Discrete Mathematics BSc Hons “Warwick University has a great community and there are always lots of opportunities to get involved in a diverse range of activities and events.” Rhys Davis (2009-2012) Discrete Mathematics (BSc Hons) What is Discrete Mathematics? Discrete Mathematics is a dynamic and interesting branch of Mathematics with many existing and emerging applications. It deals with mathematical structures and problems that are "discrete" in nature, as opposed to problems that exhibit continuous properties. Discrete Mathematics is used in all areas of algorithm and information structure design, which means our graduates are in demand in areas concerned with topics in computer security, coding theory, cryptography, electronic banking, telecommunications and web search engines. Many optimisation and planning problems also give rise to questions addressed by Discrete Mathematics. For instance, the road map in a GPS in your car is stored as a list of crossings with mutual links describing how it is possible to drive from one crossing to another. When you enter a desired destination, a complex mathematical algorithm is invoked to compute an optimal route from your current place to the destination. It is important that the algorithm computing the route is fast enough to be useful in navigation, which is exactly why we should be concerned with the complexity of algorithms. Discrete Mathematics was essential to the breaking of German codes during World War II. Bill Tutte, a key contributor to the cracking of the Enigma code, set the foundations of modern Discrete Mathematics. Discrete Mathematics is also linked to the legacy of Paul Erdos, one of the most famous mathematicians of the 20th century. It gave foundations of modern public-key cryptography used in today e-banking and e-commerce applications. Formal verification provides framework for testing logic statements about safety-essential software systems such as those in the planes. On the theoretical side, one of the greatest unsolved problems of theoretical Computer Science is the $1m Millennium Prize Problem "P vs. NP" on the existence of efficient algorithms for many intriguing Discrete Mathematics problems. Discrete Mathematics at Warwick Discrete Mathematics is an honours degree taught jointly by the Department of Computer Science and the Warwick Mathematics Institute since 2008. This distinctive course provides a firm grounding in modern Computer Science and Mathematics, building on the expertise of the Centre for Discrete Mathematics and its Applications (DIMAP), a research centre established at the University of Warwick in 2007. Our Discrete Mathematics degree equips students with a fundamental understanding of Mathematics and a comprehensive set of skills in Computer Science, which make our graduates particularly attractive to the world’s most reputable employers. Discrete Maths at Warwick “Studying was no easy task but, with the help and support of the staff, I was able to graduate with exactly the set of skills I needed to pursue a career in a competitive industry. The course also offered me great flexibility, allowing me to choose the Computer Science and Mathematics modules I found more interesting." Jess Fan (2009-2012) The course involves learning about and solving interesting problems. You will be taught how to create and analyse algorithms and have the opportunity to develop software systems individually and as part of a team. Your finalyear project will allow you to explore a topic that you find particularly interesting, building on the expertise and skills you’ve developed in previous years. DISCRETE MATHEMATICS 3 www.warwick.ac.uk/go/statistics warwick.ac.uk/dmaths Meet Our Academics Dr. Andrzej S. Murawski (Course Manager, Computer Science) Research Interests: Programming languages, software verification, automata theory 2005-2010 EPSRC Advanced Research Fellow, Oxford 2002-2006 Junior Research Fellow, St John's College, Oxford 2001 DPhil in Computation, Oxford Dr. Alina R. Ene (Computer Science) Research Interests: Discrete and combinatorial optimization, approximation algorithms 2013-14, Postdoc, Princeton University 2013 PhD in Computer Science, University of Illinois at Urbana-Champaign 2008 BSE in Computer Science (magna cum laude), Princeton University Professor Oleg Pikhurko (Mathematics) Research Interests: Extremal combinatorics, graph limits 2003-2011 Assistant and Associate Professor, Carnegie Mellon University 2000-2003 Junior Research Fellow, St John's College, Cambridge 2000 PhD in Pure Mathematics, Cambridge University Course Structure The Discrete Mathematics course develops the rigorous mathematical methods and practical skills that are demanded by a variety of technical roles, including those in scientific research. This is achieved through three years of study, where each year builds on the previous to provide breadth and depth in Mathematics and Computer Science. First year The first year of the course focuses on essential Computer Science and Mathematics skills. Students take modules in the fundamental areas of pure mathematics, such as mathematical analysis, linear algebra and probability. They also experience computer programming in Java and learn how to solve fundamental problems using algorithms and data structures. The normal load consists of around 25% Computer Science modules, 20% pure mathematics modules, 20% Discrete Mathematics modules and 5% statistics modules. 10% of the normal load is left for optional modules. Some of the modules are specifically designed for and taken only by the students studying Discrete Mathematics, whilst others are taken with the first-year students in Computer Science, Mathematics and Statistics. Second year The second year places emphasis on applying mathematical skills developed during the first year in Computer Science. The core consists of those related to Discrete Mathematics and advanced algorithm and data structure design, including theoretical concepts related to the structure of programming languages. Optional modules offer a wide range of mathematical and statistical subjects related to optimisation and stochastic methods. More than 40% of the normal load is left as optional modules for students to choose. Final year The final year gives students the freedom to choose modules to suit their interests and shape their graduate profile. The students are required to take two Computer Science modules. The first is a module on analysing the complexity of computational problems, whilst the second module is an individual project. In most cases, the projects consist of designing and implementing software to solve a studentspecified problem, though students can also choose to develop entirely mathematical projects. Careers The students graduating from Department of Computer Science and the Warwick Mathematics Institute are highly demanded by employers. Discrete Mathematics plays an essential role in modelling and in designing efficient solutions of everyday problems in the digital economy, allowing our graduates find jobs in a wide range of areas, including: • Consultancy • Finance Flexible study • Scientific Research The Discrete Mathematics course offers great flexibility, which increases from the first year to the third. Most of the modules taken by the students are taught by Department of Computer Science,Department of Statistics, Warwick Business School, and Warwick Mathematics Institute. However, there is no restriction on some of the modules taken as part of the normal load in the second and third years and the choices of the modules taken often include those offered by Department of Philosophy, Economics Department, Language Centre and others. Additional modules may also be taken by students opting to take higher than the normal load. • Software Engineering As with all degree courses offered by the Department of Computer Science, Discrete Mathematics students have the option of taking an intercalated year between the second and third years of their degree. This can be used to work in industry or to study at a university in another country for a year. Our students have spent intercalated years at companies such as Barclays Capital, BT, HSBC, IBM, Intel, PwC, Siemens and many other prestigious organisations. Another way of getting an industrial experience during the course are summer internships which sometimes lead to employment offers after the graduation. • Telecommunications Other career options include teaching mathematics and Computer Science related subjects at schools, where the shortage of Mathematics teachers remains an issue. Some graduates also continue on to study for postgraduate degrees that allow them to take up research positions within companies or universities. I started my first year of “When Discrete Mathematics, I wasn't entirely sure what to expect. However, you have the freedom to explore your options and find out what suits you best. After my second year, I took part in an Internship at PwC under Data Assurance. This allowed me to put skills developed during my degree to use and highlighted how relevant DM is in a financial and technological environment. Overall, the degree is an interesting mix of theoretical concepts and real life applications which develops your natural abilities such as problem solving." Jon Buckler (2011-2014) DISCRETE MATHEMATICS 5 www.warwick.ac.uk/go/statistics warwick.ac.uk/dmaths Mathematics and Computer Science at Warwick Since its foundation as one of the new universities in 1965, Warwick has grown into one of the strongest universities for mathematical science in Europe. Warwick Mathematics Institute is home to many internationally renowned researchers from all areas of pure and applied Mathematics. One of its members, Ian Stewart, is also a well-known author in popular science. Department of Computer Science was established in 1969. It has steadily grown in size with the increasing numbers of its undergraduate and graduate students. The Department is consistently ranked among the top UK Computer Science departments. Discrete Mathematics at Warwick In response to the growing importance of Discrete Mathematics, the University of Warwick set up the Centre for Discrete Mathematics and its Applications (DIMAP) in 2007. The centre is a joint venture of the Department of Computer Science, the Warwick Mathematics Institute and the Warwick Business School. In its first years of existence, it was supported by the EPSRC Science and Innovation Award of nearly £4M. The centre attracted many world leading experts to Warwick and turned Warwick in one of the strongest places for Discrete Mathematics in Europe. The Discrete Mathematics course is administered by the Department of Computer Science, which has a strong tradition in multidisciplinary degrees. Every student is assigned a personal tutor to provide pastoral care and help deal with any issues. The modules taught are delivered by specialists from relevant departments. Mathematical supervisions held twice a week for groups of 4 or 5 students assist the first-year students to cope with their mathematical modules. Computer Science classes are supplemented by small group tutorials. Since the first year, all students are able to shape the structure of the modules they take based on their own interests. Discrete Mathematics is one of several mathematics oriented honours degrees at the University of Warwick. Allied courses include: • Computer Science • Data Science • Mathematics • Mathematics and Physics • Mathematics and Statistics • MORSE (Mathematics, Operational Research, Statistics and Economics) The Discrete Mathematics degree focuses on (discrete) mathematics methods underpinning Computer Science applications, leading to applications in algorithm and data structure design and in planning and optimisation tasks. At the same time the degree provides large flexibility by offering a wide range of mathematical and Computer Science modules. Entry Requirements Our typical A-level offer is A*AA including an A* in Mathematics or Further Mathematics. No previous knowledge of Computer Science is required. Our minimum language requirements is a grade C in GCSE English or IELTS 6.0. About the University The University of Warwick was established in 1965 and it quickly gained a reputation for excellence in research and teaching. It has regularly been appearing among the top 20 world universities in the "under 50" lists (consisting of the universities under the age of 50) both in QS World University Rankings and the Times Higher Education World University Rankings. The Government Teaching Quality Assessment has rated as excellent both Computer Science and Mathematics. The modern main campus of the University is located in the southern part of Coventry and partly in the county of Warwickshire. Its central location means that Birmingham and its busy international airport are within easy reach. It is also one hour train ride or 90 minute motorway ride to London. The University of Warwick is a place of study for over 23,000 students with 55% being undergraduates. In January 2013, the University also opened Learning Grid in Leamington Spa to offer supportive environment to its many students living in the town. The Warwick Arts Centre is the largest arts centre in the UK outside London. It combines art gallery, cinema, a concert hall, cinema, theatres and Music Centre. The University campus also offers first class sports facilities including a Sports Centre, Games Hall, Bear Rock Climbing Centre, playing fields and running tracks. Over 200 clubs and societies are organised by a dynamic Students Union (one of the biggest in the UK) providing a vast range of activities and entertainment options for student life at Warwick. DISCRETE MATHEMATICS 7 warwick.ac.uk/datsci warwick.ac.uk/dmaths xx For further information contact: Undergraduate Admissions Computer Science University of Warwick Coventry CV4 7AL UK Tel: +44 (0)24 7652 3193 Fax: +44 (0)24 7652 3024 Email:comp-sci@dcs.warwick.ac.uk Designed by WarwickDesign, www.designatwarwick.com Printed by WarwickPrint, www.warwickprint.co.uk