- The topics to be covered include the following: Groups and subgroups: revision of de nitions with the standard examples (GLn, Z, Z mod n, dihedral
groups, Sn and An, introduction of basic properties such as uniqueness of identities and inverses, subgroup tests, Cayley tables, etc. Cosets and Lagrange's Theorem, direct products, statement without proof of the classification of fi nitely generated abelian groups. Homomorphisms. Quotient groups and normal subgroups. Group actions (orbits, stabilizers).
First Isomorphism Theorem (2nd and 3rd stated without proof). Cayley graphs.
Rings, Fields, Integral Domains. Field of fractions of an integral domain. Polynomial rings, including factorization of polynomials over a fi eld. Ring homomorphisms and quotient rings. Prime and maximal ideals.
Introduction to extension fields: Algebraic extensions (defi nition of a vector space will be briefly recalled and hopefully de finition of a module will be at least mentioned in passing).UFDs and Euclidean domains.
- This course provides an introduction to the subject of topology, sometimes called “elastic geometry”. This involves studying the most fundamental properties of geometric objects or spaces.
The course will begin with the study of metric spaces – these are sets together with a function which gives the distance between any two points. Basic properties of these spaces and functions between them will be considered, and illustrated using examples such as Euclidean space, the
taxicab metric, the chessboard metric and the railway metric. We will then see that metric spaces in turn are examples of topological spaces, and learn more about these more general objects. The course will continue with a look at surfaces (sphere, torus, and so on) and one of the crowning glories of 19th and early 20th century mathematics, the Classification of Surfaces, which tells us exactly what “and so on” means in this context.
The ideas of the course are fundamental to analysis and topology, as well as many branches of geometry.
Mechanics of Rigid and Deformable Bodies