German Studies and Mathematics

This module comprises of both oral and aural skills, to be taken alongside the Written Skills module. It builds upon skills gained in the first year.

This module aims to enhance your linguistic proficiency in spoken German in a range of formal and informal settings (both spontaneous and prepared). Specific attention will be given to developing good, accurate pronunciation and intonations well as fluency, accuracy of grammar, and vocabulary when speaking the language.

This module also aims at broadening students’ knowledge about different aspects of modern society, politics and culture, and contemporary issues and institutions in order to prepare them for residence abroad in their 3rd year.

By the end of this module, you will have had the opportunity to enhance your comprehension of the spoken language, as used in both formal speech, and in everyday life situations including those that you may encounter in German-speaking countries.

This module comprises of both oral and aural skills, to be taken alongside the corresponding Written Language module. It builds upon skills gained in the first year. Students who have taken the Intensive language course in their first year, normally follow this course throughout the second year.

The module aims to enhance your linguistic proficiency in spoken German in a range of formal and informal settings (both spontaneous and prepared). Specific attention will be given to developing good, accurate pronunciation and intonations well as fluency, accuracy of grammar, and vocabulary when speaking the language.

This module also aims at broadening your knowledge about different aspects of modern society, politics and culture, and contemporary issues and institutions in order to prepare them for residence abroad in their 3rd year.

By the end of this module, you will have had the opportunity to enhance your comprehension of the spoken language, as used in both formal speech, and in everyday life situations including those that you may encounter in German-speaking countries.

This module comprises of reading and writing skills to be taken alongside the Oral Skills module.

This module aims to consolidate skills you have hopefully developed in the first year of study, and enable them to build a level of competence and confidence required to familiarise yourselves with the culture and society of countries where your studied language is spoken.

The module aims to enhance your proficiency in the writing of German (notes, reports, summaries, essays, projects, etc.) including translation from and into German; and the systematic study of German lexis, grammar and syntax.

The module aims to enhance your linguistic proficiency, with particular emphasis on reading a variety of sources and on writing fluently and accurately in the language, in a variety of registers.

This module comprises of reading and writing skills to be taken alongside the Oral Skills module.

This module aims to consolidate skills you have hopefully developed in the first year of study, and gives you the opportunity to build a level of competence and confidence required to familiarise yourselves with the culture and society of countries where your studied language is spoken.

The module aims to enhance your proficiency in understanding spoken German, as well as in the writing of German (notes, reports, summaries, essays, projects, etc.) including translation from and into German; and the systematic study of German lexis, grammar and syntax.

The module aims to enhance your linguistic proficiency, with particular emphasis on reading a variety of sources and on writing fluently and accurately in the language, in a variety of registers.

This module is a non-credit bearing module. If you are a major student going abroad in your second or third year you are enrolled on it during the year prior to your departure, and timetabled to attend the events. These include: introduction to the International Placement Year and choice of activities; British Council English Language Assistantships and how to apply; introduction to partner universities and how they function; working in companies abroad; finance during the International Placement Year; research skills and questionnaire design; teaching abroad; curriculum writing and employability skills; and welfare and wellbeing.

What has it meant to be German over the past two hundred years? This module provides students with awareness of the ‘must-know’ moments and movements that have shaped contemporary German-language culture, while developing skills in critical-textual analysis, including film, poetry, drama, and political writing. It takes you on a journey through moments and movements across two centuries of cultural history, encountering along the way a series of radical thinkers, writers, filmmakers and creative artists. The main aim of the module is twofold: to build students' reading knowledge of German while giving them a flavour of the rich cultural output that has defined the modern German-speaking realm.

This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here, students will select a small number of properties which these and other examples have in common, and use them to define a group.

They will also consider the elementary properties of groups. By looking at maps between groups which 'preserve structure', a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same' will be discovered.

Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives students a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials; several less familiar examples will also be explored.

Complex Analysis has its origins in differential calculus and the study of polynomial equations. In this module, students will consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. They will use integral calculus of complex functions to find elegant and important results and will also use classical theorems to evaluate real integrals.

The first part of the module reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The module then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every non-constant complex polynomial has a root. Finally, the theory is employed to evaluate some definite integrals.The module ends with basic discussion of harmonic functions, which play a significant role in physics.

Students will gain a solid understanding of computation and computer programming within the context of maths and statistics. This module expands on five key areas:

• Programming and R
• Numerical solutions of equations
• Numerical differentiation and integration
• Monte Carlo methods
• Numerical solutions to ODEs

Under these headings, students will study a range of complex mathematical concepts, such as: data structures, fixed-point iteration, higher dimensions, first and second derivatives, non-parametric bootstraps, and modified Euler methods.

Throughout the module, students will gain an understanding of general programming and algorithms. They will develop a good level of IT skills and familiarity with computer tools that support mathematical computation.

Over the course of this module, students will have the opportunity to put their knowledge and skills into practice. Workshops, based in dedicated computing labs, allow them to gain relatable, practical experience of computational mathematics.

This module explores how post-war economic change has affected European societies, and how socio-political factors in turn have influenced the patterns and outcomes of economic development, over the second half of the twentieth and the beginning of the twenty first century.

The module is structured on the basis of three country-specific modules (France, Germany and Spain), examining how these countries have confronted key moments of economic change, and what the longer-term consequences of that change have been. While the module emphasis is on broad national developments, discussion also covers examples relating to particular industries and major companies.

In lectures, workshops, and seminars we will explore the context of reconstruction after World War II and the pattern of subsequent economic development; the relationship between social and economic policies; the development of the three countries' economies; the changes of the 1980s and their impact on subsequent years; and the consequences of specific momentous events, such as the re-unification of Germany and how the financial crisis of 2008 affected, and still affects, France, Germany and Spain.

This module will introduce you to the role that the language used by institutions plays in shaping individual perceptions of identity. It aims to provide you with a basic theoretical framework that allows them to understand the relationship between language and power as reflected in current language policies at regional, national, and supranational levels. It will give you the opportunity to recognise forms of prestige and stigma associated with varieties of the three main languages under study. We aim for this module to raise critical awareness of the portrayal and representation of linguistic variations in the media and in the sphere of literature.

The main topics covered in the course include Language and Power; European language policies; German as a pluricentric language and ‘Gastarbeiter’ language and policies; regional variations of France: Linguistic Diversity: A threat to French National Identity?; The languages and language attitudes of Spain (Castilian Spanish, Catalan, Basque, Galician).

This module is taught in English.

Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules. This module will give students the opportunity to study vector spaces, together with their structure-preserving maps and their relationship to matrices.

They will consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of their study will also involve looking at the concepts of length and angle with regard to vector spaces.

Probability provides the theoretical basis for statistics and is of interest in its own right.

Basic concepts from the first year probability module will be revisited and extended to these to encompass continuous random variables, with students investigating several important continuous probability distributions. Commonly used distributions are introduced and key properties proved, and examples from a variety of applications will be used to illustrate theoretical ideas.

Students will then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.

This module seeks to support you to apply your linguistic and cultural understanding in a specific professional context. This module gives you the opportunity to spend time on a work-based placement in the UK or abroad. You will be given the opportunity to develop, reflect on and articulate both the range of competences and the linguistic and cross-cultural skills that enhance employability by working in language-related professional contexts and reflecting on key issues in relation to their placement organisation. There is the opportunity to join a local work placement developed by the department, or for you to source your own placements (subject to departmental approval). Workshops before and during the placement will provide preparation and guidance on sourcing, confirming and then reflecting on academic work. Students will share their experiences and learning with each other by means of end-of-module presentations.

Project Skills is a module designed to support and develop a range of key technical and professional skills that will be valuable for all career paths. Covering five major components, this module will guide students through and explore:

• Mathematical programmes
• Scientific writing
• Communication and presentation skills
• Individual projects
• Group projects

Students will gain an excellent grasp of LaTeX, learning to prepare mathematical documents; display mathematical symbols and formulae; create environments; and present tables and figures.

Scientific writing, communication and presentations skills will also be developed. Students will work on short and group projects to investigate mathematical or statistical topics, and present these in written reports and verbal presentations.

A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at A-level.

Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.

Next, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in A-level, such as estimations of discrete sums of series.

Further possible topics include Stirling's Formula, infinite products and Fourier series.

How do films deal with topics such as immigration, environment, the posthuman and gender? Do they entertain viewers, instruct them, or both?

This module explores European, Latin American, and Chinese films in their social and historical contexts; the topics mentioned are the focus of key lectures and seminars. The module begins with introductory lectures on cinema and society and on film aesthetics and content. The main aim is to make connections between the films and such contexts not only on the level of narrative, characterisation and dialogue, but also on that of form and technique.

Statistics is the science of understanding patterns of population behaviour from data. In the module, this topic will be approached by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.

The focus will be on likelihood-based parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and linear regression techniques within the statistical modelling framework will also be considered.

Students will come to recognise the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions.

This module aims to give you a background to and insight into the diversity of twentieth and twenty-first century thought and contemporary definitions of culture.

Some key questions explored on the module include: What is 'culture' and how does it work? How do 'art' and 'culture' relate to each other? What do we mean when we talk about the production and consumption of culture? Why does popular culture arouse conflicting responses? What role does the body play in our understanding of culture? How does culture define who we are? Can a work of culture be an act of resistance?

With these questions in mind, this module focuses on texts which raise questions about class, race, gender, and subcultures.

As part of The International Placement Year you will normally spend at least eight months abroad in your third year. You will have the opportunity to:

• analyse the contemporary relevance of a tradition, contemporary social, political or economic issue, or a living part of the regional culture.
• reflect critically on cultural differences observed in everyday life such as social relationships, politics, attitudes to food, drink, religion, etc., explaining them in the context of various historical, social and cultural developments.
• think analytically about your intercultural position and understanding of the relevant culture(s).
• reflect on language use (different registers, varieties of pronunciation and accents, dialects, vocabulary and idiomatic expressions, and aspects of grammar) and the process of the acquisition of skills in the relevant language(s).

The module also aims to enhance and develop your language skills, with all assessments being written in the target language. If you have started a language as a beginner in year one you will spend a minimum of four months in a country where that language is spoken. If you are a joint honours student who is studying two languages, you may choose to spend the year in either of the two countries concerned or, if appropriate arrangements can be made, you can spend a semester in each country.

Lancaster University will make reasonable endeavours to place students at an approved overseas partner. Students conduct either a study placement at a partner University, a teaching assistantship placement with the British Council or an appropriate working placement with a vetted employer abroad or a combination of placements (please note that there are some restrictions on British Council placements which usually last for the whole of the academic year).

Joint honours degrees

If you are a joint honours student who is combining a language with a non-language subject, your placement year will provide the opportunity to develop your language skills and cultural awareness, but will not necessarily relate to the non-language aspect of your degree.

Lancaster University cannot accept responsibility for any financial aspects of your International Placement Year.

This module is integrated with the German Language: Written Skills module.

Both the oral and the written language modules focus on particular topics of cultural and contemporary interest. The general aim of these modules is to develop further the abilities you have hopefully gained during their second year and the year abroad.

By the end of this module, we aim for you to have developed an informed interest in the society and culture of the German-speaking world. We hope you will also have acquired almost native-speaker abilities in both spoken and written language.

This module is integrated with the German Language: Oral Skills module.

This module has two main aims. The first one is to enhance students’ linguistic proficiency with emphasis on the understanding of spoken and written German, the speaking of German (prepared and spontaneous) in both formal and informal settings, the writing of German, and the systematic study of German lexis, grammar, and syntax. The second aim is to increase your awareness, knowledge, and understanding of contemporary Germany.

By the end of this module, we aim for you to have developed an informed interest in the society and culture of the German-speaking world. We hope you will also have acquired almost native-speaker abilities in both spoken and written language.

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

In this module, students will be introduced to the fundamental topics of combinatorial enumeration (sophisticated counting methods), graph theory (graphs, networks and algorithms) and combinatorial design theory (Latin squares and block designs). They will also explore important practical applications of the results and methods.

Students’ knowledge of commutative rings as gained from their second year of study in Rings and Linear Algebra will be built upon, and an introduction to the fourth year Galois Theory module will be provided.

They will be introduced to two new classes of integral domains called Euclidean domains, where they have a counterpart of the division algorithm, and unique factorisation domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.

How well-known concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorisation of polynomials, carry over to this new setting, will also be explored.

This module introduces you to major themes that shape the experience of contemporary city dwellers: gender, social inequality, and practices of citizenship. These interlinking themes will be introduced through novels, poetry and films on the following European, North American (with the emphasis on immigrant communities within its cities) and Latin American cities: New York, Mexico City, Santiago de Chile, Barcelona, Berlin, and Los Angeles.

Each topic will be covered though an introductory lecture and a core text, followed by a range of additional texts for students to analyse. During workshops students will share their findings and opinions, emphasizing on identifying links between the topics studied, aiming to encourage discussion.

The format of the module encourages cross-referencing between the themes of the module (for example, gender and sexuality are relevant to an analysis of social inequality, and vice versa).

Questions relating to linear ordinary differential equations will be considered during this module. Differential equations arise throughout the applications of mathematics, and consequently the study of them has always been recognised as a fundamental branch of the subject. The module aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations, and theoretical results about the nature of solutions.

While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations, as well as allowing us to study certain properties of these solutions.

This module is assessed entirely through coursework. You are given a chance of pursuing a topic of their own interest, which is not covered in taught options. A dissertation consists of approximately 10,000 words written in English. The topic of dissertation must relate to French/German/Spanish language, or a comparison between two or more, or a general European issue. The other two restrictions on topic choice are: it must be capable of and approached from a serious academic angle and it falls within the range of expertise of a member of the Department’s staff.

Each student will be assigned a supervisor - one of the lecturers from the Department, who will provide regular supervision, and feedback on the first draft of the completed dissertation. The topic is agreed and discussed with the supervisor in the Summer Term of the second year, and preparatory research should begin during the Year Abroad.

The topic of smooth curves and surfaces in three-dimensional space is introduced. The various geometrical properties of these objects, such as length, area, torsion and curvature, will be explored and students will have the opportunity to discover the meaning of these quantities. They will then use a variety of examples to calculate these values, and will use those values to apply techniques from calculus and linear algebra.

A number of well-known concepts will be encountered, such as length and area, and some new ideas will be introduced, including the curvature and torsion of a curve, and the first and second fundamental forms of a surface. Students will learn how to compute these quantities for a variety of examples, and in doing so will develop their geometric intuition and understanding.

Students will develop the knowledge of groups that they gained in second year during the Groups and Rings module. ‘Direct products’, which are used to construct new groups, will be studied, while any finite group will be shown to ‘factor’ into ‘simple’ pieces.

Situations will be considered in which a group ‘acts’ on a set by permuting its elements; this powerful idea is used to identify the symmetries of the Platonic solids, and to help study the structure of groups themselves.

Finally, students will prove some interesting and important results, known as 'Sylow’s theorems', relating to subgroups of certain orders.

Students will examine the notion of a norm, which introduces a generalised notion of ‘distance’ to the purely algebraic setting of vector spaces. They will learn several quite natural ways to do this, both for vectors of any dimension and for functions. Focus will then be on the more special notion of an inner product which generalises angles at the same time as distances.

Once these concepts have been established, students will have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three (or even n) dimensions, notably to infinite dimensional spaces of functions. For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how people can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, students will look at some of the main results of linear algebra, which generalise very nicely to linear operators between Hilbert spaces.

This module aims at exploring the nature of the relationship between the individual and society, notions of progress and economic justice, as these are still widely debated topics in contemporary Europe in light of the current economic and political crisis.

This module will use the concepts of utopia, dystopia and ideology as a forum for discussion on the relationship between individual imagination and social discourse in the nineteenth century, as well as the relationship between fiction and political discourse. You will look at the major intellectual debates which influenced the contemporary European thought after the French Revolution.

You will explore the development of major ideologies and cultural movements such as Romanticism, Marxism, Socialism and Positivism, spanning from the period immediately following the French Revolution to the middle of the nineteenth century.

The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics modules and provide opportunities for further study. The theory of linear systems is engineering mathematics.

In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.

Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The module shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables students to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute.

The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilise a (A,B,C,D) system.

Students will be given an opportunity to consider key issues in the teaching and learning of mathematics during this module. Whilst it is an academic study of mathematics education and not a training course for teachers, it does provide an excellent foundation for a PGCE especially in preparing students to write academically.

Having studied mathematics for many years, students are 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. This module will help them with this process so that as mathematics graduates they will be able to contribute knowledgeably to future debate about the ways in which this subject is treated within the education system.

An introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications, is given during this module. Studying this module will give students a deeper understanding of continuity as well as a basic grounding in abstract topology. With this grounding, they will be able to solve problems involving topological ideas, such as continuity and compactness.

They will also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply their knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

Number theory is the study of the fascinating properties of the natural number system.

Many numbers are special in some sense, eg. primes or squares. Which numbers can be expressed as the sum of two squares? What is special about the number 561? Are there short cuts to factorizing large numbers or determining whether they are prime (this is important in cryptography)? The number of divisors of an integer fluctuates wildly, but is there a good estimation of the ‘average’ number of divisors in some sense?

Questions like these are easy to ask, and to describe to the non-specialist, but vary hugely in the amount of work needed to answer them. An extreme example is Fermat’s last theorem, which is very simple to state, but was proved by Taylor and Wiles 300 years after it was first stated. To answer questions about the natural numbers, we sometimes use rational, real and complex numbers, as well as any ideas from algebra and analysis that help, including groups, integration, infinite series and even infinite products.

This module introduces some of the central ideas and problems of the subject, and some of the methods used to solve them, while constantly illustrating the results with exercises and examples involving actual numbers.

This module lives in the space between the here-and-now and a future made possible by science. You’ll explore perceptions of science across different languages and cultures, from Asia to Europe to the Americas, and explore relationships between the spiritual and the material.

You’ll look at some intriguing questions about science and the twenty-first century human condition:

• Where is AI taking humanity and are we already robots?
• Might genetic engineering lead to animals developing souls?
• Are science fiction writers contemporary shaman?
• What unseen matters and horrors can science fiction render visible and comprehensible?
• What possibilities do modern medical advances offer for transformative queer and trans healthcare?

You’ll find out about differing views on these and other topics from a wide range of source materials, such as speculative fiction, graphic novels, film, philosophical essays, and online talks.

Typically, the module covers five main themes:

• Spirit and Matter
• Speculative Fiction
• The Post-Human
• Philosophy, Art and Neuroscience
• Biomedicine and the Hospital

This module guides students through the history of German-language speculative fiction as a way of opening up new discussions about the history and politics of German-speaking Europe. It explores the relationship between aesthetics, socio-literary engagement and the future and encompasses a range of sources from the early nineteenth century to the present day. Students also engage with philosophical texts theorising about temporality and the future.

What makes a good translation and how do translations do good? This module aims to help you understand the practice of translation as it has evolved historically from the 18th century to the present across European and American societies. The materials we study include historical textual sources (philosophical essays on the craft of translation from French, German and Hispanic authors of the 19th and 20th centuries), representative fictional texts reflecting on translation processes, and contemporary documents from the EU directorate on translation, PEN and the Translators' Association. We will also make considerable use of contemporary online resources as exemplified by Anglophone advocates of intercultural exchange such as Words Without Borders. Our aim is to look at translation as both a functional process for getting text in one language accurately into another and a culturally-inflected process that varies in its status and purpose from one context to another. We will pay particular attention to the practical role that literary translators play within the contemporary global publishing industry and consider the practicalities of following a career in literary translation in the Anglophone world.