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Hong Liu, University of Warwick

Hong Liu, University of Warwick

Embeddings in Sparse Graphs.

Hong Liu is a mathematician who received a PhD in December 2015 from the University of Illinois at Urbana-Champaign. Since then, Hong has been working at the University of Warwick.

Hong’s primary research interest is extremal combinatorics. This is a rapidly evolving field of mathematics with connections to many other research areas including algebra, mathematical analysis, optimisation, number theory, statistics, theoretical computer science and statistical physics. An archetype problem in extremal combinatorics is to study the structure of subgraphs appearing in different classes of graphs. Such problems are much better understood when the host graphs are dense thanks to a variety of tools applicable to dense graphs. In sharp contrast, there is lack of such tools for embedding problems in sparse graphs. In practice, many graphs modelling real-life networks, such as Facebook graphs, (artificial) neural networks in biology and machine learning, are sparse. Hong’s proposed research aims to fill this gap by developing a theory of certain expanders for embedding problems in sparse graphs. In a nutshell, Hong plans to design and construct versatile building blocks in sparse graphs using expanders, which can be used to embed subgraphs with various complex structures and have further applications in analogous problems in other fields. The theory developed will make decisive progress on central problems concerning embedding sparse subgraphs with additional arithmetic and topological properties. Some of such problems have remained wide open since the 60s despite active attempts from various researchers.

The duration and flexibility of the Future Leader Fellowship will provide Hong with ample time to build their own research group at Warwick and undertake this ambitious research program. Through this fellowship, Hong will also expand their current network to position themselves as a future leader in their field. Furthermore, to accelerate the impact that their research has on the wider research community, during the first four years of the fellowship, two workshops will take place to bring together researchers from both mathematics and theoretic computer science. These workshops will generate novel ideas and initiate new international collaborations.