Mathematics with Languages BSc (Hons)

​At its heart, calculus is the study of limits. Many quantities can be expressed as the limiting value of a sequence of approximations, for example the slope of a tangent to a curve, the rate of change of a function, the area under a curve, and so on. Calculus provides us with tools for studying all of these, and more. Many of the ideas can be traced back to the ancient Greeks, but calculus as we now understand it was first developed in the 17th Century, independently by Newton and Leibniz. The modern form presented in this module was fully worked out in the late 19th Century. MATH101 lays the foundation for the use of calculus in more advanced modules on differential equations, differential geometry, theoretical physics, stochastic analysis, and many other topics. It begins from the very basics – the notions of real number, sequence, limit, real function, and continuity – and uses these to give a rigorous treatment of derivatives and integrals for real functions of one real variable.​ ​

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It is the study of lines, planes, and subspaces and their intersections using algebra.

Linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, and subsequently, Cramer’s Rule for solving linear systems was devised in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination. All these classical themes, in their modern interpretation, are included in the module, which culminates in a detailed study of eigenproblems. A part of the module is devoted to complex numbers which are basically just planar vectors. Linear algebra is central to both pure and applied mathematics. This module is an essential pre-requisite for nearly all modules taught in the Department of Mathematical Sciences.

This module introduces students to powerful mathematical software packages such as Maple and Matlab which can be used to carry out numerical computations or to produce a more complicated sequence of computations using their programming features. We can also do symbolic or algebraic computations in Maple. These software packages have built-in functions for solving many kinds of equations, for working with matrices and vectors, for differentiation and integration. They also contain functions which allow us to create visual representations of curves and surfaces from their mathematical descriptions, to work interactively, generate graphics and create mathematical documents. This module will teach students many of the above-mentioned features of mathematical software packages. This knowledge will be helpful in Years 2, 3 and 4 when working on different projects, for example in the modules MATH266 and MATH371.

This module, the last one of the core modules in Year 1, is built upon the knowledge you gain from MATH101 (Calculus I) in the first semester. The syllabus is conceptually divided into three parts: Part I, relying on your knowledge of infinite series, presents a thorough study of power series (Taylor expansions, binomial theorem); part II begins with a discussion of functions of several variables and then establishes the idea of partial differentiation together with its various applications, including chain rule, total differential, directional derivative, tangent planes, extrema of functions and Taylor expansions; finally, part III is on double integrals and their applications, such as finding centres of mass of thin bodies. Undoubtedly, this module, together with the other two core modules from Semester 1 (MATH101 Calculus I and MATH103 Introduction to linear algebra), forms an integral part of your ability to better understand modules you will be taking in further years of your studies.

This module is an introduction to classical (Newtonian) mechanics. It introduces the basic principles like conservation of momentum and energy, and leads to the quantitative description of motions of bodies under simple force systems. It includes angular momentum, rigid body dynamics and moments of inertia.

A group is a formal mathematical structure that, on a conceptual level, encapsulates the symmetries present in many structures. Group homomorphisms allow us to recognise and manipulate complicated objects by identifying their core properties with a simpler object that is easier to work with. The abstract study of groups helps us to understand fundamental problems arising in many areas of mathematics. It is moreover an extremely elegant and interesting part of pure mathematics. Motivated by examples in number theory, combinatorics and geometry, as well as applications in data encryption and data retrieval, this module is an introduction to group theory. We also develop the idea of mathematical rigour, formulating our theorems and proofs precisely using formal logic.

Students will learn fundamental concepts from statistics and probability using the R programming language and will learn how to use R to some degree of proficiency in certain contexts. Students will become aware of possible career paths using statistics.

This French language module is designed for first year undergraduate students . It is for absolute beginners or students with very limited knowledge of the language. No previous knowledge of French is required. Through a variety of methods students will develop a basic competence in reading, writing, listening and speaking French and an understanding of basic French grammar.
At the end of this 12 week- module students will be able to carry out simple everyday tasks in French. Students will be able to understand and use familiar everyday expressions and basic phrases aimed at the satisfaction of needs of a concrete type. They will be able to introduce themselves and others and ask and answer questions about personal details. They will be able to interact in a simple way provided the other person talks slowly and clearly. They will have a basic understanding of significant aspects of life and culture of the country and intercultural skills necessary for their language proficiency level. This module is mapped against A1 level in French according to the Common European Framework of Reference for Languages (CEFR).

This module is mapped against A2 level in French according to the Common European Framework of Reference for Languages (CEFR).It is for students who have achieved a GCSE at foundation level or who have reached an A1 proficiency level in the Common European Framework. Through a variety of methods students will continue to develop basic competence in reading, writing, listening and speaking French and an understanding of basic French grammar. At the end of this 12 week- module students will be more confident to carry out all everyday tasks, they will start to be able to express their opinions about current affairs and function in many professional contexts. They will have an increased understanding of life and culture of the country and the intercultural skills necessary for their language proficiency level.

This module is the compulsory language module for all students enrolled in degree programmes aiming for a qualification in French. It is the first stage of a four-year learning curve and is preparation for the following year (FREN207 and FREN208).

This module is mapped against B1+ level in French according to the Common European Framework of Reference for Languages (CEFR).

This module is a compulsory language module for all students enrolled in degree programmes aiming for a qualification in French. It is the first stage of a four-year learning curve and is thus preparation for the following year (FREN207 and FREN208).

An intensive course for those who have not studied Spanish before. Through a variety of methods, students will be provided with basic competence in reading, writing, listening and speaking Spanish. Students are expected to reach a level equivalent to that of level A1.

SPAN134 is an elementary, intensive Spanish language and culture module. It aims at building language and intercultural skills, providing students with a solid understanding of the grammar, syntax, vocabulary and use of the Spanish language in context at an elementary level.

This module is mapped against B1+ level in Spanish according to the Common European Framework of Reference for Languages (CEFR).

The aim of the module is to work towards the expansion of the student’s already existent knowledge of Spanish language. Furthermore, the module places special emphasis on conversational fluency, grammatical accuracy and vocabulary acquisition. Students will receive three contact hours a week divided into two hours of integrated skills language seminars plus a one hour lab session of practical skills (listening and conversation) per week. In addition, students are expected to undertake regular independent language learning for which they will provided with materials and guidance via the University’s Virtual Learning Environment: Canvas.​

This module is mapped against B2- level in Spanish according to the Common European Framework of Reference for Languages (CEFR).

This is a first year advanced language course taught in the second semester where students attend seminars and practical sessions in small groups and focus on spoken and written Spanish. The aim is to work towards the expansion of the student’s already existent knowledge of Spanish language. Furthermore, the module places special emphasis on conversational fluency, grammatical accuracy and vocabulary acquisition. Students will receive three contact hours a week divided into two hours of integrated skills language seminars plus a one hour lab session of practical skills (listening and conversation) per week. In addition, students are expected to undertake regular independent language learning for which they will provided with materials and guidance via Canvas.

This is an accelerated beginners‘ module. You will study at A1 level in the Common European Framework of Reference (complete beginners).

This is an accelerated elementary German module. You will study at A2 level in the Common European Framework of Reference.

This compulsory language module for all students aiming for a qualification in German is designed for students who have an A-level in German, but it is also open to other students as an additional subject or as part of the Erasmus scheme. It aims to provide students with good competence in reading, writing, listening, speaking and grammar through both lessons and independent project work. Students will be introduced to basic translation and interpreting skills during grammar lessons. Students may also benefit from extracurricular activities organised by a native speaker intern, the German Society and a conversation exchange organised through the Modern languages resource centre. It is also the preparation for the following year (GRMN207 and GRMN208).

This module is designed for students with A-level German or equivalent who have successfully completed GRMN105. In this module, skills acquired in semester one will be improved and enhanced in semester two. Students will read a book in German and discuss it in an oral exam. Students will also improve their knowledge of German grammar further and have access to the languages lab for listening comprehension. The module also prepares students for GRMN207 and GRMN208 in second year. Students may benefit from extracurricular activities organised by a native speaker intern, the German Society and a conversation exchange organised through our Modern languages resource centre. Students will continue practicing their basic translating and interpreting skills.

This module offers absolute beginners a comprehensive overview of essential Chinese language functions and related cultural knowledge to develop basic competence in reading, writing, listening and speaking Chinese, and cultural sensitivity and awareness. You are expected to be active and engaged participants in the themed language classes. Computer, projector plus internet are used in on-campus class to enhance learning; Team, Zoom and other online tools are used in online classes to ensure the student learning experience under the circumstance of remote teaching/learning. Homework and self-study material is assigned weekly and is a must to achieve the expected learning outcome. Along with instructions in class, you will be given various teaching/self-learning material on Canvas ( The digital learning platform at University of Liverpool) to foster autonomy in learning the language and culture after class.

This module is the following module of CHIN112. It offers beginners a comprehensive overview of essential Chinese language functions and related cultural knowledge to develop basic competence in reading, writing, listening and speaking Chinese, and cultural sensitivity and awareness. You are expected to be active and engaged participants in the themed language classes. Computer, projector plus internet are used in on-campus class to enhance learning; Team, Zoom and other online tools are used in online classes to ensure the student learning experience under the circumstance of remote teaching/learning. Homework and self-study material is assigned weekly and is a must to achieve the expected learning outcome. Along with instructions in class, you will be given various teaching/self-learning material on Canvas ( The digital learning platform at University of Liverpool) to foster autonomy in learning the language and culture after class.

This module aims to develop students’ receptive skills in reading and listening as well as productive skills in writing and speaking at intermediate level; to deepen their understanding and appreciation of Chinese society, culture and customs.
Students are expected to be active and engaged participants in the themed language classes. Multi-media facilities plus internet are used in class to enhance learning. Homework is assigned each week. Along with face-to-face instructions in class, students will be given guidance on how to use the resources including online database, apps and blackboard to foster autonomy in learning the language and culture outside class.

This module aims to further develop students’ receptive skills in reading and listening as well as productive skills in writing and speaking at post-intermediate level; To further deepen student understanding and appreciation of Chinese society, culture and customs.

Students are expected to be active and engaged participants in the themed language classes. Homework is assigned each week. Under our guidance and supervision, students will also be prepared for year or semester abroad in China (optional for students on Chinese 25%; compulsory for students on Chinese Studies 50%)

This module is an introductory module in Italian language and will cover grammar basic aspects like noun gender and number, articles, the present and perfect tense, modal verbs, prepositions and direct pronouns.

The topics covered will include: personal information; family; education and university life.

This language module is intensive and aims to develop all the necessary skills to communicate confidently in spoken and written Italian within a range of topics, such as Italian culture and society, fashion and the "Made in Italy" industry, work and the business environment.

This module is mapped against A2 level in Italian according to the Common European Framework of Reference for Languages (CEFR).

This intermediate language module builds on existing Italian language skills. The focus is on all four areas of language competence (grammar, written, listening and oral).

This module is mapped against B1+ level in Italian according to the Common European Framework of Reference for Languages (CEFR).

This intermediate language module builds on the existing Italian language skills developed in semester one. The focus is on all areas of language competence (grammar, written, listening and oral).

This module is mapped against B2- level in Italian according to the Common European Framework of Reference for Languages (CEFR).

This introductory Portuguese language module offers absolute beginners a comprehensive overview of basic grammatical functions and linguistic skills that will provide students with basic competence in reading, writing, listening and speaking Portuguese at A1+/A2 level according the Common European Framework of Reference.

This module is mapped against A2 level in Portuguese according to the Common European Framework of Reference for Languages (CEFR).

This module is a continuation of PORT112 and improves upon the linguistic skills acquired in that module. It offers beginners of Portuguese a comprehensive overview of basic grammatical functions and linguistic skills that will provide students with a sound competence in reading, writing, listening and speaking Portuguese.

The principal aim of this module is to achieve greater proficiency in written and speaking Catalan and to provide a solid grammatical foundation.

The student will also have the opportunity to achieve an extra qualification by taking the International Catalan Certificate issued by the Institut Ramon Llull and held at the University of Liverpool.

This module is mapped against A2 level in Catalan according to the Common European Framework of Reference for Languages (CEFR).

This is an introductory intensive module which aims to provide students with a working knowledge of modern Catalan, written and spoken, roughly at A-level standard. The student will also have the opportunity to achieve an extra qualification by taking the International Catalan Certificate issued by the Institut Ramon Llull and held at the University of Liverpool.

This module provides an introduction to the subjects of fluid mechanics and electromagnetism, to the various vector integrals, the operators div, grad and curl and the relations between them and to the many applications of vector calculus to physical situations.

This module introduces students to a surprising, very beautiful theory having intimate connections with other areas of mathematics and physical sciences, for instance ordinary and partial differential equations and potential theory.

Linear algebra provides a toolbox for analysing phenomena ubiquitous in many areas of mathematics: linear maps, or linearity in general. In all of these situations it is essential to first identify the kind of objects which are mapped or behave in a linear way. To cover the many different possibilities the concept of an abstract vector space is introduced. It generalizes the real vector spaces introduced in MATH103 (Introduction to Linear Algebra) and the calculational techniques developed there can still be used. Applications of ideas from Linear Algebra appear in Geometry, in Algebra, in solving Differential Equations, which in turn model many physical systems, in Physics, especially Quantum Mechanics, in Biology and in Statistics.

Analysis of data has become an essential part of current research in many fields including medicine, pharmacology, and biology. It is also an important part of many jobs in e.g. finance, consultancy and the public sector. This module provides an introduction to statistical methods with a strong emphasis on applying and interpreting standard statistical techniques. Since modern statistical analysis of real data sets is performed using computer power, a statistical software package is introduced and employed throughout.

This module is an introduction to classical (Newtonian) mechanics. It introduces the basic principles like conservation of momentum and energy, and leads to the quantitative description of motions of bodies under simple force systems. It includes angular momentum, rigid body dynamics and moments of inertia.

A group is a formal mathematical structure that, on a conceptual level, encapsulates the symmetries present in many structures. Group homomorphisms allow us to recognise and manipulate complicated objects by identifying their core properties with a simpler object that is easier to work with. The abstract study of groups helps us to understand fundamental problems arising in many areas of mathematics. It is moreover an extremely elegant and interesting part of pure mathematics. Motivated by examples in number theory, combinatorics and geometry, as well as applications in data encryption and data retrieval, this module is an introduction to group theory. We also develop the idea of mathematical rigour, formulating our theorems and proofs precisely using formal logic.

Students will learn fundamental concepts from statistics and probability using the R programming language and will learn how to use R to some degree of proficiency in certain contexts. Students will become aware of possible career paths using statistics.

Differential equations play a central role in mathematical sciences because they allow us to describe a wide variety of real-world systems and the mathematical techniques encountered in this module are useful to a number of later modules; this is why MATH201 is compulsory for a number of degree programmes. The module will aim to stress the importance of both theory and applications of ordinary differential equations (ODEs) and partial differential equations (PDEs), putting a strong emphasis on problem solving and examples. It has broadly 5 parts and each part contains two types of equations: those that can be solved by specific methods and others that cannot be solved but can only be studied to understand some properties of the underlying equations and their solutions. The main topics are first order ODEs, second order ODEs, systems of ODEs, first-order PDEs and some of the most well-known second-order PDEs, namely the wave, heat and Laplace equations.

Most problems in modern applied mathematics require the use of suitably designed numerical methods. Working exactly, we can often reduce a complicated problem to something more elementary, but this will often lead to integrals that cannot be evaluated using analytical methods or equations that are too complex to be solved by hand. Other problems involve the use of ‘real world’ data, which don’t fit neatly into simple mathematical models. In both cases, we can make further progress using approximate methods. These usually require lengthy iterative processes that are tedious and error prone for humans (even with a calculator), but ideally suited to computers. The first few lectures of this module demonstrate how computer programs can be written to handle calculations of this type automatically. These ideas will be used throughout the module. We then investigate how errors propagate through numerical computations. The focus then shifts to numerical methods for finding roots, approximating integrals and interpolating data. In each case, we will examine the advantages and disadvantages of different approaches, in terms of accuracy and efficiency.

This module is concerned with the motion of physical bodies both in everyday situations and in the solar system. To describe motion, acceleration and forces you will need background knowledge of calculus, differentiation, integration and partial derivatives from MATH101 (Calculus I), MATH102 (Calculus II) and MATH103 (Introduction to Linear Algebra). Classical mechanics is important for learning about modern developments such as relativity (MATH326), quantum mechanics (MATH325) and chaos and dynamical systems (MATH322). This module will make you familiar with notions such as energy, force, momentum and angular momentum which lie at the foundations of applied mathematics problems.

This is a foundational module aimed at providing the students with the basic concepts and techniques of modern real Analysis. The guiding idea will be to start using the powerful tools of analysis, familiar to the students from the first year module MATH101 (Calculus I) in the context of the real numbers, to vectors (multivariable analysis) and to functions (functional analysis). The notions of convergence and continuity will be reinterpreted in the more general setting of metric spaces. This will provide the language to prove several fundamental results that are in the basic toolkit of a mathematician, like the Picard Theorem on the existence and uniqueness of solutions to first order differential equations with an initial datum, and the implicit function theorem. The module is central for a curriculum in pure and applied mathematics, as familiarity with these notions will help students who want to take several other subsequent modules as well as many projects. This module is also a useful preparation (although not a formal prerequisite) for MATH365 Measure theory and probability, a very useful module for a deep understanding of financial mathematics.

​The module provides an introduction to the theory and methods of the modern commutative algebra (commutative groups, commutative rings, fields and modules) with some applications to number theory, algebraic geometry and linear algebra.

This module provides an introduction to probabilistic methods that are used not only in actuarial science, financial mathematics and statistics but also in all physical sciences. It focuses on discrete and continuous random variables with values in one and several dimensions, properties of the most useful distributions (e.g. geometric, exponential, and normal), their transformations, moment and probability generating functions and limit theorems. This module will help students doing MATH260 and MATH262 (Financial mathematics). This module complements MATH365 (Measure theory and probability) in the sense that MATH365 provides the contradiction-free measure theoretic foundation on which this module rests.

​Mathematical Finance uses mathematical methods to solve problems arising in finance. A common problem in Mathematical Finance is that of derivative pricing. In this module, after introducing the basic concepts in Financial Mathematics, we use some particular models for the dynamic of stock price to solve problems of pricing and hedging derivatives. This module is fundamental for students intending to work in financial institutions and/or doing an MSc in Financial Mathematics or related areas.

The term "Operational Research" came in the 20th century from military operations. It describes mathematical methods to achieve the goal (or to find the best possible decision) having limited resources. This branch of applied mathematics makes use of and has stimulated the development of optimisation methods, typically for problems with constraints. This module can be interesting for any student doing mathematics because it concentrates on real-life problems.

This compulsory module is designed for students who have successfully completed the beginners’ modules FREN112 and FREN134, or have reached an A2+ proficiency level in the Common European Framework, and who plan to go abroad in their third year. It aims to provide students with an advanced competence in reading, writing, listening, speaking and grammar through both lessons and independent work. At the end of this 12 weeks- module students will have reached a B1 proficiency level in the Common European Framework of Reference and will confidently carry out all everyday tasks, they will be able to express their opinions about current affairs and function in many professional contexts. They will have a good understanding of life and culture of the country and the intercultural skills necessary for their language proficiency level.

This compulsory module is designed for students who have successfully completed the beginners courses FREN112 and FREN134, as well as FREN256 in semester one of their second year, and plan to go abroad in their third year. It aims to further provide students with good competence in reading, writing, listening, speaking and grammar through both lessons and independent work.
At the end of this 12 weeks- module students will have reached a B1 proficiency level in the Common European Framework of Reference and will confidently carry out all everyday tasks, they will be able to express their opinions about current affairs and function in many professional contexts. They will have a good understanding of life and culture of the country and the intercultural skills necessary for their language proficiency level. This module is mapped against B2 level in French according to the Common European Framework of Reference for Languages (CEFR).

This module is mapped against B2 level in French according to the Common European Framework of Reference for Languages (CEFR).

This module aims to develop further the skills of reading, listening, writing and speaking in French, as well as translating. This module will concentrate on fluency and accuracy in written and spoken French, through text analysis and language exercises. The module also provides a cultural preparation for the period of study in a French speaking country, focusing on job application, the world of work, the various registers and accents in French and practical information on accommodation, etc.

This module aims to develop further the skills of reading, listening, writing and speaking in French, as well as translating and interpreting both in and out of French. The module also provides a cultural preparation for the period of study in a French speaking country, focusing on the Education system, aspects of French life, practical information on banks, accommodation, transport, health system.

This module is mapped against B1 level in Spanish according to the Common European Framework of Reference for Languages (CEFR).

Students will build on the ab initio modules (SPAN112 / SPAN134), developing and consolidating their speaking, listening, reading and writing skills, especially through the coverage given to important elements of Spanish Grammar. Vocabulary acquisition will be consolidated.

Students will build on the SPAN256 syllabus, developing and consolidating their speaking, listening, reading and writing skills.

This Module is mapped against the B2 level in Spanish according to the Common European Framework of Reference for Languages (CEFR).

This second-year advanced Spanish language course consists of small group tuition (on-campus and/or online) in spoken and written Spanish. The module aims to consolidate and extend students’ existing language skills, providing them with an enhanced understanding of Spanish grammar, and significantly improved oral and writtencommunication skills. SPAN207 and SPAN208 are mapped against the B2.2 level (CEFR). Specific language objectives include: Writing objectives: Writing clear, detailed texts on a wide range of subjects related to the students’ interests; writing essays or reports, passing on information or giving reasons in support of or against a particular point of view; writing letters highlighting the personal significance of events and experiences.

Reading objectives: Reading articles and reports concerned with contemporary problems in which the writers adopt particular attitudes or viewpoints; understanding contemporary literary prose.

Speaking objectives: Presenting clear, detailed descriptions on a wide range of subjects related to the student’s field(s) of interest; explaining a viewpoint on a topical issue giving the advantages and disadvantages of various options; interacting with a degree of fluency and spontaneity that makes regular interaction with native speakers quite possible; taking an active part in discussions in familiar contexts, accounting for and sustaining the students’ views. Listening objectives: Understanding extended speech and lectures and following even complex lines of argument provided the topic is reasonably familiar; understanding most TV news and current affairs programs; understanding films in standard dialect.

This second-year advanced Spanish language course consists of small group tuition (on-campus and/or online) in spoken and written Spanish. The module aims to consolidate and extend students’ existing language skills, providing them with an enhanced understanding of Spanish grammar, and significantly improved oral and written communication skills.

SPAN207 and SPAN208 are mapped against the B2.2 level (CEFR). Specific language objectives include:

Writing objectives:
Writing clear, detailed texts on a wide range of subjects related to the students’ interests; writing essays or reports, passing on information or giving reasons in support of or against a particular point of view; writing letters highlighting the personal significance of events and experiences.

Reading objectives:
Reading articles and reports concerned with contemporary problems in which the writers adopt particular attitudes or viewpoints; understanding contemporary literary prose.

Speaking objectives:
Presenting clear, detailed descriptions on a wide range of subjects related to the students’ field(s)of interest; explaining a viewpoint on a topical issue giving the advantages and disadvantages of various options; interacting with a degree of fluency and spontaneity that makes regular interaction with native speakers quite possible; taking active part in discussions in familiar contexts, accounting for and sustaining the students’ views.

Listening objectives:
Understanding extended speech and lectures and following even complex lines of argument provided the topic is reasonably familiar; understanding most TV news and current affairs programmes; understanding the majority of films in standard dialect.

This required module is mapped against B1 level in German according to the Common European Framework of Reference for Languages (CEFR), and is designed for students who have successfully completed the beginners’ modules GRMN112 and GRMN134 and plan to go abroad in their third year. In four weekly taught hours it aims to provide students with good competence in reading, writing, listening, speaking and grammar through both lessons and independent study. For their independent project, students can choose from a range of tasks to improve specific language skills such as listening, speaking or reading, about which they have to deliver a written report. Students will be introduced to first translation skills in dedicated lessons. Students may also benefit from extracurricular activities organised by a native speaker intern or Language Advisor in the Language Lounge, the Modern Languages Society and a conversation exchange organised through our Language Lounge or via our online exchange scheme EUniTa.

This German language module is designed for students at ex-beginner’s level who have successfully completed GRMN256 and plan to go abroad. In four weekly taught hours, it aims to enhance further skills acquired in semester one through both lessons and independent project work. Students may benefit from extracurricular activities organised by a native speaker intern, the German Society and a conversation exchange organised through our languages centre. Students will also continue practicing basic translating and interpreting skills.

This module is for students who have successfully completed GRMN105 and GRMN106 at level B2 as specified in the Common European Framework of Reference. In this module, students will be introduced to Austrian and Swiss history, culture and language in one hour per week while also preparing students for their year abroad and its different pathways. Listening and speaking skills will be practiced in one hour per week. Audio and video listening skills will be improved through both class exercises and independent study, and students will prepare oral debates and presentations. Students will be introduced to research skills in preparation of their year abroad and write longer essays in German. In their third hour, students will also be introduced to general translation skills from German into English and English into German in a variety of genres and continue practicing advanced grammar skills. They will be introduced to basic interpreting skills in the languages lab.

This module is for students of German at an advanced level who successfully completed GRMN207. In this module, students will tackle topics of the history and culture of Germany in one hour per week, and receive detailed practical information for their year abroad and continue to practice listening, speaking and presentation skills in one hour per week. Students will continue to practice general translation skills from German into English in a variety of genres and English into German at a basic level, continue basic interpreting skills and continue practicing advanced grammar skills. In preparation for their year abroad, students will be prepared for the various pathways to cope with work placements, assistantships and studentships.

This module aims to develop students’ receptive skills in reading and listening as well as productive skills in writing and speaking at intermediate level; to deepen their understanding and appreciation of Chinese society, culture and customs.
Students are expected to be active and engaged participants in the themed language classes. Multi-media facilities plus internet are used in class to enhance learning. Homework is assigned each week. Along with face-to-face instructions in class, students will be given guidance on how to use the resources including online database, apps and blackboard to foster autonomy in learning the language and culture outside class.

This module aims to further develop students’ receptive skills in reading and listening as well as productive skills in writing and speaking at post-intermediate level; To further deepen student understanding and appreciation of Chinese society, culture and customs.

Students are expected to be active and engaged participants in the themed language classes. Homework is assigned each week. Under our guidance and supervision, students will also be prepared for year or semester abroad in China (optional for students on Chinese 25%; compulsory for students on Chinese Studies 50%)

This module aims to consolidate understanding of the structure and functions of Chinese and related cultural knowledge and seeks to enable students to achieve an advanced level in Chinese that enables near fluent communication.

This module aims to further consolidate understanding of the structure and functions of Chinese and related cultural knowledge and seeks to enable students to achieve post-advanced level in Chinese that enables near fluent communication.

This module will expand students’ knowledge of the Italian language by looking at grammar aspects like the tenses of the past and their contrast, the use of the subjunctive, and conditional sentences. Topics covered will relate to students’ interests and may be chosen with them, typically including young people, language variety, Italian cinema and literature, music.

This module is intensive and aims to develop further skills acquired in ITAL256 to enable students to communicate confidently and effectively in spoken and written Italian.

This module is mapped against B2 level in Italian according to the Common European Framework of Reference for Languages (CEFR).

The module aims to build on advanced oral and written language skills and to develop more specialised competences of the kind required in the year abroad.

This module is mapped against B2 level in Italian according to the Common European Framework of Reference for Languages (CEFR).

The module builds on advanced listening, reading, oral and written skills in the target language and develops more specialised competences of the kind required in translation and interpreting tasks and in preparation for the year abroad.

This module is mapped against B2+ level in Italian according to the Common European Framework of Reference for Languages (CEFR).

PORT256 is a continuation of PORT112 and PORT134, and improves upon the linguistic skills, grammatical functions and vocabulary acquired on these courses. This module is mapped against B1 level descriptors in Portuguese according to the Common European Framework of Reference for Languages (CEFR). It aims to achieve the level of fluency required for those students intending to spend their year abroad in a Portuguese-speaking country.

PORT278 is a continuation of PORT256 and improves upon the linguistic skills acquired on that module. This module is mapped against B2 level descriptors in Portuguese according to the Common European Framework of Reference for Languages (CEFR) which is the level of fluency required for those students intending to spend their year abroad in Portugal or Brazil.

This module is mapped against B1 level in Catalan according to the Common European Framework of Reference for Languages (CEFR).

This module builds upon the year one beginners’ modules CATL112 and CATL134, with greater emphasis placed on written, as opposed to spoken, Catalan. While continuing to use audiovisual methods with a view to further increasing oral fluency, the principal aim will be to achieve greater proficiency in written Catalan and to impart a solid grammatical foundation.

This module builds upon Intermediate Catalan 5+6.

The principal aim will be to achieve greater proficiency in written Catalan and to impart a solid grammatical foundation.

Ordinary and partial differential equations (ODEs and PDEs) are crucial to many areas of science, engineering and finance. This module addresses methods for, or related to, their solution. It starts with a section on inhomogeneous linear second-order ODEs which are often required for the solution of higher-level problems. We then generalize basic calculus by considering the optimization of functionals, e.g., integrals involving an unknown function and its derivatives, which leads to a wide variety of ODEs and PDEs. After those systems of two linear first-order PDEs and second-order PDES are classified and reduced to ODEs where possible. In certain cases, e.g., `elliptic’ PDEs like the Laplace equation, such a reduction is impossible. The last third of the module is devoted to two approaches, conformal mappings and Fourier transforms, which can be used to obtain solutions of the Laplace equation and other irreducible PDEs.

​This module provides an introduction to basic concepts and principles of continuum mechanics. Cartesian tensors are introduced at the beginning of the module, bringing simplicity and versatility to the analysis. The module places emphasis on the importance of conservation laws in integral form, and on the fundamental role integral conservation laws play in the derivation of partial differential equations used to model different physical phenomena in problems of solid and fluid mechanics.

The development of Quantum Mechanics, requiring as it did revolutionary changes in our understanding of the nature of reality, was arguably the greatest conceptual achievement of all time. The aim of the module is to lead the student to an understanding of the way that relatively simple mathemactics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world.

Einstein’s theories of special and general relativity have introduced a new concept of space and time, which underlies modern particle physics, astrophysics and cosmology. It makes use of, and has stimulated the development of modern differential geometry. This module develops the required mathematics (tensors, differential geometry) together with applications of the theory to particle physics, black holes and cosmology. It is an essential part of a programme in theoretical physics.

Number theory begins with, and is mainly concerned with, the study of the integers. Because of the fundamental role which integers play in mathematics, many of the greatest mathematicians, from antiquity to the modern day, have made contributions to number theory. In this module you will study results due to Euclid, Euler, Gauss, Riemann, and other greats: you will also see many results from the last 10 or 20 years.Several of the topics you will study will be familiar from MATH142 (Numbers, groups, and codes). We will go into them in greater depth, and the module will be self-contained from the point of view of number theory. However, some background in group theory (no more than is in MATH142) will be assumed.

The module provides an introduction to the modern theory of finite non-commutative groups. Group Theory is one of the central areas of Pure Mathematics. Being part of Algebra, it has innumerable applications in Geometry, Number Theory, Combinatorics and Analysis, but also plays a very important role in Theoretical Physics, Mechanics and Chemistry. The module starts with basic definitions and some well-known examples (the symmetric group of permutations and the groups of congruence classes modulo an integer) and builds up to some very interesting and non-trivial constructions, such as the semi-direct product, which makes it possible to construct more complicated groups from simpler ones. In the final part of the course, the Sylow theory and its applications to the classification of groups are considered.

Differential geometry studies distances and curvatures on manifolds through differentiation and integration. This module introduces the methods of differential geometry on the concrete examples of curves and surfaces in 3-dimensional Euclidean space. The module MATH248 (Geometry of curves) develops methods of differential geometry on examples of plane curves. This material will be discussed in the first weeks of the course, but previous familiarity with these methods is helpful. Students following a pathway in theoretical physics might find this module interesting as it discusses a different aspect of differential geometry, and might take it together with MATH326 (Relativity). MATH410 (Manifolds, homology and Morse theory) and MATH446 (Lie groups and Lie algebras).​

This module studies discrete-time Markov chains, as well as introducing the most basic continuous-time processes. The basic theory for these stochastic processes is considered in detail. This includes the Chapman Kolmogorov equation, communication of states, periodicity, recurrence and transience properties, asymptotic behaviour, limiting and stationary distributions, and an introduction to Poisson processes. Applications in different areas, in particular in insurance, are considered.

This module extends earlier work on linear regression and analysis of variance, and then goes beyond these to generalised linear models. The module emphasises applications of statistical methods. Statistical software is used throughout as familiarity with its use is a valuable skill for those interested in a career in a statistical field.

Statistical Physics is a core subject in Physics and a cornerstone for modern technologies. To name just one example, quantum statistics is informing leading edge developments around ultra-cold gases and liquids giving rise to new materials. The module will introduce foundations of Statistical Physics and will develop an understanding of the stochastic roots of thermodynamics and the properties of matter. After successfully completing this module students will understand statistical ensembles and related concepts such as entropy and temperature, will understand the properties of classical and quantum gases, will be know the laws of thermodynamics and will be aware of advanced phenomena such as phase transition. The module will also develop numerical computer programming skills for the description of macroscopic effects such as diffusion by an underlying stochastic process.

In this module you will explore, from a game-theoretic point of view, models which have been used to understand phenomena in which conflict and cooperation occur and see the relevance of the theory not only to parlour games but also to situations involving human relationships, economic bargaining (between trade union and employer, etc), threats, formation of coalitions, war, etc.

Many real-world systems in mathematics, physics and engineering can be described by differential equations. In rare cases these can be solved exactly by purely analytical methods, but much more often we can only solve the equations numerically, by reducing the problem to an iterative scheme that requires hundreds of steps. We will learn efficient methods for solving ODEs and PDEs on a computer.

Combinatorics is a part of mathematics in which mathematicians deal with discrete and countable structures by means of various combinations, such as permutations, ordered and unordered selections, etc. The seemingly simple methods of combinatorics can raise highly non-trivial mathematical questions and lead to deep mathematical results, which are, in turn, closely related to some fundamental phenomena in number theory

A “dynamical system” is a system that changes over time according to a fixed rule. In complex dynamics, we consider the case where the state of the system is described by a single (complex) variable, and the rule of evolution is given by a holomorphic function. It turns out that this seemingly simple setting leads to very rich, subtle and intricate problems, some of which are still the subject of ongoing mathematical research, both at the University of Liverpool and internationally. This module will provide an introduction to this fascinating subject, and introduce students to some of these problems. In the course of this study, we will encounter many results about complex functions that may seem “magic” when compared with what might be expected from real analysis. A highlight of this kind is the theorem that every polynomial is “chaotic” on its Julia set. We will also see how this “magic” can help us understand phenomena that at first seem to have no connection with complex numbers at all.

Topology is the mathematical study of space. It is distinguished from geometry by the fact that there is no consideration of notions of distance, angle or other similar quantities. For this reason topology is sometimes popularly referred to as ‘rubber sheet’ geometry. It was introduced by Poincaré, under the name of analysis situs, in 1895 and became one of the most successful areas of 20th century mathematics. It continues to be an active research area to this day, and its insights and methods underlie many areas of modern mathematics. More recently, new applications of topological ideas outside mathematics have been developed, in particular to provide qualitative analysis of large data sets. This module introduces the basic notions of topological space and continuous map, illustrating them with many examples from different areas of mathematics. It also introduces homotopy theory, the study of paths in a space, which has become one of the most fundamental areas of modern mathematics.

​Stochastic processes are ways of quantifying the dynamic relationships of sequences of random events. Stochastic models play an important role in elucidating many areas of the natural and engineering sciences. They can be used to analyse the variability inherent in biological and medical processes, to deal with uncertainties affecting managerial decisions and with the complexities of psychological and social interactions, and to provide new perspectives, methodology, models and intuition to aid in other mathematical and statistical studies. This module is intended as a beginning course in introducing continuous-time stochastic processes for students familiar with elementary probability.  The objectives are: (1) to introduce students to the standard concepts and methods of stochastic modelling; (2) to illustrate the rich diversity of applications of stochastic processes in the science; and (3) to provide exercises in the applications of simple stochastic analysis to appropriate problems.

This module introduces fundamental topics in mathematical statistics, including the theory of point estimation and hypothesis testing. Several key concepts of statistics are discussed, such as sufficiency, completeness, etc., introduced from the 1920s by major contributors to modern statistics such as Fisher, Neyman, Lehmann and so on. This module is absolutely necessary preparation for postgraduate studies in statistics and closely related subjects.

In recent years a culture of evidence-based practice has become the norm in the medical profession. Central to this is the medical statistician, who is required to not only analyse data, but to design research studies and interpret the results. The aim of MATH364 is to provide the student with the knowledge to become part of a “team” to enhance and improve medical practice. This is done by demonstrating the design of studies, methods of analysis and interpretation of results through a number of real-world examples, covering epidemiology, survival analysis, clinical trials and meta-analysis.

This module is important for students who are interested in the abstract theory of integrating and in the deep theoretical background of the probability theory. It will be extremely useful for those who plan to do MSc and perhaps PhD in Probability, including financial applications. If you plan to take level 4 module(s) on Financial Mathematics next year, MATH365 can be very helpful.

To provide an understanding of the mathematical risk theory used in practise in non-life actuarial depts of insurance firms, to provide an introduction to mathematical methods for managing the risk in insurance and finance (calculation of risk measures/quantities), to develop skills of calculating the ruin probability and the total claim amount distribution in some non ‐ life actuarial risk models with applications to insurance industry, to prepare the students adequately and to develop their skills in order to be exempted for the exams of CT6 subject of the Institute of Actuaries (MATH366 covers 50% of CT6 in much more depth).

​MATH367 aims to develop an appreciation of optimisation methods for real-world problems using fundamental tools from network theory; to study a range of ‘standard problems’ and techniques for solving them. Thus, network flow, shortest path problem, transport problem, assignment problem, and routing problem are some of the problems that are considered in the syllabus. MATH367 is a decision making module, which fits well to those who are interested in receiving knowledge in graph theory, in operational research, in economics, in logistics and in finance. 　

The final year language module is a course in communicative French at level C1 or C2, as specified in the Common European Framework of Reference. It introduces students to the principles of debate, reporting and presentation. Students also have the opportunity of developing their interpreting skills. Alongside the consolidation of their oral skills, students will also work on improving their writing skills, practise their grammatical accuracy, translation skills, as well as producing an extended piece of prose , focusing on report writing. Listening and reading skills will be developed during the contact hours and while preparing for assignments.

This is the second module which makes up the final-year programme in communicative French language. Like FREN311, it focuses on improving communication skills in French both orally and in writing. The aim is to increase linguistic confidence and to equip students with skills useful for social and professional life in a French environment.

This is the first of two core Spanish language modules in Final Year. During the first semester, students will work on achieving a C1 level (proficiency) as specified in the CEFR. They will be able to use the target language flexibly and effectively for social, academic, and professional purposes, using an appropriate range of registers. More specifically, students will be able to read and analyse appropriate examples of academic, professional, and literary texts in the target language and recognise implicit meaning. They will be able to produce clear, well structured, detailed texts on complex subjects, showing controlled use of organisational patterns, connectors, and cohesive devices.
In oral interactions, students will be able to express themselves fluently and spontaneously. Overall, their speaking, listening, reading, and writing skills will be improved, and their ability to translate texts and interpret short interviews/talks will be developed.

This is the second of two core Spanish language modules in Final Year. On successful completion of this module, students will have achieved at least a C1 level (proficiency) or a C2 level (mastery) as specified in the CEFR. They will be able to use the target language flexibly and effectively for social, academic, and professional purposes, using an appropriate range of registers. More specifically, students will be able to read and analyse appropriate examples of academic, professional, and literary texts in the target language and recognise implicit meaning. They will be able to produce clear, wellstructured, detailed texts on complex subjects, showing controlled use of organisational patterns, connectors, and cohesive devices.
In oral interactions, students will be able to express themselves fluently and spontaneously. Overall, their speaking, listening, reading, and writing skills will be improved, and their ability to translate texts and interpret short interviews/talks will be developed.

This final year module is a module for very advanced students who are approaching a near native-speaker level of German at level C1/C2 as specified in the Common European Framework of Reference. Students will learn to write in a variety of genres in German. Students will further improve their interpreting skills in class. Listening skills will be practiced both in class and in their own time. Students will be introduced to debate and present a topic and use various oral skills, often inspired by a previous listening exercise.

This is the second module which makes up the final-year programme in the German language. Like GRMN311, it focuses on improving communication skills in written and spoken German. Further writing genres as well as listening skills will be practiced during Textarbeit, and more debating skills will as well as employability skills and presenting will be practiced during the oral class and assessed in an oral exam. The third hour is dedicated to translating.

This module aims to consolidate understanding of the structure and functions of Chinese and related cultural knowledge and seeks to enable students to achieve an advanced level in Chinese that enables near fluent communication.

This module aims to further consolidate understanding of the structure and functions of Chinese and related cultural knowledge and seeks to enable students to achieve post-advanced level in Chinese that enables near fluent communication.

This module aims to consolidate understanding of the structure and functions of Chinese and related cultural knowledge and seeks to enable students to achieve an effective proficiency in Chinese that enables fluent communication.

This module aims to further consolidate understanding of the structure and functions of Chinese and related cultural knowledge and seeks to enable students to achieve an effective proficiency in Chinese that enables fluent communication. This module also offers HSK (the Chinese Proficiency Test) training based on HSK official guidelines to prepare you for the test which you can sit before graduation. Under our guidance and supervision, you could also prepare a CV for China-related jobs.  

The module aims to bring the students to a level of linguistic competence that is of degree standard and that will allow them to deal with a wide range of linguistic, intercultural and professional contexts confidently and competently.

This module is mapped against C1 level in Italian according to the Common European Framework of Reference for Languages (CEFR)

The module aims to bring the students to a level of linguistic competence that is of degree standard and that will allow them to deal with a wide range of linguistic, intercultural and professional contexts confidently and competently.

This module is mapped against C1+ level in Italian according to the Common European Framework of Reference for Languages (CEFR).

This module builds on the linguistic skills acquired in previous Portuguese modules and aims to consolidate students’ knowledge of Portuguese at an advanced level. The module will focus on perfecting accurate and appropriate use of Portuguese in speech, listening, reading, and writing at C1 level according the Common European Framework of Reference for Languages (CEFR).

This module builds on the linguistic skills acquired in previous Portuguese modules (PORT256, PORT278 and PORT311), and aims to consolidate students’ knowledge of Portuguese at an advanced level. The module will focus on perfecting accurate and appropriate use of Portuguese in speech, listening, reading, and writing at C1 level according the Common European Framework of Reference .

The course aims at further improving the students’ fluency in spoken Catalan and their mastery of written Catalan in all the areas of comprehension, writing and grammar.

The module builds upon Proficient Catalan 11.

The course aims at further improving the students’ fluency in spoken Catalan and their mastery of written Catalan in all the areas of comprehension, writing and grammar.

The student will also have the opportunity to achieve an extra qualification by taking the International Catalan Certificate issued by the Institut Ramon Llull and held at the University of Liverpool.

In the current age of big data, mathematics is becoming indispensable in order for us to make sense of experimental results and in order to gain a deeper understanding into mechanisms of complex biological systems. Mathematical models can provide insights that cannot be gained through experimental work alone. This module will focus on teaching students how to construct and analyse models for a wide range of biological systems. Mathematical approaches covered will be widely applicable.

Networks are familiar to us from many real-world systems such as the internet, power grids, transportation and biological networks. The underpinning mathematical concept is called a graph an it is no surprise that the same issues arise in each area, whether this is to identify the most important or influential individuals in the network, or to prevent dynamics on the network (e.g. epidemics) or to make the network robust to the dynamics it supports (e.g. power grids and transportation). In this module, we learn about different classes of networks and how to quantify and describe them including their structures and their nodes. Much of our detailed understanding of networks and their features will come from analysis of idealised random networks which nevertheless are often good representations of those seen in the real world. We will consider real-world biological applications of network theory, in particular focusing on epidemics.