Mathematics with German MMath

Core modules:

Introduction to Probability and Statistics

The module provides an introduction to the fields of probability and statistics, which form the basis of much of applicable mathematics and operations research. The theory behind probability and statistics will be introduced, along with examples occurring in diverse areas. Some of the computational statistical work may make use of the statistics package R.

20 credits

Mathematics Core 1

The module explores topics in mathematics which will be used throughout many degree programmes. The module will consider techniques for solving equations, special functions, calculus (differentiation and integration), differential equations, Taylor series, complex numbers and finite and infinite series.

20 credits

Mathematics Core II

The module continues the study of core mathematical topics begun in MS4F1015, which will be used throughout many degree programmes. The module will discuss 2-dimensional co-ordinate geometry, discussing the theory of matrices geometrically and algebraically, and will define and evaluate derivatives and integrals for functions which depend on more than one variable, with an emphasis on functions of two variables.

20 credits

Numbers and Groups

The module provides an introduction to more specialised Pure Mathematics. The first half of the module will consider techniques of proof, and these will be demonstrated within the study of properties of integers and real numbers. The second semester will study symmetries of objects, and develop a theory of symmetries which leads to the more abstract study of groups.

20 credits

German Advanced 1

Assuming a good A level in the language or equivalent, this unit aims to provide an initial preparation for a prolonged professional, academic or recreational stay in a country where the language is spoken and introduces the full range of linguistic and cultural skills required to engage in authentic and spontaneous interaction with native speakers (CEF level B2-). Based on 33 hours of small group (15-20) interactive seminars and tutorials predominantly delivered in the foreign language, the unit also comprises 67 hours of monitored private study. https://www.sheffield.ac.uk/mltc/lfa/courses/german/stage_3a

10 credits

German Advanced 2

Assuming successful completion of the Advanced 1 unit or equivalent, this unit aims to consolidate an initial preparation for a prolonged professional, academic or recreational stay in a country where the language is spoken and introduces the full range of linguistic and cultural skills required to engage in authentic and spontaneous interaction with native speakers (CEF level B2). Based on 36 hours of small group (15-20) interactive seminars and tutorials predominantly delivered in the foreign language, the unit also comprises 64 hours of monitored private study. https://www.sheffield.ac.uk/mltc/lfa/courses/german/stage_3b

10 credits

You'll then take one of these two groups of modules, as determined by the Modern Languages Teaching Centre.

German-English Tandem Advanced 1

This unit aims to enable students to develop their spoken proficiency and intercultural awareness in relation to the language they are studying through weekly meetings with a native-speaker partner and to acquire independent language learning skills through regular supervision from a language tutor and the completion of a personal planning and reflexive diary. Partners will reciprocally support each other in achieving their own pre-agreed goals, alternatively acting as learner and teacher, taking responsibility for their own learning and providing constructive opportunities for language practice and feedback to their partner. https://www.sheffield.ac.uk/mltc/lfa/courses/german/tandem

10 credits

German Project Advanced 2

This unit aims to enable students at an 'Advanced' level of competence in the language to work semi-independently, exploring critically a specialised topic normally related to their main field of study, in order to acquire subject-specific terminology and basic discursive skills in the language, as well as develop sound research skills. Through weekly tutorials, students will be guided and suppported in their completion of a written portfolio (1,500 words) and their preparation of an oral presentation summarizing their research and demonstrating their acquisition of relevant academic and linguistic skills. https://www.sheffield.ac.uk/mltc/lfa/courses/german/project_advanced

10 credits

Or:

German Project Advanced 1

This unit aims to enable students at an 'Advanced' level of competence in the language to work semi-independently, exploring critically a specialised topic normally related to their main field of study, in order to acquire subject-specific terminology and basic discursive skills in the language, as well as develop sound research skills. Through weekly tutorials, students will be guided and suppported in their completion of a written portfolio (1,500 words) and their preparation of an oral presentation summarizing their research and demonstrating their acquisition of relevant academic and linguistic skills.  https://www.sheffield.ac.uk/mltc/lfa/courses/german/project_advanced

10 credits

German-English Tandem Advanced 2

This unit aims to enable students to develop their spoken proficiency and intercultural awareness in relation to the language they are studying through weekly meetings with a native-speaker partner and to acquire independent language learning skills through regular supervision from a language tutor and the completion of a personal planning and reflexive diary. Partners will reciprocally support each other in achieving their own pre-agreed goals, alternatively acting as learner and teacher, taking responsibility for their own learning and providing constructive opportunities for language practice and feedback to their partner. https://www.sheffield.ac.uk/mltc/lfa/courses/german/tandem

10 credits

Core modules:

Advanced Calculus and Linear Algebra

Advanced Calculus and Linear Algebra are basic to most further work in pure and applied mathematics and to much of statistics. This course provides the basic tools and techniques and includes sufficient theory to enable the methods to be used in situations not covered in the course. The material in this course is essential for further study in mathematics and statistics.

20 credits

Analysis

This course is a foundation for the rigorous study of continuity, differentiation and integration of functions of one real variable. As well as providing the theoretical underpinnings of calculus, we develop applications of the theory and examples that show why the rigour is needed, even if we are focused on applications. The material in this course is vital to further studies in metric spaces, measure theory, parts of probability theory, and functional analysis.

20 credits

Statistical Inference and Modelling

This unit develops methods for analysing data, and provides a foundation for further study of probability and statistics at Level 3. It introduces some standard distributions beyond those met in MAS113, and proceeds with study of continuous multivariate distributions, with particular emphasis on the multivariate normal distribution. Transformations of univariate and multivariate continuous distributions are studied, with the derivation of sampling distributions of important summary statistics as applications. The concepts of likelihood and maximum likelihood estimation are developed. Data analysis is studied within the framework of linear models. There will be substantial use of the software package R.

20 credits

German Higher Advanced 1

Assuming a one-year post-A-level study of the language or equivalent, this module aims to provide a full linguistic and cultural preparation for learners wishing to undertake an extended professional, academic or recreational stay in a country where the language is spoken and to expand the range of linguistic and cultural skills required to operate effectively in most situations encountered when engaging with native speakers (CEF level B2+). Based on 33 hours of small group (15-20) interactive seminars almost exclusively delivered in the foreign language, the unit also comprises 67 hours of monitored private study. https://www.sheffield.ac.uk/mltc/lfa/courses/german/stage_4a

10 credits

German Higher Advanced 2

Assuming successful completion of the Higher Advanced 1 unit or equivalent, this module aims to consolidate the full linguistic and cultural preparation of learners wishing to undertake an extended professional, academic or recreational stay in a country where the language is spoken and to consolidate the full range of linguistic and cultural skills required to operate effectively in most situations encountered when engaging with native speakers (CEF level C1). Based on 36 hours of small group (15-20) interactive seminars almost exclusively delivered in the foreign language, the unit also comprises 64 hours of monitored private study. https://www.sheffield.ac.uk/mltc/lfa/courses/german/stage_4b

10 credits

German Placement Preparation

The aim of this unit is to provide linguistic, cultural and practical preparation to students about to undertake an extended work placement or study exchange abroad. Students will research their destination (region, institution, etc.) and learn how to carry out a broad range of practical and study-related tasks using the language, including securing a residence or work permit, travelling throughout the country, arranging accommodation, registering for a course of study, opening a bank account, registering with and consulting a doctor, translating forms and other documents, taking lecture and seminar notes, writing letters, reports and/or essays, etc. https://www.sheffield.ac.uk/mltc/lfa/courses/german/placement_preparation

10 credits

Optional modules:

Algebra

This unit continues the study of abstract algebra begun in MAS114, going further with the study of groups, and introducing the concepts of a ring, which generalises the properties of the integers, and a vector space, which generalises the techniques introduced in linear algebra to many more examples. As well as demonstrating the interest and power of abstraction, this course is vital to further studies in most of pure mathematics, including algebraic geometry and topology, functional analysis and Galois theory.

20 credits

Mathematics and Statistics in Action

This module will demonstrate, in a series of case studies, the use of applied mathematics, probability and statistics in solving a variety of real-world problems. The module will illustrate the process of mathematical and statistical modelling, whereby real-world questions are translated to mathematical and/or statistical questions. Students will see how techniques learned earlier in their degree, as well as simple computer programming, can be used to explore these problems. There will be a mix of individual and group projects, and some projects will involve the use of R or Python, but MAS115 is not a prerequisite.

10 credits

Probability Modelling

The course introduces a number of general models for processes where the state of a system is fluctuating randomly over time. Examples might be the length of a queue, the size of a reproducing population, or the quantity of water in a reservoir. The aim is to familiarize students with an important area of probability modelling.

10 credits

You'll then take one of these two groups of modules, as determined by the Modern Languages Teaching Centre.

German-English Tandem Higher Advanced 1

Students will be required to work in collaboration with a native-speaker with whom they will communicate in the target language. At the start of the module, students will exercise responsibility for the organisation of their own learning, establish and maintain contact with their partners, negotiate and set objectives; and seek and offer the correction of language errors. They should give proof of effective time-management, sequence sessions logically to demonstrate management of learning and demonstrate use of reviewing and evaluating procedures. They will be required to sign a learning contract and to keep a learner diary, in which they will record progress made, plan their next steps and reflect on their work during the semester. Their progress will be monitored in advisory and counselling sessions with MLT Centre Tutors. https://www.sheffield.ac.uk/mltc/lfa/courses/german/tandem

10 credits

German Project Higher Advanced 2

The aims are to enable students to work independently and collaboratively in German, to foster the acquisition of subject-specific terminology and discourse in German; to encourage interaction with native speakers of German, by e-mail or face-to-face. Students will be expected to investigate an agreed topic closely related to their main course of study. This will entail devising a project proposal and plan, gathering information from a range of written and spoken sources (normally including German native-speakers), and synthesising this to form the basis of a formal written report and a structured oral presentation. On completion, students should be able to: collaborate successfully with native-speakers of German: use international e-mail links to do so where appropriate; analyse information, distinguishing fact from opinion and main ideas from supporting detail; and present findings in speech and writing using formal register and - where appropriate - subject-specific language. https://www.sheffield.ac.uk/mltc/lfa/courses/german/project_higher_advanced

10 credits

Or:

German Project Higher Advanced 1

The aims are to enable students to work independently and collaboratively in German, to foster the acquisition of subject-specific terminology and discourse in German; to encourage interaction with native speakers of German, by e-mail or face-to-face. Students will be expected to investigate an agreed topic closely related to their main course of study. This will entail devising a project proposal and plan, gathering information from a range of written and spoken sources (normally including German native-speakers), and synthesising this to form the basis of a formal written report and a structured oral presentation. On completion, students should be able to: collaborate successfully with native-speakers of German: use international e-mail links to do so where appropriate; analyse information, distinguishing fact from opinion and main ideas from supporting detail; and present findings in speech and writing using formal register and - where appropriate - subject-specific language.  https://www.sheffield.ac.uk/mltc/lfa/courses/german/project_higher_advanced

10 credits

German-English Tandem Higher Advanced 2

Students will be required to work in collaboration with a native speaker with whom they will communicate in the target language. At the start of the module, students will exercise responsibility for the organisation of their own learning, establish and maintain contact with their partners, negotiate and set objectives; and seek and offer the correction of language errors. They should give proof of effective time-management, sequence sessions logically to demonstrate management of learning and demonstrate use of reviewing and evaluating procedures. They will be required to sign a learning contract and to keep a learner diary, in which they will record progress made plan their next steps and reflect on their work during the semester. Their progress will be monitored in advisory and counselling sessions with MLT Centre tutors.  https://www.sheffield.ac.uk/mltc/lfa/courses/german/tandem

10 credits

Core modules:

Study Abroad

contact convenor for more infortmation

100 credits

German Year Abroad 1

This unit aims to enable students to heighten their awareness of cultural and linguistic aspects of the country where they are studying, encouraging their social integration and fostering their independent and/or collaborative study in the foreign language. Students will complete a diary in which they will record and reflect upon their linguistic and cultural integration progress over a period of at least 10 weeks. They will also contribute to a blog intended to share language learning advice and practical tips about living in the region and country with other students currently abroad or preparing for a year abroad. https://www.sheffield.ac.uk/mltc/lfa/courses/german/year_abroad

10 credits

German Year Abroad 2

This unit aims to enable students to heighten their awareness of cultural and linguistic aspects of the country where they are studying, encouraging their social integration and fostering their independent and/or collaborative study in the foreign language. Students will complete a diary in which they will record and reflect upon their linguistic and cultural integration progress over a period of at least 10 weeks. They will also contribute to a blog intended to share language learning advice and practical tips about living in the region and country with other students currently abroad or preparing for a year abroad. https://www.sheffield.ac.uk/mltc/lfa/courses/german/year_abroad

10 credits

Core modules:

Mathematics and Statistics Project II

This unit forms the final part of the SoMaS project provision at Level 4 and involves the completion, under the guidance of a research active supervisor, of a substantial project on an advanced topic in Mathematics and Statistics.

30 credits

German Proficient 1

Assuming a solid, two year post-A-level study of the language and, ideally, a prolonged stay in a country where the language is spoken, this unit aims to provide the linguistic and cultural skills required to operate as a near-native speaker in the target country, whether for professional, academic or recreational purposes, and to develop strategies and techniques to become a fully autonomous, life-long learner of the language and culture (CEF level C1+). Based on 33 hours of small group (15-20) interactive seminars almost exclusively delivered in the foreign language, the unit also comprises 67 hours of monitored private study. https://www.sheffield.ac.uk/mltc/lfa/courses/german/stage_5a

10 credits

German Proficient 2

Assuming successful completion of the Proficient 1 unit or equivalent, this unit aims to perfect the linguistic and cultural skills required to operate as a near-native speaker in the target country, whether for professional, academic or recreational purposes, and to consolidate the strategies and techniques to become a fully autonomous, life-long learner of the language and culture (CEF level C2). Based on 36 hours of small group (15-20) interactive seminars almost exclusively delivered in the foreign language, the unit also comprises 64 hours of monitored private study. https://www.sheffield.ac.uk/mltc/lfa/courses/german/stage_5b

10 credits

Project Presentation in Mathematics and Statistics

The unit provides training and experience in the use of appropriate computer packages for the presentation of mathematics and statistics and guidance on the coherent and accurate presentation of technical information.

10 credits

Optional modules:

Bayesian Statistics and Computational Methods

This module introduces the Bayesian approach to statistical inference. The Bayesian method is fundamentally different in philosophy from conventional frequentist/classical inference, and has been the subject of some controversy in the past, but is now widely used. The module also presents various computational methods for implementing both Bayesian and frequentist inference, in situations where obtaining results ‘analytically’ would be impossible. The methods will be implemented using the programming languages R and Stan, and some programming is taught alongside the theory lectures.

30 credits

Algebraic Topology

This unit will cover algebraic topology, following on from MAS331: Metric Spaces. Topology studies the shape of space, with examples such as spheres, the Mobius Band, the Klein bottle, the torus and other surfaces. The first task is to formalise this notion of space, and to work out when a given space can be smoothly deformed into another, where stretching and bending is allowed, but cutting and glueing is not. Algebraic topology builds a powerful bridge between space and algebra, enabling the use familiar algebraic techniques from group theory to study spaces and their deformations.

20 credits

Commutative Algebra and Algebraic Geometry

This module develops the theory of algebraic geometry, especially over complex numbers, from both a geometrical and algebraic point of view. The main ingredient is the theory of commutative algebra, which is developed in the first part of the module.

20 credits

Functional Analysis

Functional analysis is the study of infinite-dimensional vector spaces equipped with extra structure. Such spaces arise naturally as spaces of functions. As well as being a beautiful subject in its own right, functional analysis has numerous applications in other areas of both pure and applied mathematics, including Fourier analysis, study of the solutions of certain differential equations, stochastic processes, and in quantum physics. In this unit we focus mainly on the study of Hilbert spaces- complete vector spaces equipped with an inner product- and linear maps between Hilbert spaces. Applications of the theory considered include Fourier series, differential equations, index theory, and the basics of wavelet analysis.

20 credits

Stochastic Processes and Finance

A stochastic process is a mathematical model for phenomena unfolding dynamically and unpredictably over time. This module studies two classes of stochastic process particularly relevant to financial phenomena: martingales and diffusions. The module develops the properties of these processes and then explores their use in Finance. A key problem considered is that of the pricing of a financial derivative such as an option giving the right to buy or sell a stock at a particular price at a future time. What is such an option worth now? Martingales and stochastic integration are shown to give powerful solutions to such questions.

20 credits

Generalised Linear Models

This module introduces the theory and application of generalised linear models. These models can be used to investigate the relationship between some quantity of interest, the “dependent variable”, and one or more “explanatory” variables; how the dependent variable changes as the explanatory variables change. The term “generalised” refers to the fact that these models can be used for a wide range of different types of dependent variable ,continuous, discrete, categorical, ordinal etc. The application of these models is demonstrated using the programming language R.

15 credits

Machine Learning

Machine learning lies at the interface between computer science and statistics. The aims of machine learning are to develop a set of tools for modelling and understanding complex data sets. It is an area developed recently in parallel between statistics and computer science. With the explosion of “Big Data”, statistical machine learning has become important in many fields, such as marketing, finance and business, as well as in science. The module focuses on the problem of training models to learn from training data to classify new examples of data.

15 credits

Medical Statistics

This module introduces an important application of statistics: medical research, specifically, the design and analysis of clinical trials. For any new drug to be approved by a regulator (such as the Medicines and Healthcare products Regulatory Agency in the UK) for use on patients, the effectiveness of the drug has to be demonstrated in a clinical trial. This module explains how clinical trials are designed and how statistical methods are used to analyse the results, with a particular focus on ‘survival’ or ‘time-to-event’ analysis.

15 credits

Sampling Theory and Design of Experiments

Whereas most statistics modules are concerned with the analysis of data, this module is focussed on the collection of data. In particular, this module considers how to collect data efficiently: how to ensure the quantities of interest can be estimated sufficiently accurately, using the smallest possible sample size. Three settings are considered: sample surveys (for example when conducting an opinion poll), physical experiments, as may be used in industry, and experiments involving predictions from computer models, where there is uncertainty in the computer model prediction.

15 credits

Advanced Operations Research

Mathematical Programming is concerned with the algorithms that deal with constrained optimisation problems. We consider only constraints which are linear, and for which the objective function to be maximised or minimised is also linear. These problems are not continuously differentiable and so they do not fall into the category of problems considered in organisation; special algorithms have to be developed. The module considers not only the solution of such problems but also the important area of post-optimality analysis; i.e. given the solution can one answer questions about the effect of small changes in the parameters of the problem (such as values of the cost coefficients)? Additional topics will include the transportation and assignment problems.

10 credits

Analytic Number Theory

The module will discuss the distribution of prime numbers (Bertrand's Postulate, prime counting function, the statement of the Prime Number Theorem and some of its consequences), basic properties of the Riemann zeta function, and Euler products of L-series. A big chunk of the module will be dedicated to Dirichlet's Theorem on primes in arithmetic progressions and it's proof.

10 credits

Fields

A field is set where the familiar operations of arithmetic are possible. It is common, particularly in the study of equations, that a field may need to be extended. This module will study the idea of field extension and the various problems that may arise as a result. Particular use is made of this to answer some of the classical problems of Greek geometry, to ask whether certain geometrical constructions such as angle trisection or squaring the circle are possible.

10 credits

Financial Mathematics

The discovery of the Capital Asset Pricing Model by William Sharpe in the 1960's and the Black-Scholes option pricing formula a decade later mark the beginning of a very fruitful interaction between mathematics and finance. The latter obtained new powerful analytical tools while the former saw its knowledge applied in new and surprising ways. (A key result used in the derivation of the Black-Scholes formula, Ito's Lemma, was first applied to guide missiles to their targets; hence the title 'rocket science' applied to financial mathematics). This course describes the mathematical ideas behind these developments together with their application in modern finance, and includes a computational project where students further explore some of the ideas of option pricing.

10 credits

Galois Theory

Given a field K (as studied in MAS333/MAS438) one can consider the group G of isomorphisms from K to itself. In the cases of interest, this is a finite group, and there is a tight link (called the Galois correspondence) between the structure of G and the subfields of K. If K is generated over the rationals by the roots of a polynomial f(x), then G can be identified as a group of permutations of the set of roots. One can then use the Galois correspondence to help find formulae for the roots, generalising the standard formula for the roots of a quadratic. It turns out that this works whenever the degree of f(x) is less than five. However, the fifth symmetric group lacks certain group-theoretic properties that lie behind these formulae, so there is no analogous method for solving arbitrary quintic equations. The aim of this course is to explain this theory, which is strikingly rich and elegant.

10 credits

Measure and Probability

The module will give students an additional opportunity to develop skills in modern analysis as well as providing a rigorous foundation for probability theory. In particular it would form a companion course to MAS436 (Functional Analysis) and MAS452 (Stochastic Processes and Finance), the latter of which is fundamentally dependent on measure theoretic ideas.

10 credits

Optics and Symplectic Geometry

This course is an introduction to some of the areas of pure mathematics which have evolved from the mathematical study of optics. Optics provides a unifying thread, but no prior knowledge of the properties of light is required. Mathematical topics covered include symplectic structures on vector spaces, symplectic maps and matrices, Lagrangian subspaces and characteristic functions and, if time permits, an introduction to the Maslov class and/or Symplectic manifolds. In terms of optics we cover Gaussian, linear and geometrical optics and (if time permits) an introduction to aberration.

10 credits

Signal Processing

The transmission reception and extraction of information from signals is an activity of fundamental importance. This course describes the basic concepts and tools underlying the discipline, and relates them to a variety of applications. An essential concept is that a signal can be decomposed into a set of frequencies by means of the Fourier transform. From this grows a very powerful description of how systems respond to input signals. Perhaps the most remarkable result in the course is the celebrated Shannon-Whittaker sampling theorem, which tells us that, under certain conditions, a signal can be perfectly reconstructed from samples at discrete points. This is the basis of all modern digital technology.

10 credits

You'll then take one of these two groups of modules, as determined by the Modern Languages Teaching Centre.

German Project Proficient 1

This unit aims to enable students at a 'Proficient' level of competence in the language to engage critically and independently with a highly specialised topic normally closely related to their main field of study in order to acquire expert use of subject-specific terminology and discourse and to consolidate advanced research skills in relation to the language and field of study. Working under regular supervision, students will complete a written portfolio (2,500 - 3,000 words) and prepare a structured oral presentation summarizing authoritatively their research for a specialist audience and demonstrating their acquisition of relevant academic and linguistic skills. https://www.sheffield.ac.uk/mltc/lfa/courses/german/project_proficient

10 credits

German-English Tandem Proficient 2

Students will be required to work in collaboration with a native-speaker with whom they will communicate in the target language. At the start of the module, students will exercise responsibility for the organisation of their own learning, establish and maintain contact with their partners, negotiate and set objectives; and seek and offer the correction of language errors. They should give proof of effective time-management, sequence sessions logically to demonstrate management of learning and demonstrate use of reviewing and evaluating procedures. They will be required to sign a learning contract and to keep a learner diary in the target language in which they will record progress made, plan their next steps and reflect on their work during the semester. Their progress will be monitored in advisory and conselling sessions with MLT Centre Tutors. https://www.sheffield.ac.uk/mltc/lfa/courses/german/tandem

10 credits

Or:

German-English Tandem Proficient 1

Students will be required to work in collaboration with a native-speaker with whom they will communicate in the target language. At the start of the module, students will exercise responsibility for the organisation of their own learning, establish and maintain contact with their partners, negotiate and set objectives; and seek and offer the correction of language errors. They should give proof of effective time-management, sequence sessions logically to demonstrate management of learning and demonstrate use of reviewing and evaluating procedures. They will be required to sign a learning contract and to keep a learner diary in the target language in which they will record progress made, plan their next steps and reflect on their work during the semester. Their progress will be monitored in advisory and counselling sessions with MLT Centre Tutors. https://www.sheffield.ac.uk/mltc/lfa/courses/german/tandem

10 credits

German Project Proficient 2

This unit aims to enable students at a 'Proficient' level of competence in the language to engage critically and independently with a highly specialised topic normally closely related to their main field of study in order to acquire expert use of subject-specific terminology and discourse and to consolidate advanced research skills in relation to the language and field of study. Working under regular supervision, students will complete a written portfolio (2,500 - 3,000 words) and prepare a structured oral presentation summarizing authoritatively their research for a specialist audience and demonstrating their acquisition of relevant academic and linguistic skills. https://www.sheffield.ac.uk/mltc/lfa/courses/german/project_proficient

10 credits