# Faculty of Mathematics

Please take this page in conjunction with the Part III Guide to Courses Geometry and Topology section.

The three Part III courses Algebraic Geometry, Algebraic Topology, Differential Geometry don't strictly require any previous knowledge of those areas, but because of the speed they go at, some previous experience is very helpful to give some background and framework, especially for the first two courses. All do have some essential prerequisites though as well, which you can find under the First Level Prerequisites in the points below.

### Prerequisite areas

Back to main page

## Basic Algebra

This section is on the Algebra page.

## Basics of Classical Algebraic Geometry

In theory, the Algebraic Geometry course usually starts from scratch, but you will find it impossible to keep up if you are not already familiar with basic algebra and point-set topology. It is also well worth gaining some exposure to simple concepts in classical algebraic geometry. Note that in 2016/17 there is no Commutative Algebra course, so getting a head start in reading about that would also be helpful.

You will need this for the following Part III courses:

• Algebraic Geometry
• (Elliptic Curves)

### Reality check

• Elementary point-set topology: topological spaces, continuity, closure of a subset etc. See e.g. Ib Metric and Topological Spaces. You can check if you are at the required level by doing the following exercises: Met & Top 2015-16 Example Sheet 1.
• Commutative algebra, at roughly the level mentioned in prerequisites of Basic Algebra: rings, ideals (including prime and maximal) and quotients, algebras over fields (in particular, some familiarity with polynomial rings over fields). See e.g. Ib Groups, Rings and Modules. You can check if you are at the required level by doing the following exercises: GRM 2015-16 Example Sheet 2.
• For all above example sheets, we recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.

### Prerequisites

Basic affine algebraic geometry, in particular:

• affine space and algebraic sets;
• the Hilbert basis theorem and applications;
• the Zariski topology on affine space;
• irreducibility;
• the Nullstellensatz;
• morphisms of affine varieties.

Some exposure to projective varieties may also be useful. You can use the following exercises to check your present level and to guide your study.

• Algebraic Geometry exercises provided by Jack Smith (thank you!).

### Useful books and resources

• Notes from previous catch-up workshop on Algebraic Geometry, provided by Jack Smith (thank you!).
• Ravi Vakil's online notes Foundations of Algebraic Geometry (linked hopefully to latest version, labelled Dec 2015).
• Eisenbud Commutative Algebra with a view toward algebraic geometry (covers all the algebra you might need, with a geometric flavour---it has pictures).
• Atiyah, Macdonald Introduction to Commutative Algebra is given as reference in the Algebraic Geometry Guide to Courses page.
• Pelham Wilson's online notes for the Preliminary Chapter 0' of his Part III Algebraic Geometry course from 2014 cover much of this catch-up material but are pretty brief (warning: the 2015/16 course had a different lecturer so will have been different). They do give further resources and book suggestion.
• Hartshorne Algebraic Geometry' (classic textbook, on which I think the 2015/16 course was based, although it's quite dense; the workshop (notes above) mainly tried to match terminology and notation with Chapter 1 of this book).

## (Simplicial) Homology

In principle, the Algebraic Topology course does not require any Algebraic Topology prerequisites, but as the corresponding page in the Guide to Courses mentions, some knowledge of simplicial homology, or some other homology theory, and/or the fundamental group, is helpful in order to keep up with the lecture material.

You will need this for the following Part III courses:

• Algebraic Topology

### Reality check

• Standard undergraduate group theory, including the structure theorem for finitely generated abelian groups. See e.g. Ia Groups and Ib Groups, Rings and Modules. You can check if you are at the required level by going through the example sheets for those courses. For all example sheets, we recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.
• Fluency with definitions in basic point-set topology, such as compactness and connectedness. See e.g. Ib Metric and Topological Spaces. You can check if you are at the required level by going through the example sheets. We recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.

### Prerequisites

It is very advisable that you are comfortable with the concept of homotopy equivalence, and have a good intuition about which spaces are homotopically equivalent. If you have no experience of homology, then some familiarity with the fundamental group can help. The following topics are covered for example in the notes by Dr Randal-Williams (link below):

• Homotopic maps and homotopy equivalence of spaces. (see Chapter 2)
• Chain complexes and exact sequences. (see Sections 7.2 and 7.4)

Furthermore, some experience of some version of homology in algebraic topology is helpful (but not strictly necessary) in order to keep up with the speed of the course, and have some intuition for the singular homology theory which will be developed in the course. For example some experience of the following topics, which are also covered in the notes by Dr Randal-Williams:

• Simplicial homology. (Or another type of homology.)
• The Mayer-Vietoris sequence.
• Applications, including Brouwer's fixed-point theorem.

You can use the following exercises to check your present level and to guide your study.

• Workshop exercises provided by Ben Barrett (thank you). Include questions on Homotopies, Homological Algebra, Simplicial Homology, Mayer-Vietoris.

### Useful books and resources

• The lecture notes on Part II Algebraic Topology by Dr Randal-Williams are a good source for learning about homotopy equivalence, and also simplicial homology.
• The book Algebraic Topology by Hatcher (CUP 2001) is suitable for learning about the Fundamental Group. You can find online versions here.

## Basics of Classical Differential Geometry

The course generally starts from scratch, and since it is taken by people with a variety of interests (including topology, analysis and physics) it is usually fairly accessible. It is an important stepping stone for many other geometry courses.

You will find this helpful for the following Part III courses:

• Differential Geometry
• (Algebraic Topology)
• Other geometry and geometric analysis courses which change from year to year (eg Riemannian Geometry)
• Theoretical Physics courses (eg General Relativity, Symmetries, Fields and Particles, Applications of Differential Geometry to Physics)

### Reality check

• Linear algebra: abstract vector spaces and linear maps, bilinear forms. See e.g. Ib Linear Algebra. You can check if you are at the required level by going through the example sheets of that course. We recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.
• Multi-variable calculus: derivatives of functions as linear maps, the chain rule, partial derivatives, Taylor's theorem in several variables. See e.g. Ib Analysis II. You can check if you are at the required level by doing the following exercises: Analysis II 2015-16 Sheet 4 (Questions 4, 5, 11).
• Elementary point-set topology: topological spaces, continuity, compactness etc. See e.g. Ib Metric and Topological Spaces. You can check if you are at the required level by doing the following exercises: Met & Top 2015-16 Example Sheet 1. We recommend that you restrict yourself to the main questions, excluding any "extra", "additional" or "starred" questions.

### Prerequisites

It is useful to have had exposure to some of the ideas of classical differential geometry. See possible books and notes below.

### Useful books and resources

• Notes from the Part II Differential Geometry Course.
• Milnor's classic book "Topology from the Differentiable Viewpoint" is a terrific introduction to differential topology as covered in Chapter 1 of the Part II course. It is quite different in feel from the Part III course but would be great to look at in preparation.
• Nakahara "Geometry, Topology and Physics". This is not a pure maths book, so comes with a warning that it is not always completely precise and rigorous. It also covers lots of material outside the Part III course. However, it is excellent for giving an intuitive picture of the concepts, and may be particularly helpful to physicists taking the course.