Moodle@Warwick

The Science of Music module (IL016) aims to introduce students (in all subject areas and with any level of musical, mathematical or scientific expertise) to the relationships between science, music and mathematics. The module will explore multiple facets of Music by combining tools from a variety of disciplines, from Physics and Maths to Psychology and History, with contributions from a range of professional musicians.Link opens in a new window

Teaching for the module will be based around 9 2-hour workshops and a field trip (on a Saturday mid-February) to Royal Birmingham Conservatoire.

Lecturer: Christian Boehning

Term(s): Term 1

Status for Mathematics students: List A for Mathematics

Commitment: 30 lectures plus weekly worksheets

Assessment: The weekly worksheets carry 15% assessed credit; the remaining 85% credit by 2-hour examination.

Prerequisites: None, but an understanding of MA125 Introduction to Geometry will be helpful.

Leads To: Third and fourth year courses in Algebra and Geometry, including:  MA3D9 Geometry of Curves and Surfaces, MA3E1 Groups and Representations, MA4A5 Algebraic Geometry, MA4E0 Lie Groups, MA473 Reflection Groups, MA4H4 Geometric Group Theory, MA448 Hyperbolic Geometry and others

Content: Geometry is the attempt to understand and describe the world around us and all that is in it; it is the central activity in many branches of mathematics and physics, and offers a whole range of views on the nature and meaning of the universe. 

Klein's Erlangen program describes geometry as the study of properties invariant under a group of transformations. Affine and projective geometries consider properties such as collinearity of points, and the typical group is the full  matrix group. Metric geometries, such as Euclidean geometry and hyperbolic geometry (the non-Euclidean geometry of Gauss, Lobachevsky and Bolyai) include the property of distance between two points, and the typical group is the group of rigid motions (isometries or congruences) of 3-space. The study of the group of motions throws light on the chosen model of the world. 

Aims: To introduce students to various interesting geometries via explicit examples; to emphasize the importance of the algebraic concept of group in the geometric framework; to illustrate the historical development of a mathematical subject by the discussion of parallelism. 

Objectives: Students at the end of the module should be able to give a full analysis of Euclidean geometry; discuss the geometry of the sphere and the hyperbolic plane; compare the different geometries in terms of their metric properties, trigonometry and parallels; concentrate on the abstract properties of lines and their incidence relation, leading to the idea of affine and projective geometry. 

Books: 

M Reid and B Szendröi, Geometry and Topology, CUP, 2005 (some Chapters will be available from the General office). 

E G Rees, Notes on Geometry, Springer 

HSM Coxeter, Introduction to Geometry, John Wiley & Sons

Aims: To develop further and to continue the study of linear algebra, which was begun in Year 1.

To point out and briefly discuss applications of the techniques developed to other branches of mathematics, physics, etc.

Objectives: By the end of the module students should be familiar with: the theory and computation of the the Jordan canonical form of matrices and linear maps; bilinear forms, quadratic forms, and choosing canonical bases for these; the theory and computation of the Smith normal form for matrices over the integers, and its application to finitely generated abelian groups.

Lecturer: Christian Böhning

Term(s): Term 2

Status for Mathematics students: List C

Commitment: 30 lectures plus assignments

Assessment: Assignments (15%), 3 hour written exam (85%).

Prerequisites: 
A background in algebra (especially MA249 Algebra II) is essential. The module develops more specialised material in commutative algebra and in geometry from first principles, but MA3G6 Commutative Algebra will be useful. More than technical prerequisites, the main requirement is the sophistication to work simultaneously with ideas from several areas of mathematics, and to think algebraically and geometrically. Some familiarity with projective geometry (e.g. from MA243 Geometry) is helpful, though not essential.

Leads To: 
A first module in algebraic geometry is a basic requirement for study in geometry, number theory or many branches of algebra or mathematical physics at the MSc or PhD level. Many MA469 projects are on offer involving ideas from algebraic geometry.

Content: 
Algebraic geometry studies solution sets of polynomial equations by geometric methods. This type of equations is ubiquitous in mathematics and much more versatile and flexible than one might as first expect (for example, every compact smooth manifold is diffeomorphic to the zero set of a certain number of real polynomials in R^N). On the other hand, polynomials show remarkable rigidity properties in other situations and can be defined over any ring, and this leads to important arithmetic ramifications of algebraic geometry.

Methodically, two contrasting cross-fertilizing aspects have pervaded the subject: one providing formidable abstract machinery and striving for maximum generality, the other experimental and computational, focusing on illuminating examples and forming the concrete geometric backbone of the first aspect, often uncovering fascinating phenomena overlooked from the bird's eye view of the abstract approach.

In the lectures, we will introduce the category of (quasi-projective) varieties, morphisms and rational maps between them, and then proceed to a study of some of the most basic geometric attributes of varieties: dimension, tangent spaces, regular and singular points, degree. Moreover, we will present many concrete examples, e.g., rational normal curves, Grassmannians, flag and Schubert varieties, surfaces in projective three-space and their lines, Veronese and Segre varieties etc. 

Books:

• Atiyah M.& Macdonald I. G., Introduction to commutative algebra, Addison-Wesley, Reading MA (1969)
• Harris, J., Algebraic Geometry, A First Course, Graduate Texts in Mathematics 133, Springer-Verlag (1992)
• Mumford, D., Algebraic Geometry I: Complex Projective Varieties, Classics in Mathematics, reprint of the 1st ed. (1976); Springer-Verlag (1995)
• Reid, M., Undergraduate Algebraic Geometry, London Math. Soc. Student Texts 12, Cambridge University Press (2010)
• Shafarevich, I.R., Basic Algebraic Geometry 1, second edition, Springer-Verlag (1994)
• Zariski, O. & Samuel, P., Commutative algebra, Vol. II, Van Nos- trand, New York (1960)

This is the third module in the Warwick algebraic topology sequence.  

The abelian group structure on homology is upgraded via dualization to a natural ring structure on cohomology. This additional structure makes cohomology a more powerful invariant than homology. Cohomology is also a more natural invariant in many contexts, and various cohomology theories play a key role in a number of fields, including differential geometry, mathematical physics, algebraic geometry, and number theory, among others.

Poincaré duality is a relation between the homology and cohomology of an oriented manifold. The ring structure of cohomology plays a critical role in its proof.   

The two laboratory mini-projects conclude your MSc year and prepare you for your PhD projects. MD979 is the first one. Due to the highly inter-disciplinary nature of the IBR DTP, you have a great variety of choice for these projects. You will have an 11-week period for each of the two mini-project modules (MD979 and the subsequent MD980). Normally there would be one experimental biology project and one either theoretical biology project (e.g., bioinformatics, computational biology) or experimental project in chemistry, physics or engineering. The supervisor pool is accordingly expanded to include colleagues across the departments of the Faculty of Science in addition to WMS.

The projects will either be developed by the you together with an academic from the supervisor pool, or initially by the supervisors alone, who will submit projects directly to the mini-project call for presentation to the student cohort at a mini-project “fair”. You should be aware that all projects will first be vetted in the IBR DTP management team before inclusion in the course.

The modules Research Topics in Interdisciplinary Biomedical Research [MD978] and Laboratory Project 1 [MD979] are a pre-requisite for this module.

Students will undertake two laboratory projects in two different disciplines. In most cases, this will be a biology-focused project and one in either chemistry, physics, mathematics, engineering or computer science. If you are a student on the Quantitative Imaging programme, your projects should focus on imaging and image analysis. Projects can be undertaken in WMS or a department within the Faculty of Science at Warwick.

Students are encouraged to develop a project proposal together with a member of staff from the supervisor pool (www2.warwick.ac.uk/fac/med/study/mrcdtp/supervisorsandprojects/). In addition, the supervisor pool will be invited to submit potential projects for consideration by the IBR DTP management committee. Projects will be reviewed for fit to the scientific brief and will be then offered to the students. The final choice of project will be made by the student in consultation with the MSc Director.