# Meet the PhD Graduate: Natalie Behague

Natalie Behague was a PhD student in the Combinatorics group between 2016 and 2020 supervised by Robert Johnson and Mark Walters. We caught up with her to find out about her time at QMUL and what she has been doing since.

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What have you been up to since graduating from QMUL?

I am currently doing my first postdoc at Ryerson University in Toronto. I've been collaborating with a few different groups within the department on a variety of interesting combinatorial problems, including list colourings of graphs, graph searching and subgraphs in semi-random graphs. Some of these projects have met with success and there are some papers in the pipeline! Next year I'll still be in Canada but on the far side, at the University of Victoria.

Your PhD thesis title is Topics in Extremal and Probabilistic Combinatorics. Tell us something about those areas.

Combinatorics is a broad umbrella but it can be roughly described as the study of discrete mathematical objects: in contrast to the continuous objects of geometry, combinatorics is all about individual points.  Examples of objects studied in combinatorics include graphs (in the sense of a collection of nodes and edges between them), two-player games (think tic-tac-toe), set systems and automata. These objects of study are usually finite (though arbitrarily large) and in some cases are randomly generated, which is one way a probabilistic element can come in.

A typical problem in extremal combinatorics asks how large or small some parameter can be, given a structural property the object satisfies. A simple example might be to ask what is the maximum number of edges a graph can have, given it contains no triangles. Often solutions to these problems require two complementary parts: an argument that shows how possible behaviour is limited, along with a construction to show that these limits can be attained.

Can you describe one specific problem you worked on?

A big area within combinatorics is graph theory (a graph in the combinatorial sense is just a collection of nodes and edges between them). Suppose I tell you that I have a graph that doesn't contain any triangles, but adding any new edge would create a triangle. We call such a graph triangle-saturated, by analogy with the notion of saturation from chemistry. If my graph has n vertices, what is the minimum number of edges it can have?

The answer to that particular question is known, but if you replace triangle by some other fixed subgraph G then we do not know in general what the minimum number of edges can be. It is conjectured that asymptotically the minimum number of edges is n multiplied by some constant depending on G, and I investigated this conjecture and generalisations of it. To get technical, I showed that the conjecture fails if we try to generalise to saturation with families of hypergraphs, which are like graphs but where edges contain more than two vertices.

What good memories do you have from your time at QMUL?

Queen Mary has a lovely mathematical community, particularly among the PhD students. I fondly remember leisurely lunches crammed around one table, evenings playing boardgames in the bar and many Friday nights at the pub. This friendliness wasn't restricted to the PhD students either, as the combinatorics group's weekly post-seminar discussion over a meal goes to show. (I'm writing this having been exclusively working from home for over a year, so this question is a dangerous invitation to wallow in nostalgia.)

From a more mathematical perspective, I can recall with clarity the moment I had my first big breakthrough on a problem I had been wrestling with for months (I was, fairly idyllically, ambling along the canal towpath on my way to the office). Doing research gives you this wonderful space to just turn these cool concepts over and over in your head until, if you're lucky, something goes click, like a final puzzle piece dropping perfectly into place.

I'm not claiming combinatorics has a monopoly on neat and satisfying proofs, and I'm definitely not claiming that all proofs in combinatorics are neat and satisfying, but it is certainly an area of maths where a clever trick or an inspired definition can suddenly bring illumination. This is what I love about it.

I enjoy that in combinatorics you can get stuck in straight away. Unlike some areas of maths, problems in combinatorics don't usually require heavy theories to understand and you can hit the ground running, drawing small examples and playing around with them. That is is another thing I enjoy: it can be a very visual field (think graphs, games, colourings) where you can draw pictures to get a feel for the objects you're studying.

What are your interests outside maths?

I enjoy all that a city has to offer: a relaxed weekend afternoon walking around food markets and flower markets in good weather, or visiting museums and playing pub board games when the weather is bad. I enjoy food, both cooking it and trying new restaurants and cuisines. I'm an occasional runner and while living in London I got very into tag rugby -- I remain only passably good at the sport but my team became close friends. During lockdown, I've read many books and done some knitting, which is a great way to feel productive, while watching Netflix.

Any advice for students considering applying for a PhD place?

The two most important things to consider are your field and your supervisor. Pick an area of maths that you are excited about and a supervisor that not only works on problems you find interesting but that you believe you would work well with. Definitely talk to your potential supervisor before you apply! It might help to talk to current or former PhD students too and to visit the department if you can.