For example, many open problems of (finite) combinatorics were recently solved by using limits of discrete structures. Roughly, the idea here is to approximate, say, a large graph G by some object W of bounded complexity. 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 "limits" W for dense graphs can be described in many ways: using two-variable symmetric measurable real functions (called graphons), random infinite graphs or infinite positive semi-definite connection matrices. This gives a general way to apply tools from e.g. real analysis to finite graphs (and also makes the analytic limit structures into objects of great independent interest).
In the other direction, combinatorial methods and ideas have been very successfully applied to various questions of analysis and descriptive set theory such as, for example, to find a constructive proof of Tarski's circle-squaring problem, where one has to divide a disk in the plane into finitely many "definable" pieces and rearrange them to form a square.
Finally, yet another possible direction for PhD research is to use computers to attack questions of extremal combinatorics within the framework of flag algebras. Although there are a few freely available software packages for this, a new problem often requires quite extensive coding. If you are interested in this kind of work, please make sure to state this and describe your previous programming experience in detail in your application.
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. Also, the Department of Computer Science has the active and large Theory and Foundations Group.
You have to apply via the central university system (http://www2.warwick.ac.uk/study/postgraduate/apply/). See also the information page for prospective PhD applicants prepared by the Warwick Mathematics Institute.
Keep the following in mind:
Email: O dot Pikhurko at warwick dot ac dot uk