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Combinatorics of the Discrete Hypercube

  • Full or part time
  • Application Deadline
    Applications accepted all year round
  • Competition Funded PhD Project (Students Worldwide)
    Competition Funded PhD Project (Students Worldwide)

Project Description

The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing either in September 2020 for students seeking funding, or in January 2020 or April 2020 for self-funded students. The deadline for funded applications is the 31st of January 2020. The deadline for China Scholarship Council Scheme applications is 12th January 2020.

This project will be supervised by Dr Robert Johnson.

The discrete hypercube Q_n has vertex set {0,1\}^n, with two vertices joined by an edge if they differ in a single coordinate. It can also be thought of as the covering relation of the partial order on subsets of an n-element set given by containment, or as the n-dimensional analogue of the usual geometric cube.

Despite its simple to describe construction, the hypercube has a surprisingly complicated structure and there are many tantalising open questions about it. Many questions in extremal set theory (see [B] for instance) are naturally expressed in terms of the hypercube. Graph theoretic concepts such as isoperimetric inequalities and Ramsey results often have interesting analogues in the context of the hypercube. Properties of the hypercube have found application in social choice, computer science and coding theory, often leading to further mathematical problems.

There are numerous possible directions that a PhD project in this area could take. Here are a few areas (one extremal, one structural, and one probabilistic) to give a flavour of the kind of possibilities with a representative problem in each. The exact direction would depend on the background and interests of the candidate.


For full details of this project, please see: https://www.qmul.ac.uk/maths/media/maths/postgraduate/phd-projects/Johnson_2020.pdf

The application procedure is described on the School website. For further inquiries please contact Dr Robert Johnson at . This project is eligible for full funding, including support for 3.5 years’ study, additional funds for conference and research visits and funding for relevant IT needs. Applicants interested in the full funding will have to participate in a highly competitive selection process.

Funding Notes

This project is eligible for full funding, including support for 3.5 years’ study, additional funds for conference and research visits and funding for relevant IT needs.

This project can be undertaken as a self-funded project. Self-funded applications are accepted year-round for a January, April or September start.

We welcome applicants through the China Scholarship Council Scheme (deadline for applications 12th January 2020).

The School of Mathematical Sciences is committed to the equality of opportunities and to advancing women’s careers. As holders of a Bronze Athena SWAN award we offer family friendly benefits and support part-time study.

References

B. Bollobas, Combinatorics: Set Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability, (1985), CUP.

J.R. Johnson and T. Pinto (2019), The Q_2-free process in the hypercube, preprint: https://arxiv.org/abs/1804.09029

J.R. Johnson, and J. Talbot, Vertex Turan problems in the hypercube, (2010), J Combin. Theory, Ser. A, 117, 454--465

I. Leader and E. Long, Long geodesics in subgraphs of the cube, (2014), Discrete Math. 326, 29--33

N. Morrison, J. Noel and A. Scott, Saturation in the hypercube and bootstrap percolation}, (2017), Combin. Probab. Comput. 26, 78--98

T. Mutze, Proof of the middle levels conjecture, (2016), Proc. LMS, 112, 677--713

Related Subjects

How good is research at Queen Mary University of London in Mathematical Sciences?

FTE Category A staff submitted: 34.80

Research output data provided by the Research Excellence Framework (REF)

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