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
- 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
- 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.
Multivariable Analysis, Differential 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
- – 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
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)
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"