Propagating rumors and spreading gossip (with combinatorics)! Meet our PIMS PDF at the University of Victoria, Natalie Behague.
Ruth A. Situma, PIMS Programs & Communications
When Natalie moved to Victoria in January, she was relieved that things were just beginning to open up. Starting her new position and meeting everyone in-person, rather than online, was wonderful, both for the ease of communicating maths and for making friends with her colleagues. Natalie completed her first PDF in Toronto, where working entirely from home, made her feel isolated. “I am still working online with some long-distance collaborators and it is useful to have developed good ways of working remotely (meeting on zoom, writing on tablets, working asynchronously) but everything runs so much more smoothly and seamlessly when you are in a room together,” she says. A native of England, She was quite nervous about choosing another post that was far from home and potentially as hard as it was living in Toronto. But with Vancouver Island opening up she could explore places and meet people in the new city. She still misses her Toronto flatmates, who were a welcome dose of sunshine and a constant source of baked dough!
Tell us about your academic journey: What field are you in and how did you get there?
I work in probabilistic and extremal combinatorics, in which we study discrete mathematical objects such as graphs, two-player games (think tic-tac-toe), set systems, and automata. These objects of study are usually finite but 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.
I completed my Ph.D. in 2020 at Queen Mary University of London under the supervision of Robert Johnson, working on a variety of different problems under the broad umbrella of extremal combinatorics. During my Ph.D., I published two single-author papers: ‘Hypergraph saturation irregularities’ and ‘Semi-perfect 1-factorizations of the hypercube’. The former answers a question of Pikhurko’s on a generalization of Tuza’s conjecture to hypergraphs — Tuza’s conjecture being the big open problem in the theory of graph saturation. The latter is a clever construction showing that a hypercube of any dimension can be broken into pieces that relate to each other in a way that is ‘semi-perfect’, which is best possible under certain restrictions.
I followed my Ph.D. with my first postdoctoral position at Ryerson University in Toronto, working with graphs together with the Ryerson group. Despite having to work exclusively from home I collaborated on several successful projects with the group. This has already resulted in two published papers: ‘The localization capture time of a graph’ with Anthony Bonato, Melissa A. Huggan, Trent Marbacht, and Brittany Pittman; and ‘Tight bounds on probabilistic zero forcing on hypercubes and grids’ with Paweł Prałat and Trent Marbacht. Probabilistic zero forcing is a model for rumour (or infection or fake news) spreading across a network, where a node is more likely to spread the rumour if a large proportion of their neighbours also believe it. We investigated how long it takes for the rumour to propogate. There are also more papers in the works including an interesting project on semi-random graphs with Paweł Prałat, Trent Marbacht, and Andrzej Ruciński.
I first met my current postdoctoral fellowship supervisors, Natasha Morrison and Jonathon Noel, back when they were Ph.D. students and I was a masters student applying for Ph.D. positions. Natasha gave me a tour of the department at Oxford University where she was studying, and we clicked instantly due to some combination of shared interests and gossip about Cambridge University — she had studied for her masters there the same as me. I didn’t end up doing my Ph.D. at Oxford but my path crossed with both Tash and Jon’s paths several times at conferences and events in the small UK combinatorics scene. When I saw that they were advertising a postdoctoral position I lept at the chance to work with two people who work on problems so closely aligned to my interests and who I knew I could be friends with. In the few months I have been here we have been working on applying flag algebras to get results about ‘commonness’ of graphs — watch this space!
What do you do to balance your research and life? What does a typical Sunday look like for you?
Having just returned from a holiday to Banff, I thoroughly recommend taking time off to spend with loved ones, especially when it is in scenery as stunning as the Rockies! It felt like a real adventure, with many firsts for me: using crampons, trying cross-country skiing and seeing elk.
When I came to Victoria, I set myself a goal of doing one new thing every week, and, bar one extremely wet weekend, have so far succeeded. I’ve explored downtown — the tourist hotspots, the bookshops, the cafes, and going on hikes in local nature reserves and beauty spots. I am starting to exhaust things that are nearby, but fortunately, the weather is warming up and I can start cycling further afield. I have made a small group of friends since moving here, and we often explore together, going for a hike, playing pool or trying out a new pub.
A typical Sunday usually starts with a catch-up with my family and my boyfriend back home. Sometimes I get eggs benedict with Tash and Jon at our favourite brunch place, or I might go exploring. If the weather is terrible I will probably buy some groceries and try out an interesting new recipe, bake a cake, or just read a book. On Sunday evenings there is a regular meet-up in a board game cafe, which is how I met my friends. It is a fun and competitive way to finish the weekend!
What is your best discovery since arriving in Victoria? Anything you miss from your time in Ontario?
There are many things I enjoy about Victoria: the famous west coast sushi, the bike trails that take me from my house to UVic, to downtown or out of the city, the brunch spots, the bug zoo… I think the best thing is probably the beautiful nature and wildlife on our doorstep, of which there is so much to explore. I have seen seals, sea lions, and even orcas!
Without a doubt the best parts of my time in Toronto were my two flatmates, without whom living in lockdown in a strange new city would have been unbearable. One was always taking me on an adventure to explore a ravine or climb a tree, the other supplied delicious freshly baked goods on an almost daily basis, and both were a joy to spend time with.
Natalie Behague completed her PhD in 2020 at Queen Mary University of London under the supervision of Robert Johnson. Prior to this, she completed both her Bachelors and Masters degrees at the University of Cambridge. After finishing her PhD she spent a year at the University of Ryerson in Toronto with the Graphs at Ryerson research group. She has worked on various problems under the broad umbrella of probabilistic and extremal combinatorics, including automata, graph saturation, graph factorization and probabilistic zero-forcing (a model for infection or rumour spreading across networks). Since the start of 2022 she has been a PIMS postdoctoral fellow at the University of Victoria, working with Natasha Morrison and Jonathan Noel.
Natalie will be speaking at the PIMS Emergent Research Seminar Series, on May 25 , 2022, at 9:30 AM Pacific. Details on her talk, Subgraphs in Semi-random Graphs can be found here.
