector calculus.

An introduction to the mathematical and computational toolsets for modelling the randomness of the world. You will learn about probability, the language used to describe random fluctuations, and statistical techniques. This will include exploring how computing tools can be used to solve challenges in scientific research, artificial intelligence, machine learning and data science.

You will develop the axiomatic theory of probability and discover the theory and uses of random variables, and how theory matches intuitions about the real-world. You will then dive into statistical inference, learning to select appropriate probability models to describe discrete and continuous data sets.

You will gain the ability to implement statistical techniques to draw clear, informative conclusions. Throughout, you will learn the basics of R or Python, and their use within probability and statistics. This will equip you with the skills to deploy statistical methods on real scientific and economic data.

Symmetry is central to our understanding of a range of subjects, from the structure of molecules to the roots of polynomials. In this module, you will see how group theory naturally appears whenever we look at symmetry.

Using familiar examples, including the symmetry of regular polygons, rotations and reflection matrices, roots of 1 in the complex plane, and permutations, you will define what makes a group and how this can provide a unifying language, highlighting connections between seemingly different subjects.

You will then transition into mathematical analysis, developing an approach to sequences, limits, and continuity that provides the foundation for calculus. Examining a range of examples, you will build your understanding of precise mathematical reasoning and gain an appreciation for the importance of proof, generalisation and abstraction.

Ever wondered about the hidden structures that govern mathematics? Algebra is more than just equations, it's the language of symmetry and structure, underpinning subjects ranging from geometry and quantum mechanics to number theory and cryptography. The main frameworks for modern algebra are group theory and ring theory.

Group theory topics include classifying symmetries, the symmetric group, Lagrange's theorem and the first isomorphism theorem. Similarly, ring theory explores the notions of subrings, ideals, and homomorphisms in an example-driven methodology, using abstract number systems, polynomial structures, and matrices.

This module introduces the essential theory and techniques for algebra, laying a solid foundation for further study in mathematics, physics, and other related fields.

The success of Newton/Leibniz’s calculus raises the question: what happens if we replace the real numbers with the complex numbers? After all, their arithmetic structure is similar, and we can measure distances between points in both. You will learn how to define the derivative of a complex function as usual and explore the behaviour of functions that are complex differentiable. Everything resembles the real case, ultimately leading to the astonishing result that if a complex function can be differentiated once, it can be differentiated infinitely often and is expressed by its Taylor series. Integral calculus for complex functions opens a route towards evaluating definite integrals that cannot be reached by real variables.

Applications of these results include a proof of the fundamental theorem of algebra, which states that every non-constant complex polynomial has a root.

This module lays foundations for further studies of mathematical analysis, pure and applied.

Throughout your degree you gain a unique skills set based on your understanding of the interdisciplinary nature of sciences. In this module we develop your self-awareness of these skills and how to make the most of graduate-level employment opportunities.

We introduce you to the University’s employability resources including job search techniques and search engine use. We develop your skills in writing CVs and cover letters, and we draw on the expertise of employers and alumni. Your ability to effectively use these resources will enhance your employability skills, your communication skills and help you to develop a short-term career plan.

Building on your knowledge of vectors and matrices, this module explores the elegant framework of linear algebra, a powerful mathematical toolkit with remarkably diverse applications across statistical analysis, advanced algebra, graph theory, and machine learning.

You'll develop a comprehensive understanding of fundamental concepts, including vector spaces and subspaces, linear maps, linear independence, orthogonality, and the spectral decomposition theorem.

Through individual exploration, small-group collaboration, and computational exercises, you'll gain both theoretical insight and practical skills. The module emphasises how these abstract concepts translate into powerful problem-solving techniques across multiple disciplines, preparing you for advanced studies while developing your analytical reasoning abilities.

Continuing with your study into real numbers, you will explore their completeness (the idea that there are no ‘gaps’, unlike in the rationals). This completeness will be used to understand the limits of sequences, convergence of series, and power series.

This framework will allow for precision when exploring continuity, differentiability, and integrability of functions of a real variable, providing an improved foundation for calculus. That will enable you to understand when it is appropriate to use calculus; for instance, in proving theorems in other areas of mathematics, such as mathematical physics, probability and number theory.

The cornerstone of mathematical analysis is the construction of proofs involving arbitrarily small numbers, so-called epsilons and deltas. You will have opportunities to practise and improve your management of these quantities, in the process developing your skills in logic, communication and problem-solving.

In this module we continue to develop your employability skills. We focus on your ability to communicate your scientific learning to reflect the interdisciplinary nature of your degree and empower you when it comes to job applications and interviews. This includes practice for assessment centres and associated tasks such as psychometric testing and skills testing, and 1-1 recruitment selection or panel-based interviews.

Researching, writing and presenting are key skills for all Natural Sciences students. This module develops these skills and gives you the opportunity to produce an individual project on a chosen mathematical or statistical theme. You will receive support from an appropriate supervisor to design, research and deliver the project. You will learn how to format and structure professional scientific reports and papers, understand how to research them, discover how to typeset them using the specialised package LaTeX and find out how to correctly include citations and references. You will deploy established techniques of mathematical or statistical analysis to your chosen project, applying knowledge and skills from previous years, and you will communicate your findings to others by producing a report as well as a presentation.

Commutative rings generalise both integers and polynomials and they play a very important role in a wide area of mathematics. As well as being important in algebra, they sit at the heart of algebraic approaches including geometry and number theory, in part because rings of functions occur so naturally there, as they do in analysis.

At this stage, you will already know how to factor and divide integers and polynomials. Therefore, a crucial question is to understand the factorisability and divisibility properties in more general commutative rings. For example, what is the analogue of the set of prime integers, or which are the invertible elements?

You will seek to answer these questions, beginning by looking at rings with certain properties and finding the key examples of these, continuing to describe several constructions that allow us to produce rings with properties we would like. You will conclude by discussing the applications to the areas mentioned above.

The study of graphs (mathematical objects used to model networks and pairwise relations between objects) is a cornerstone of discrete mathematics. Graphs can represent important real-world situations, and the study of algorithms for graph-theoretical problems has strong practical significance.

You will learn about structural and topological properties of graphs, including graph minors, planarity and colouring. We will introduce several theoretical tools, including matrices relating to graphs and the Tutte polynomial. We will also study fundamental algorithms for network exploration, routing and flows, with applications to the theory of connectivity and trees, considering implementation, proofs of correctness and efficiency of algorithms.

You will gain experience in following and constructing mathematical proofs, correctly and coherently using mathematical notation, and choosing and carrying out appropriate algorithms to solve problems. The module will enable you to develop an appreciation for a range of discrete mathematical techniques.

An inner product space is a real or complex vector space, equipped with certain extra structure that formalises the geometrical notion of orthogonality. It turns out that each inner product space has an intrinsic notion of distance, allowing us to discuss convergence and completeness. Complete inner product spaces are known as Hilbert spaces.

The theory of Hilbert spaces blends linear algebra and (real) analysis. It is a natural and powerful tool for studying problems of quantitative approximation. Furthermore, it provides an abstract framework that can be applied to diverse areas of maths, from differential equations and spectral theory to quantum mechanics and stochastic processes.

This module will introduce you to the theory of Hilbert spaces and prepare for advanced study in functional analysis, approximation theory, signal processing, and statistical learning.

Knots play a fundamental role in many areas of mathematics, from pure topology and algebra through to quantum field theory and protein-folding.

Develop tools to measure knottedness, including geometrical ideas like curvature, knot invariants like the Jones polynomial, and the crucial concept of the fundamental group, which has applications in topology far beyond detecting knots.

Linear systems of differential and integral equations provide a mathematical model for a wide range of real-world devices, including communication systems, 5G networks, electrical circuits, heating systems and economic processes. Mathematical analysis of these models gives insight into the behaviour of these devices, with applications in automatic control, signal processing, wireless communications and numerous other areas.

Linear systems are considered in continuous time that reduce to a standard (A,B,C,D) state space representation. Via the Laplace transform, these are reduced further to rational transfer functions. Linear algebra enables us to classify and solve (A,B,C,D) models, while we describe their properties via diagrams in standard computer software. You will consider feedback control for linear systems, describing the rational controllers that stabilise an (A,B,C,D) system. Alongside the development of analytic methods to study linear systems, you will also gain experience in modelling real-world devices by such systems.

The module commences by looking at classical methods of encryption, discussing their advantages, disadvantages and efficiency. You will also investigate statistical attacks on these methods of encryption and the need for better methods.

After this, you will explore modern methods of encryption that are used in the real-world and rely on the robustness of modular arithmetic. While most encryption methods are still considered secure, you will review potential attacks on these systems (e.g. factorisation algorithms) and situations where bad key generation or implementation has occurred.

Production of a big enough quantum computer renders the above schemes useless. Therefore, you will dive into a short introduction to post-quantum cryptography, including the production of next-gen cryptographic schemes considered to be impenetrable to both classical and quantum computers. You will also explore the theory of lattices and see how these can be used to produce new schemes that may be quantum secure (e.g. NTRU).

Consider the key issues in the teaching and learning of mathematics. Develop an excellent foundation for a PGCE by engaging with educational literature and gain experience in writing academically.

Having studied mathematics for many years, you will be well-placed to reflect upon that experience and attempt to make sense of it in the light of theoretical frameworks developed by researchers in the field. Throughout this module, you will prepare to become a mathematics graduate who can contribute knowledgeable to future debates about the ways in which maths is treated within the education system.

Put theory into practice as you participate in a semester-long, part-time placement in a local primary or secondary school. During your placement you will have the opportunity to take part in classroom observation and assistance, develop classroom resources, host one-on-one or small group support sessions and possibly even teaching parts of a lesson to the class.

This module is recommended if you have an interest in, or are curious about, a career in teaching or educational research. During regular meetings, an academic will help you to develop your skills in critical reflection as you relate your placement experiences to theoretical frameworks. Alongside your personal development, it is also imperative that you provide genuine assistance to local teachers by bringing your mathematical knowledge and enthusiasm into their classroom to help encourage future mathematicians.

A metric space consists of a set, whose elements are called points, and a notion of distance between points governed by three simple rules, abstracted from basic properties of Pythagorean distance in the Euclidean plane. In examples, ‘points’ may be functions where uniformity of convergence can be captured, or binary sequences with applications in computer science, or even subsets of a Euclidean space delivering fractal sets as limits.

Topology goes further, abstracting the notions of continuity and convergence, rendering a teacup and doughnut indistinguishable. A topological space equips each of its ‘points’ with its so-called ‘neighbourhoods’. The few simple principles governing these unlock a robust theory that now pervades the mathematical sciences and theoretical physics.

You will learn the fundamental concepts of completeness, total boundedness for metric spaces, compactness, and the Hausdorff property and metrisability for topological spaces.

Study the structure of intricate mathematical objects, such as groups and rings, by looking at linear approximations of them. Linear approximation is such a fundamental idea that it extends throughout mathematical sciences, cropping up in quantum physics and topological data analysis.

Explore representations of finite groups before passing to algebras and modules, which are ‘vector spaces’ over rings. You will look at the atomic theory of representations: the simple and indecomposable representations that are their building blocks. Can we describe all the building blocks? Attempting to answer this leads us to complete reducibility for representations of finite groups (Maschke's theorem) and to representations of directed graphs. We see how Jordan Normal Form in linear algebra is a first theorem of representation theory and methods for complete classification of the building blocks when complete reducibility fails, culminating in connections to Lie theory.

Do you want to entertain and inspire children and the public in STEM? With an introduction to teaching as well as wider engagement opportunities, learn how to understand your audience and how to engage and enliven them. You will also learn how to balance this with educating them and presenting science in a way that’s appropriate to your audience. We include an introduction to pedagogy, how to inspire school pupils and how to use traditional and new media for science communication.

You will deliver an activity of your choosing to an audience. This could be a lesson at school, engaging with children at a large outreach event or delivering a public lecture. In addition, you will also reflect on your activity to discuss what you’ve learnt and what changes you would make. You can deliver this by either video, podcast or article.

A highlight of your degree will be a significant individual project that will be undertaken with the guidance of a supervisor. Prior to starting your final year, you will be given a list of potential projects according to the current research interests of our academic staff and based on your preferences you will be allocated a supervisor.

Once the year commences, you will begin to explore an area of mathematics or statistics that is of particular interest to you, while receiving support through regular meetings with your supervisor. You will find relevant resources and steer the direction of your project, which will run throughout the academic year. By the end you will be able to showcase your findings, both in written and oral form.

Your dissertation represents the culmination of years of mathematical study and may even provide an entry point for a PhD, if you are interested in further study.

Combinatorics is a core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous, such as graphs, lattices, designs and codes. While combinatorics is a huge subject, with deep and important connections to many areas of modern mathematics, it is a very accessible one.

Explore the fundamental topics of combinatorial enumeration (sophisticated counting methods) and combinatorial design theory (Latin squares and block designs). Alongside this, you will learn additional combinatorial topics chosen from areas such as set systems, error-detecting and error-correcting codes, and combinatorial geometry.

Throughout the module, you will gain an understanding of how combinatorial results and methods may be applied, both within mathematics and in real-world settings.

Galois theory is the study of roots of polynomials and symmetries of these roots.

A basic example of such a symmetry is complex conjugation, which swaps the roots of the polynomial x^2+1 (or any irreducible real quadratic). Polynomials of a degree higher than 2 can have much more complicated symmetries. In fact, any finite group is possible! Galois theory provides you with a framework for understanding how the group of symmetries encodes very deep information about the polynomial itself. A famous application, which will be covered in the module, is a proof of the Abel-Ruffini theorem. Unlike lower degrees, a general polynomial of degree 5 or higher has no ‘solution in radicals’ (i.e. obtainable using only the arithmetic operations and n-th roots).

The first time you meet groups, they tend to be finite: the symmetry group of a triangle, or a cube, or a permutation group. Lie theory is the study of continuous groups of transformations, like rotations of 2-, 3-, 4- or higher-dimensional spaces. It underpins most of modern geometry and particle physics, with applications from solving differential equations, to understanding matter made of quarks, to classifying polynomials in pure algebra. You will explore the foundations of this powerful subject and gain the skills to perform the complex calculations needed to understand the applications.

Have you ever wondered how to define the size of an unusually shaped subset of Euclidean space, such as the Cantor set? Or how to compute the integral of a function that is not piecewise continuous?

This module will introduce the concept of a measure of a set, that generalises the idea of the length of an interval or the area of a rectangle to a much bigger class of sets - the measurable sets. You will then be able to generalise the idea of integrals of functions and lead to new methods for computing integrals.

You will develop this theory using countability arguments, set theory, openness and closedness, as well as sequences of numbers or functions. You will learn to explore this theory using a range of examples and counterexamples. These results have applications in other areas of mathematics such as probability theory and Hilbert spaces.

Operator theory can be seen as an infinite-dimensional version of matrix theory, where matrices act linearly on vectors, they can be added and multiplied, and one can determine eigenvalues and Jordan normal forms etc.

But, while some things are similar to finite dimensions, others are completely different, for example not every symmetric operator can be diagonalised. Due to infinite dimensionality, delicate convergence and completeness issues arise and require input from analysis. If you enjoyed both linear algebra and real analysis, this module is for you. It will prepare you for further postgraduate study in operator algebras, differential operators, mathematical quantum mechanics and other advanced topics.

An introduction to analytic and algebraic techniques for studying problems in number theory. You will explore how methods for analysis can be useful in studying the distribution of prime numbers, the asymptotics of arithmetic functions and the solubility of Diophantine equations. You will also study the solubility of Diophantine equations using methods from algebra, particularly the concept of unique factorisation in certain rings (or lack of). By the end of the module, you will have developed a working knowledge of analytic and algebraic number theory.