Geometry, paintings, and Lance

‘One can argue that a painting is only a shadow of an imagined object that exists in the mind of the painter’, says Paul Arne Østvær, Professor of Mathematics at the University of Oslo.

If you ask two painters to paint the same landscape, he continues, the results will most likely be different, and it can even be impossible to recognise which landscape it is.

‘It is sort of the same with the subject we call Motivic Homotopy Theory’, Østvær explains, referring to the fact that mathematicians attach different numbers or structures to the same object, just as painters will each portray a motif differently.

His CAS project, Motivic Geometry, gathers leading scholars in the field from France, Germany, Italy, Japan, Poland, Russia, the UK and the US. 

‘We study the motives of mathematical objects that are supposed to capture all the information about a given object that one can possibly imagine.’

This idea, Østvær says, goes back to the German-French mathematician Alexander Grothendieck (1928-2015), and was developed in detail by the Russian-American mathematician Vladimir Voevodsky (1966-2017).

Helping Lance to return home

The project participants study zeros of polynomial equations, which one learns about in high school. A first degree polynomial is an expression of the form x, x+1, x+y, x+y+z, and so on, whereas xx-xy and xx+yy-1 are examples of second degree polynomials. There are infinitely many possibilities.

‘Obviously, those types of polynomials describe a lot about the world. We observe geometric figures everywhere, for instance in this room’, Østvær demonstrates by pointing at the table we sit around. 

‘Now, your coffee mug is in some sense much more interesting than the table, because it has a circle described by the polynomial equation xx+yy=1. Mathematically, you cannot deform your mug to a single point in a continuous way because it has a hole.’

You wrote in your project abstract that you expect a jump in knowledge in the field of motivic geometry.

‘Motivic Geometry is a rapidly developing area of research in pure mathematics. The project aims to solve central computational problems, enhance counting techniques in algebraic geometry, and lay the foundation for new directions of research based on a few key ideas’, Østvær responds.

One idea is on the theme called compactification. This is where Lance the biker comes in.

‘You can sort of think of it as taking an object that stretches out in infinite directions, and then we want to make it into an object that looks and feels like something that is finite’, Østvær explains.

Every physical object we observe is finite. This table; your coffee cup; the earth.

‘But when we work with this table mathematically, we imagine that it stretches out infinitely’.

Østvær, an eager powerlifter, naturally chose a sports icon for his analogy when explaining compactification.

‘My friend Lance likes to bike’, he begins. ‘The longer the track, the better for him, so he wants to ride his bicycle along an infinite track.’

However, Lance’s family is not too happy about him being away for that long.

‘Compactification is sort of a trick that brings Lance back home. You take the two ends of an infinite stretch of road and join them at a point, at infinity, where the two ends meet and form a circle.’

Lance can now return home by biking past the point at infinity.

‘This is the basic idea of a compactification. We hope to employ this notion to formulate and solve interesting problems.’

The railway track that Lance is often biking along is in mathematical jargon called an affine line. It describes polynomials in one variable. For polynomials of more variables, one can study affine spaces of arbitrary dimension. 

‘But every so often he bikes along the affine line or cylinder on some unknown path. If the cylinder on the unknown path turns out to be an affine space, then Zariski's cancellation conjecture from the 1950s predicts that the unknown path is also an affine space,’ Østvær says, referring to one of the problems the research group expects to deepen knowledge of. 

A conjecture, Østvær explains, is a notably important mathematical statement that is suspected to be true, but whose proof or disproof is still pending, even if the supporting evidence might be overwhelming.

‘Despite numerous attempts, Lance has not found any path that contradicts Zariski's conjecture. This gives overwhelming evidence for the conjecture, but mathematicians have not been able to provide a proof.’

Lance is therefore trying to find other paths in different corners of the universe. A candidate is the so-called Koras-Russell threefold in dimension three.

‘The fact that its cylinder has four dimensions represents a challenge, even for Lance.’

Another problem the CAS scholars expect a deeper understanding of is an emerging motivic Poincaré conjecture, named after the father of algebraic topology, Henri Poincaré. The topological Poincaré conjecture is one of the deepest conjectures in mathematics – it took nearly a century to find a solution.

You say that you want to understand the motivic Poincaré conjecture. What do you mean?

‘In the simplest case, we try to use motivic homotopy to distinguish between the Koras-Russell threefold and the affine space of dimension three.  All the known motivic shadows of these two objects are the same. This means that if we zoom in on these two objects with our motivic glasses, then they look identical to us. But with better technology we should be able to say they are genuinely different. To that end we need to find the right motive that enables us to distinguish the two.’

Østvær says that sometimes it can be difficult to prove that two mathematical objects are different. In the motivic Poincaré conjecture, we are basically asking for a characterization of affine spaces among infinitely many other objects of the same dimension.

‘There are so many possibilities, which makes it interesting and challenging.’

‘Every researcher’s dream’

Why is CAS a suitable arena for your research?

‘I’ve enjoyed extended visits to foreign research institutes, such as the Institute for Advanced Study in Princeton and Institut Mittag-Leffler in Stockholm. This place is unique in Norway and it is in my hometown. The program runs for one academic year, and it includes a generous budget for inviting fellows. During the program, I can enjoy complete academic freedom and the opportunity to map out future research endeavours. It is every researcher’s dream; I find it very rewarding to organize a program at CAS’, Østvær answers. 

How are you dealing with the corona restrictions?

‘These days we have several activities on Zoom with participants across different time zones.  We try to make the best out of the current situation, and it works quite well. We have a weekly seminar and lecture series online’, he says, and explains that the project group has their own YouTube channel, enabling people from all over the world to follow some of the activities in the program.

‘But of course, this is not popular science’, Østvær concludes. He hopes to welcome more fellows in the spring.