# 4-Year PhD Scholarship

A 4-year PhD scholarship is available at the University of Warwick, funded by Oleg Pikhurko's ERC advanced grant "Finite and Descriptive Combinatorics". The starting date is 1st October 2022.

## Possible research directions

This broad project allows for a wide range of possible directions for PhD research, depending on the applicant's background and interests. In addition to extremal (finite) combinatorics, other possibilities are to explore deep emerging connections between combinatorics and other fields, such as analysis, descriptive set theory, distributed algorithms, measured group theory, semi-definite programming, etc, with applications going both ways.

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.

## Combinatorics at Warwick

The Warwick Mathematics Institute has an increasing number of people who directly work in combinatorics (our current permanent staff members include Agelos Georgakopoulos, Hong Liu, Vadim Lozin, and Oleg Pikhurko) as well as a number of research groups in related areas. We have the weekly 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. Also, the Department of Computer Science has the active and large Theory and Foundations Group.

## Scholarhip Particulars

The scholarship will pay a stipend for 4 years (subject to candidate's making good progress and meeting some standard requirements) and all tuition fees (at home or overseas level). It also privides 2,500 euro per year for research-related travel and 1,500 euro for buying a laptop.

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:

• For full consideration, please make sure that your application is complete (including all reference letters) before 1 February 2022.
• Do not forget to mention my name as a possible PhD supervisor at Warwick (so that your application reaches me).
• In your research statement, please do not forget to describe your mathematical background, interests, and your previous research or programming experience (if any). If there are already some areas, or even concrete open questions, that interest you as possible PhD topics, then please also elaborate on them in your statement.
• ## Enquiries

I will be happy to answer informal enquiries (those that cannot be answered by the University or Mathematics Admissions Teams).

Thank you!

Oleg Pikhurko
Email: O dot Pikhurko at warwick dot ac dot uk