Introduction to additive combinatorics (TCC)In a nutshell
Synopsis  Syllabus  Organisation  Assessment  BibliographySynopsisThis course serves as a first introduction to additive combinatorics, a subject that has a substantial history but has gained much attention in recent years as a result of numerous highprofile breakthroughs such as the GreenTao theorem on arithmetic progressions in the primes. Historically, additive combinatorics addresses problems regarding the additive structure of the integers. For example, what can we say about a finite set A of integers if we know that its sumset A+A={a+a': a, a' in A} is small? It turns out that such sets are highly structured, and we shall see various quantitative results to this effect in the 'toy setting' of F_p^n. We shall also study other additive structures such as 3term arithmetic progressions in dense subsets of this vector space. The aforementioned results about arithmetic structure are intimately related to structural results about large graphs, and in particular Szemerédi's celebrated regularity lemma. After proving it and deriving some graphtheoretic as well as arithmetical consequences, we shall give some examples of applications to theoretical computer science. The methods employed throughout are a mix of analytic and algebraic techniques, together with some elementary probabilistic arguments and some handson combinatorics. For a precise list of topics please see the detailed syllabus below. The prerequisites for this course are minimal. We shall need the fundamental notions of (discrete) Fourier analysis as well as basic concepts from linear algebra and discrete probability. Feel free to contact me at julia.wolf at bristol.ac.uk with any questions prior to the start of the course. Syllabus
OrganisationThis course will not take place on Tuesday, 29th October 2013. AssessmentAt the end of the course, participants will choose from a list of original research articles and write up an exposition of the chosen result. This exposition should place the result in the context of what has been discussed in the course, and should be sufficiently detailed for other course participants to be able to follow the main steps of the argument. The completion of the short weekly problem sets is optional but strongly encouraged. BibliographyThe following reading material may be helpful to course participants. You may also wish to consult my personal roadmap for interesting surveys and original research articles in this general area.


This page was last updated 23rd October 2013.  © 20032024 Julia Wolf 