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Geometry and Topology

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

Prerequisite areas

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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)

Relevant undergraduate courses are:

First level prerequisites

Second level 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 and affine varieties;
  • the Nullstellensatz;
  • morphisms of affine varieties;
  • projective varieties.

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).

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Algebraic Topology

You will need this for the following Part III courses:

  • Algebraic Topology

Relevant undergraduate courses are:

First level prerequisites

  • 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 doing the following exercises: coming soon.
  • 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 doing the following exercises: coming soonish .

Second level prerequisites

Some experience of some version of homology in algebraic topology. For example you should know about:

  • Homotopic maps and homotopy equivalence of spaces.
  • Chain complexes and exact sequences.
  • Simplicial homology. (Or another type of homology.)
  • The Mayer-Vietoris sequence.
  • Applications, including Brouwer's fixed-point theorem.

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Useful books and resources

  • Can put here last year's workshop notes.

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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:

  • Complex Manifolds
  • (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:

First level prerequisites

  • 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 doing the following exercises: [to be filled in prob by Julia].
  • 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

Second level prerequisites

Useful to have had exposure to some of the ideas of classical differential geometry.

Useful books and resources

  • Notes from the Part II 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.

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