Courses for 2017-2018

The plan for 2017–2018 is given below. Note that this is still subject to change and dates and times have not been finalised yet. The current timetable is available here.

Term 1

The following courses will run all term and all LSGNT students are expected to attend.

  • Topics in geometry (every week - see here to see what was covered last year).
    • I'd suggest they do lots of exercises maybe. But one thing that I consider fundamental (and often missing) is basic algebraic topology, including – in fact especially – Poincaré duality. There are millions of books covering this so I won't recommend a particular one; they will find one that suits their style. - R. Thomas.

  • Topics in number theory (every week - see here to see what was covered last year).

  • Computing for geometry and number theory (J. Armstrong).
    • This course will introduce programming in both Mathematica and in Python through projects and examples taken from geometry and number theory. We recommend that if you have no prior programming experience, you try to learn a little Python before you start (you could start by looking at some of the tutorials listed on The Hitch-Hiker's Guide to Python).

The following courses will run for part of the term and LSGNT student should choose which to attend.

  • Low-dimensional topology (R. Casals)

  • Elliptic curves (V. Dokchitser)
    • The aim of the course is to cover the classical arithmetic theory of elliptic curves, from the group law to the proof of the Mordell-Weil theorem. The core material will be presented in a "low tech" manner with few prerequisites. The course will also include more advanced material, aimed at graduate students in algebraic number theory. There will be exercise sheets and numerical examples illustrating the theory.
    • Core topics: Weierstrass equations, Group law on elliptic curves, Elliptic curves over C, Heights, Elliptic curves over p-adic fields, Mordell-Weil Theorem, Explicit 2-descent, Further topics (a subset of the following): Formal groups, Elliptic curves over finite fields and zeta-functions, Selmer groups and Galois cohomology, Tate module, Reduction types of elliptic curves, The Birch-Swinnerton-Dyer conjecture.
    • Prerequisites: Galois theory, Familiarity with p-adic numbers, Some knowledge of algebraic curves or Riemann surfaces may be helpful, but is not essential.
    • Recommended books: J. W. S. Cassels, "Lectures on Elliptic Curves", LMS Student Texts 24, J. H. Silverman, "The Arithmetic of Elliptic Curves", Springer Graduate Texts in Mathematics

  • Class field theory (M. Kakde)
    • It would be good if students could have some familiarity with algebraic number theory, for example at the level of Milne's notes or Daniel Marcus's book "Number fields".

  • TBA (J. Rodrigues and C. Williams)

  • Mean curvature flow (F. Schulze)
    • The aim of this course is to give an introduction to mean curvature flow. It will cover the by now 'classical' results on curve shortening flow and the long time behaviour of the flow of closed convex hypersurfaces in Euclidean space, but include also some more recent results like Huisken-Sinestrari's surgery procedure for 2-convex mean curvature flow and Brendle's classification of simply connected, embedded self-shrinkers in R3.
    • Prerequisites: Multivariable Analysis, Differential Geometry, Riemannian Geometry.
    • Suggested preparatory reading: 1) Manfredo P. do Carmo, Riemannian Geometry, Chapter 6: Isometric Immersions, 2) Lawrence C. Evans, Partial Differential Equations, Chapter 6.4: Maximum principles and Chapter 7.1.4: Maximum principles

  • Schemes (A. Yafaev)

  • Elliptic operators and index theory (M. Singer and S. Scott)
    • I shall want to assume familiarity with topological spaces, manifolds (e.g. Spivak's "Calculus on manifolds"), and vector bundles. A good reference for vector bundles is Sections 1.1-1.5 of Atiyah's book "K-theory". I shall try to do the analysis more-or-less from scratch.

Term 2

The following courses will run all term. By this point you will have figured out how many courses you can keep up with at a time, but we expect all students to keep attending the two topics courses.

  • Topics in geometry (alternate weeks).

  • Topics in number theory (alternate weeks).

The following courses will run for part of the term.

  • Elliptic operators and index theory, continued (M. Singer and S. Scott)

  • The Cremona group (N. Shepherd-Barron)

  • Birational geometry (P. Cascini)

  • Arithmetic homotopy (A. Pal)
    • Outline syllabus. Quadratic forms over fields, Grothendieck-Witt ring, Pfister local-global principle, Milnor conjecture. Topological K-theory, Atiyah-Hirzebruch spectral sequence. Simplicial sets, simplicial homotopy, infinity categories. Stable homotopy theory, spectra. Motivic cohomology and motivic homotopy.
    • Suggested reading: Hartshorne "Algebraic Geometry" and Hatcher "Algebraic Topology".

  • L-functions (K. Buzzard)
    • Prerequisites for the course would be a good grounding in (what I would call undergraduate-level) algebraic number theory, for example the contents of Marcus' book "number fields" (or at least the first half of it) or Froehlich-Taylor, or any other book called "algebraic number theory"

Summer reading recommendations from students

The following topics were suggested by our current LSGNT students as things they wish they had known better before arriving. For example, topics courses are taught by a variety of lecturers, who may well make a variety of assumptions about what you already know. If you don't know something, it's always best to ask (better than being miserably confused!). But if you want to fill in some of the gaps over the summer, here are some things that it would be good to know (and some places you could read about them)...

  • – Manifolds (Spivak's "Calculus on manifolds")

  • – Bundles, connections, differential forms, Poincaré duality (Madsen and Tornehave "From calculus to cohomology", Bott and Tu "Differential forms in algebraic topology")

  • – Line bundle-divisor correspondence (e.g. Griffiths and Harris "Principles of algebraic geometry")

  • – Cohomology, characteristic classes (Madsen and Tornehave "From calculus to cohomology", Milnor and Stasheff "Characteristic classes"

  • – Commutative algebra (Atiyah and MacDonald "Commutative algebra")

  • – Sheaves/sheaf cohomology/schemes (Vakil's "Foundations of algebraic geometry")

  • – Representation theory (Fulton and Harris "Representation theory: a first course")

  • – Multiplicative number theory (Davenport "Multiplicative number theory")

Summer reading recommendations from faculty

What follows is a selection of books which have been recommended by the LSGNT faculty for those keen to start reading over the summer. We don't suggest that you read them all (!) but browsing through a couple, maybe on topics you have not met before, will be excellent preparation for some of next year's courses.

Another good place to start looking for stuff to read (with some overlap) is Burt Totaro's Books for Beginning Research.

  • – J. W. S. Cassels "Elliptic Curves", LMS Student Texts 24, 1991.

  • – F. Diamond and J. Shurman, "A first course in modular forms", Springer GTM, 2007.

  • – M. Mendès-France and G. Tenenbaum "The prime numbers and their distribution", AMS Student Mathematical Library, Vol. 6, 2000.

  • – J.-P. Serre, "A course in arithmetic", Springer GTM, 1973.

  • – S. Donaldson, "Riemann surfaces", Oxford GTM, 2011.

  • – R. Bott and L. Tu, "Differential forms in algebraic topology", Springer GTM, 1995.

  • – G. E. Bredon, "Topology and geometry", Springer GTM, 1993.

  • – J. Jost, "Riemannian geometry and geometric analysis", Springer Universitext, 2011.

  • – R. K. Lazarsfeld, "Positivity in algebraic geometry, I and II", Springer, 2004.

  • – P. Griffiths and J. Harris, "Principles of algebraic geometry", Wiley Classics Library, 1994.

  • – R. Hartshorne, "Algebraic Geometry", Springer GTM, 1977.

  • – J. Milnor, "Morse theory", Princeton University Press, 1963.

  • – J. Milnor and J. Stasheff, "Characteristic classes", Princeton University Press, 1974.