Moodle@Warwick

A module that examines the relationship between the self and its mediation in a converged media landscape, through the production of an online mediated self project and an exploration of the very latest thinking on digital representations of the self.

Notes of the course: https://moodle.warwick.ac.uk/pluginfile.php/946222/course/summary/LectureNotesMA125.pdf

Notes of last week of the course: https://moodle.warwick.ac.uk/pluginfile.php/946222/course/summary/LectureOnStereographicProjection.pdf

Problem sheet 1: this is suggested at the end of week 2/week 3. This is not compulsory but strongly recommended to train yourself and check if the basic concepts are clear. https://moodle.warwick.ac.uk/pluginfile.php/946222/course/summary/Problem%20Sheet%201.pdf

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

The course will start by introducing the concept of a manifold (without recourse to an embedding into an ambient space). In the words of Hermann Weyl (Space, Time, Matter, paragraph 11):

“The characteristic of an n-dimensional manifold is that each of the elements composing it (in our examples, single points, conditions of a gas, colours, tones) may be specified by the giving of n quantities, the “co-ordinates,” which are continuous functions within the manifold. This does not mean that the whole manifold with all its elements must be represented in a single and reversible manner by value systems of n co-ordinates (e.g. this is impossible in the case of the sphere, for which n = 2); it signifies only that if P is an arbitrary element of the manifold, then in every case a certain domain surrounding the point P must be representable singly and reversibly by the value system of n co-ordinates.”

Thus the points on the surface of a sphere form a manifold. The possible configurations of a double pendulum (one pendulum hung off the pendulum bob of another) is a manifold that is nothing but the surface of a two-torus: the surface of a donut (a triple pendulum would give a three-torus etc.) The possible positions of a rigid body in three-space form a six-dimensional manifold. Colour qualities form a two-dimensional manifold (cf. Maxwell’s colour triangle).

It becomes clear that manifolds are ubiquitous in mathematics and other sciences: in mechanics they occur as phase-spaces; in relativity as space-time; in economics as indifference surfaces; whenever dynamical processes are studied, they occur as “state-spaces” (in hydrodynamics, population genetics etc.)

Moreover, in the theory of complex functions, the problem of extending one function to its largest domain of definition naturally leads to the idea of a Riemann surface, a special kind of manifold. 

Although it seems so natural from a modern vantage point, it took some time and quite a bit of work (by Gauss, Riemann, Poincare, Weyl, Whitney, …) till mathematicians arrived at the concept of a manifold as we use it today. It is indispensable in most areas of geometry and topology as well as neighbouring fields making use of geometric methods (ordinary and partial differential equations, modular and automorphic forms, Arakelov theory, geometric group theory…)

Some buzz words suggesting topics which we plan to cover include:

-The notion of a manifold (in different setups), examples of constructions of manifolds (submanifolds, quotients, surgery)

-The tangent space, vector fields, flows/1-parameter groups of diffeomorphisms

-Tangent bundle and vector bundles

-Tensor and exterior algebras, differential forms

-Integration on manifolds, Stokes’ theorem

-de Rham cohomology, examples of their computation (spheres, tori, real projective spaces...)

-Degree theory, applications: argument principle, linking numbers, indices of singularities of vector fields

We will also discuss a lot of concrete and interesting examples of manifolds in the lectures and work sheets, such as for example: tori, n-holed tori, spheres, the Moebius strip, the (real and complex) projective plane, higher-dimensional projective spaces, blow-ups, Hopf manifolds… 

The nature of the material makes it inevitable that considerable time must be devoted to establishing the foundations of the theory and defining as well as clarifying key concepts and geometric notions. However, to make the content more vivid and interesting, we will also seek to include some attractive and non-obvious theorems, which at the same time are not too hard to prove and natural applications of the techniques introduced, such as, for instance, Ehresmann's theorem on differentiable fibrations, or that a sphere cannot be diffeomorphic to a product of (positive-dimensional) manifolds. 

This Module is mathematically closely related to, but formally completely independent of MA3D9 Geometry of Curves and Surfaces. 

Books:

Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer (1983) (esp. Chapters 1, 2, 4)

M. Berger, B. Gostiaux, Differential Geometry: Manifolds, Curves, and Surfaces, Springer (1988) (esp. Chapters 2-6)

Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1, Publish or Perish, Inc. (2005) (esp. Chapters 1-8)

John Lee, Introduction to smooth manifolds, Springer (2012)

Loring W. Tu, An Introduction to Manifolds, Springer (2011)

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)