PhD Studentships in Extremal Combinatorics
Two 4-year PhD studentships starting in October 2013 are available in connection with the ERC grant
"Extremal Combinatorics" at the University of Warwick.
What is Extremal Combinatorics?
A typical problem of Extremal Combinatorics is to maximise or
minimise a certain parameter given some combinatorial
restrictions. The structures that we consider are
usually graphs and set systems but they may also be, for example,
sets of integers or faces of a polytope. This area experienced a remarkable growth in the last
few decades, having a wide range of applications
(in number theory, algebra, geometry, logic,
information theory, theoretical computer science, etc).
A very recent and actively developing area is graph limits. Roughly, the idea here is to approximate a large graph G by some object W. Now we can deal with W alone (in some sense, we take the "limit" as the number of vertices of G tends to infinity). Remarkably, the set of possible "graph limits" W can be described in many ways: using two-variable measurable functions, random infinite graphs and so-called flag algebras. This gives a general way to apply tools from analysis, ergodic theory and semi-definite programming to finite graphs.
Combinatorics at Warwick
The Warwick Mathematics Institute has an
increasing number of people who directly work in combinatorics (our current
staff members include Endre Csoka, Agelos Georgakopoulos, Roman Glebov, Jan Hladky, Dan Kral', Vadim Lozin,
Andras Mathe, Oleg Pikhurko, and Juraj Stacho) as well as a number of
research groups in related areas (statistical mechanics, tropical geometry, probability, etc). We have the Combinatorics Seminar and the (general) Mathematics Colloquium.
Additionally, the Centre for Discrete Mathematics and its Applications (DIMAP) promotes multidisciplinary research spanning three departments (Business School, Computer Science and Mathematics). The centre has many affiliated researchers and runs the DIMAP seminar.
Each studentship will pay a stipend for 4 years (subject to candidate's making good progress and meeting some standard requirements). It also provides 1500 Euro per year for travel to conferences.
I will be looking for candidates that have an interest in combinatorics and graph theory. While a previous knowledge and close familiarity with these areas will be an advantage, it is not required. Rather, I will be looking for your depth of knowledge, willingness to learn new areas and your general
potential as a successful researcher.
One aspect of this broad project is to apply tools from analysis, algebra, convex optimisation and probability theory to problems of discrete mathematics; candidates with a strong background in one of these areas are also encouraged to apply.
Another direction of the project is to generate mathematical proofs by computer within the framework of
flag algebras. Although there are a number of freely available software packages for this, a new problem often requires quite extensive coding. If you are interested in this direction, please make sure to state this and describe your previous programming experience in detail in your application.
How to Apply
You have to apply via the central university system (http://www2.warwick.ac.uk/study/postgraduate/apply/). Keep the following in mind:
Do not forget to mention my name as a possible PhD supervisor at Warwick (so that your application reaches me).
Although there is no deadline, it is advised to submit your application as soon as possible so that it receives full consideration.
A research statement is an important part of your application.
Please describe what you would like to work
on during your PhD, which open problems you consider important
and why, your previous research experience, etc
I will be happy to answer informal enquiries (those that cannot be answered by the Postgraduate Admissions Team).