COURSES: COMPULSORY

COURSES: OPTIONAL

SUMMER READING: STUDENTS

SUMMER READING: FACULTY

Courses for 2023-2024


The plan for 2021–2022 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 Yanki Lekili. - see here to see what was covered last year - this page may change!).
    • (From 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 (Every week in Term 1).
    • The course may change this year, but some things are still up in the air. From last year: 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. Yyou 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.

  • – TBD (S. Donaldson)

  • – Elliptic curves (V. Dokchitser)
    • The Elliptic Curves course is deliberately light on prerequisites, as it's meant to be accessible to both number theorists (algebraic and analytic) and geometers.
    • Prerequisites: A first course in Galois theory (e.g. Stewart's book); - Some familiarity with p-adic numbers. Some background knowledge of algebraic curves or Riemann surfaces may be helpful, but is certainly not essential.

  • – Spin geometry or twistor theory (TCC) (M-A. Lawn)
    • – REC

  • – How to talk to physicists (Independent) (E. Segal)
    • – REC

  • – TBD, possibly K-theory and Monodromy in Algebraic Geometry (Independent) (D. Beralodo)
    • – Thomason, Trobaugh; Higher Algebraic K-theory of schemes/li>
      • – SGA7 I and II "Groupes de Monodromie en Geometrie Algebrique"

  • – Gowers norms and the Mobius function (LTCC) (A. Walker)
    • Students will benefit from having seen a proof of Vinogradov's three primes theorem, and more generally having had a brief acquaintance with the circle method and with Vaughan's identity. This material can be found in many different books: for instance, The Hardy-Littlewood Method by R. C. Vaughan, the third edition of Multiplicative Number Theory by H. Davenport, and (for a modern perspective) The Distribution of Prime Numbers by D. Koukoulopoulos.
    • – The Distribution of Prime Numbers, D. Koukoulopoulos, AMS, 2019
    • – Analytic Number Theory, H. Iwaniec and E. Kowalksi, AMS, 2004
    • – Multiplicative Number Theory I, Classical Theory, H. Montgomery and R. C. Vaughan, CUP, 2006

  • – Algebraic cycles and motives (LTCC) (O. Patashnik)
    • – REC

  • – Beilinson's conjectures, special values of L-functions, algebraic cyclies (LTCC) (Gregory)
    • – REC

  • – Birational Geometry (LTCC) (Spicer)
    • – REC

  • – Introduction to spectral geometry (LTCC) (Lagace)
    • – REC

  • – Modular forms and representations of GL2 (LTCC) (Rockwood)
    • – REC

  • – Topics in probabilistic number theory (S. Lester and I. Wigman)
    • – Students should be familiar with the foundations of analytic number theory, and we recommend reading Introduction to Analytic and Probabilistic Number Theory, Chapter 1-2.4,2,8, by Tenenbaum.
    • –The lectures will be based in part on topics from the following texts: An Introduction to Probabilistic Number Theory, Kowalski; Introduction to Analytic and Probabilistic Number Theory, Chapters 2.5-3, by Tenenbaum. Reading these texts is optional.
    • –Further optional reading: Analytic Number Theory, by Iwaniec and Kowalski.

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.

  • –Lectures on Symplectic Geometry by Ana Cannas Cannas da Silva. (Lekili)
  • –Fukaya Categories and Picard-Lefschetz Theory by Paul Seidel (Lekili)
  • – 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.