All LSGNT students are expected to attend the following courses.
- – Topics in geometry (coordinated by R. Thomas: every week in Term 1 and alternate weeks in Term 2 - 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 (coordinated by P. Kassaei: every week in Term 1 and alternate weeks in Term 2 - see here to see what was covered last year).
- – Computing for geometry and number theory (J. Armstrong: every week in Term 1).
- 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 either Term 1 or Term 2 and LSGNT students should choose which to attend.
- – Galois representations (F. Diamond)
- – Kähler and hyperKähler geometry (M. Singer)
- – Integrable Hamiltonian systems and Lagrangian torus fibrations (J. Evans)
- This course will cover some subset of the following topics:
Hamiltonian mechanics, examples of classical integrable systems (pendulum, spinning tops), toric manifolds and convexity, the Arnold-Liouville theorem and action-angle coordinates,
integral affine manifolds, focus-focus singularities (Ngoc's invariant, Symington's almost toric manifolds, mutation of polygons), singularities of SYZ fibrations, Ruan's method (Lagrangian torus fibrations on the quintic 3-fold),
Gelfand-Cetlin systems, Newton-Okounkov bodies.
- – Quantum ergodicity and number theory (S. Lester and Y. Petridis
- – Euler systems and Iwasawa theory (S. Zerbes)
- – 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
The following optional courses and titles have not yet been confirmed, but at least some are expected to run.
- – Introduction to p-adic cohomology (A. Pal - only one of his two courses will run)
- Syllabus: De Rham cohomology for smooth manifolds, algebraic de Rham cohomology, Grothendieck's comparison theorem. The Gauss-Manin connection.
Basics of formal schemes and rigid analytic spaces. The definition of p-adic cohomology via frames. The strong fibration theorem, computations of basic cases. Isocrystals. The p-adic monodromy theorem, finiteness of cohomology.
- – Introduction to motivic homotopy (A. Pal - only one of his two courses will run)
- This course would be a continuation of this year's course Arithmetic and Homotopy, but without relying much on what is covered.
The aim would be cover cover some of the homotopical background of Morel's computation of the 0th stable motivic homotopy group and then move onto covering parts of Morel's book.
- Syllabus: Cobordism and Bordism. Thom-Pontryagin construction, killing spaces, Thom's computation of the unoriented cobordism ring. Oriented generalised cohomology theories.
Relations to formal group laws, elliptic cohomology. Definition of the A1-homotopy category, unramified Milnor-Witt K-theories,A1-homotopy and A1-homology sheaves. The motivic Hurewitz theorem.
- – Low-dimensional topology (S. Sivek)
- The first part of this course would be devoted to foundational material about smooth 3- and 4-manifolds,
and the second half would be an introduction to gauge theory and its application to some major problems in the field.
- Prerequisites: The main prerequisites would be some algebraic topology (homology and cohomology) and differential geometry (smooth manifolds, vector bundles), nothing too advanced will be assumed.
- – Symplectic resolutions and singularities (T. Schedler)
- Symplectic resolutions generalise Springer resolutions and quiver varieties (connected also to Cherednik/symplectic reflection algebras).
It's connected to representation theory, D-modules, quantum cohomology and Gromov-Witten theory.
- – Differential geometry (S. Donaldson)
- This will be an elementary course in differential geometry, with a focus on Riemannian geometry.