Recent trends VII: Richard Montgomery receives the ECM Prize 2024

01 Apr 2025

Richard Montgomery was one of the ten recipients of the prestigious European Mathematical Society (EMS) prize, given in during the 9^th European Congress of Mathematics, in Sevilla. This prize is awarded every four years by the EMS to mathematicians under the age of 35 in recognition of exceptional contributions to mathematics. His award announcement credits him for his solution of the Ringel tree packing conjecture, development of distributive absorption techniques with applications to graph embedding problems, and resolution of several classical conjectures of Erdős and others on cycle lengths in sparse graphs using the novel machinery of sublinear expanders. Let us briefly describe some of these achievements.

A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$ — in which case we say that $H$ packs into $G$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that every tree with $n$ edges packs $2n+1$ times into the complete graph $K_{2n+1}$. The earliest result about spanning tree decompositions is perhaps due to Walecki (1882), who showed that complete graphs on $2n$ vertices can be decomposed into spanning paths. After this, a number of increasingly sophisticated approaches, often using probabilistic techniques and the Regularity method, confirmed the decomposition for relaxations of the problem (such as considering specific classes of trees or approximate decompositions). Finally, in 2021, Alexey Pokrovskiy, Richard Montgomery and Benny Sudakov solved Ringel’s conjecture for all sufficiently large cliques. Their argument involves, among many others, the absorption method (introduced by Röodl, Ruciński and Szemerédi [5]). This method plays an essential role in modern combinatorics, lying at the heart of breakthroughs such as Keevash’s proof of the existence of designs.

Another major contribution of Montgomery is, incidentally, the development of a particular type absorption technique, known as distributive absorption (see, e.g., [3, Section 5]). Absorption, roughly speaking, is a general name given to strategies for decomposing a large object $O$: firstly, one reserves a small substructure of $O$ that has some flexibility in it; on a second step, the leftover structure is nearly completely decomposed; finally, the flexibility of the initially reserved structure is used to convert the approximate decomposition into a perfect one. Absorbers are often built using a combination of probabilistic and structural methods, and we shall not attempt to give details here (instead we refer the interested reader to the Montgomery’s survey article [3, Section 5]).

Distributive absorption was developed by Montgomery in 2019, in an article confirming a conjecture of Kahn [2] about spanning trees of the binomial random graph $G(n,p)$. More precisely, he proved that for each $\Delta > 0$ there exists $C=C(\Delta)$ such that $G(n,C\log n/n)$ almost surely contains a copy of every tree with $n$ vertices and maximum degree at most $\Delta$.

We conclude this brief account by stating some ground-breaking progress obtained by Montgomery and collaborators on an old problem of Erdős. In 1975, Erdős os asked what is the maximum number of edges that an $n$-vertex graph can have if it does not contain two edge-disjoint cycles on the same vertex set. Untill recently, the best known lower and upper bounds were, respectively, $\Omega(n\log\log n)$ and $\Omega(n^{3/2})$. Recently, Debsoumya Chakraborti, Oliver Janzer, Abhishek Methuku, and Richard Montgomery obtained an upper bound of order $n\cdot \mathrm{polylog} (n)$, closing the problem up to the polylog term. Their proof combines a number of novel methods, and an approach to regularizing sparse graphs that might find applications in a number of related problems.

Altogether, the contributions mentioned above are but a small sample of many impressive contributions of Richard Montgomery to Mathematics, certainly worthy of wide recognition!

References

1. D. Chakraborti, O. Janzer, A. Methuku, and R. Montgomery. Edge-disjoint cycles with the same vertex set. arXiv preprint arXiv:2404.07190, 2024.
2. R. Montgomery. Spanning trees in random graphs. Advances in Mathematics, 356:106793, 2019.
3. R. Montgomery. Transversals in latin squares. arXiv preprint arXiv:2406.19873, 2024.
4. R. Montgomery, A. Pokrovskiy, and B. Sudakov. A proof of ringel’s conjecture. Geometric and Functional Analysis, 31(3):663–720, 2021.
5. V. Rödl, A. Ruciński, and E. Szemerédi. A dirac-type theorem for 3-uniform hypergraphs. Combinatorics, Probability and Computing, 15 (1-2):229–251, 2006.

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The author of this post, Tássio Naia, is a Postdoctoral Researcher at the Centre de Recerca Matemàtica.