The magnificence of objectivity and a couple of solid proofs

“I was bored at school,” he admits, so his father, an engineer, looked for ways to further support his highly gifted son and came across the Hagen Distance-Learning University. “This meant that I could study from home while continuing to go to school as usual, so no one noticed anything different,” Steiner says. At the age of 18, he earned his Master’s degree in mathematics at the same time he graduated from secondary school.

He had already caused a stir among experts with his Bachelor’s thesis. Together with a mathematics professor in Berlin, he wrote a scientific paper prior to completing his Master’s degree and then chose to do his doctorate at TU Berlin. In 2021, he received his doctorate summa cum laude at the age of just 21. “After that, I wanted to stay close to my home in southern Germany,” he says, “so ETH, as an excellent university, was an obvious choice.”

He contacted Angelika Steger, a professor at ETH’s Institute of Theoretical Computer Science, and received a coveted ETH Fellowship. He is particularly pleased that the Ambizione funding also worked out after that. “I’m delighted because this means four years of security,” he says. “Being an academic researcher is great, but as a postdoc, you’re usually already worrying about your future, too.”

Proof with a computer’s help

The topic he’s working on with the support of the Ambizione grant is also not an easy one for the average person to understand. It deals with a conjecture put forth by Swiss mathematician Hugo Hadwiger in 1943 regarding graphs with certain structural properties. The Hadwiger conjecture is that these graphs can be broken down into simple substructures in a very specific way, while at the same time excluding certain other substructures. “That’s what I’m trying to prove, or at least make progress towards proving,” Steiner says. “It’s a major problem; a lot of thinkers have racked their brains over this.”

The Hadwiger conjecture is about graph colouring, Steiner explains. Each node of a graph is assigned a colour. The only condition is that nodes that are connected, i.e. have an edge, must not have the same colour. Part of this conjecture is what’s known as the four-colour theorem. It states the following: if you want to colour a map of different countries so that no two neighbouring countries have the same colour, you will need no more than four colours. Although the statement of the four-colour theorem was conjectured as early as the mid-19th century, it wasn’t proved for over 100 years. “The approach was controversial in mathematics,” Steiner says, “because the proof required computer assistance. That was something new.”

Up to then, mathematicians used to analyse the various possible cases of a conjecture by hand. But in the four-colour theorem, there were so many problematic cases that only a computer could manage it all. “People were sceptical at the time as to whether the program code was actually bug-free,” Steiner says. The proof has now been checked with modern programming languages and there is no longer any doubt. “Still, not everyone is happy about this kind of proof,” Steiner says. “If you solve a problem with a computer, then it becomes a fact, but intuitively you don’t really understand why the solution works.” This challenge is likely to be exacerbated in the near future when AI-based proof assistants are deployed as well.

Intermediate goal within reach

There are also various approaches to proving the much more general Hadwiger conjecture. The starting point could be a new, non-computer-assisted proof of the four-colour theorem, which could serve as a basis for generalisation. “I’ve tried this, too, but it’s tricky,” Steiner admits. Now he’s taking a different approach: “When we can’t solve a mathematical problem, we often simplify it a bit and then try to get closer to the real problem step by step.” An intermediate goal appears to be within reach, as previous work has shown.

Keeping the larger goal in mind, but focusing on intermediate steps – Steiner also does this when indulging in one of his favourite pastimes, chess. “I really enjoy it,” he says. But he also occasionally unwinds by taking part in ETH’s fitness training or by going running. “I also like to sing and I’m thinking about finding a band to join,” he says.

But how does such a high-flyer cope when things don’t work out right away? Steiner readily admits that there are some days at work when the ideas for solutions don’t simply bubble up as from a spring. But the fascination for his work far outweighs this frustration. “The beauty of mathematics is that it also gives definite answers to definite questions,” he explains. “If a well-founded proof of something exists, then even someone who doesn’t like you has to accept it. I think this kind of objectivity is magnificent.”