# Faculty of Mathematics

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

The three Michaelmas 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. All do have some essential prerequisites though as well, which you can find under the "Reality Checks" in the points below.

### Prerequisite areas

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## 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.

You will need this for the following Part III courses:

• Algebraic Geometry

Relevant undergraduate courses are:

### 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!).
• 2017 lecturer Pelham Wilson's online notes for the Preliminary Chapter 0' of his Part III Algebraic Geometry course cover much of this catch-up material but are pretty brief. They do give further resources and book suggestion.
• Ravi Vakil's online notes Foundations of Algebraic Geometry (linked hopefully to latest version, labelled Dec 2015).
• Pelham Wilson's lecture notes on Algebraic Curves (referred to a lot in his Preliminary Chapter 0).
• Miles Reid Undergraduate Algebraic Geometry, Cambridge University Press (1988), is listed as introductory reading in Pelham Wilson's preliminarly chapter.
• 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.
• Hartshorne Algebraic Geometry' (classic textbook, although it's quite dense; the workshop (notes above) mainly tried to match terminology and notation with Chapter 1 of this book).

## Basic Algebraic Topology

In principle, the Algebraic Topology course does not require any Algebraic Topology prerequisites, but some knowledge of the fundamental group and concepts related to it, is helpful in order to keep up with the lecture material.

You will need this for the following Part III courses:

• Algebraic Topology

Relevant undergraduate courses are:

### 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. Some familiarity with the fundamental group can also help. In particular, the notion of functoriality (whether in the context of fundamental group or homology) is extremly useful to know about. The following topics are covered for example in the notes by Dr Randal-Williams (link below):

• The notion of a homotopy and homotopy equivalence. (see Chapter 2)
• The notion of functoriality (see e.g. Proposition 2.3.3)
• Chain complexes and exact sequences. (see Sections 7.2 and 7.4)

Furthermore, some experience of some version of homology in algebraic topology can be 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:

• (Some version of a homology theory.)
• The Mayer-Vietoris sequence and examples of computations with it.
• (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, but moves quite fast. 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)

Relevant undergraduate courses are:

### 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.