# Courses for 2018-2019

The plan for 2018–2019 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.

## Compulsory courses

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).

## Optional courses

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 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

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. Arithmetic applications.
• 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.

# Summer reading recommendations from students

The following topics were recommended by our current LSGNT students as things they suggest one should know before arriving, arranged into three categories: essential, useful, and for interest.

As a preliminary, it is suggested that you look at Jonny Evan's blog post on reading mathematics.

## Essential

Be familiar with the definition and first properties of the following (eg. read Wikipedia page):
• – Varieties
• – Manifolds
• – Differentials
• – Bundles
• – A cohomology theory
• – Field extensions
• – p-adic numbers
• – Projective space
• – Categories

## Useful

Be familiar with the topics below (many of which are contained in the book recommendations below):
• – Classical modular forms
• – Elliptic curves
• – Number fields, ring of integers, and ramification of primes
• – Schemes and sheaves
• – Riemann curvature
• – De Rham cohomology

## For interest

Here are some books for interest. Those marked with a * are highly recommended.
• –*D.A. Marcus, "Number Fields".
• –I. Stewart and D.O. Tall, "Algebraic Number Theory".
• –*J. Serre, "A Course in Arithmetic", Ch. 1-2,6-7.
• –J.H. Silverman, "The Arithmetic of Elliptic Curves".
• –R. Vakil, "Foundations of Algebraic Geometry", Lecture Notes, Ch. 2-5.
• –I.G. Macdonald, "Algebraic Geometry: Introduction to Schemes".
• –*A. Gathmann, "Algebraic Geometry", Lecture Notes.
• –R. Hartshorne, "Algebraic Geometry", Ch. 1-3.
• –*J.M. Lee, "Introduction to Smooth Manifolds", Ch. 1-4,9-11.
• –I.R. Shafarevich, "Basic Algebraic Geometry 1", Ch. 1.

# 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 (you will notice some overlap with the student suggestions above). We definitely don't suggest that you read them all (!) but browsing through a couple, maybe on topics you have not met before (and in light of the student recommendations above), 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.