Spectral multipliers on 2-step groups: topological versus homogeneous dimension

Mathematical breakthroughs are rare. Solutions to longstanding pure mathematics problems are rarer still. So much so that partially solving a problem – such as showing what a solution definitely is not – can be a major achievement. This is what Birmingham’s Dr Alessio Martini has done in his recently published paper.

Alessio, a lecturer and a member of the Mathematical Analysis Research Group in the School of Mathematics, won the College’s Paper of the Month award earlier this year for the article he co-wrote with Professor Detlef Müller from the University of Kiel in Germany, entitled ‘Spectral multipliers on 2-step groups: topological versus homogeneous dimension’.

Published in the prestigious journal GAFA (Geometric and Functional Analysis), the paper deals with sub-Riemannian geometry and sub-Laplacians, which are pervasive in many areas of mathematics and have increasing importance in wider applications such as control theory and neurobiology.

‘The general setting is harmonic analysis and partial differential equations (PDEs),’ explains Italian-born Alessio. ‘Much is still unknown about the regularity of solutions to PDEs, but it turns out that properties of these solutions are dependent in an essential way on the geometric context.’

In studying physical phenomena, one is led to consider geometries with different characteristics. One of these characteristics is the ‘dimension’ of the configuration space; that is, the number of parameters or coordinates needed to describe the configuration of the studied object. For example, a bead moving on a piece of string has a one-dimensional (1D) geometry.

‘Dimension is an important geometric property, but there are others, such as curvature,’ explains Alessio, whose main area of research is harmonic analysis on Lie groups and their homogeneous spaces. ‘Commonly we think of 1D, 2D and 3D, but depending on the physical phenomena, one may need to work with spaces of arbitrarily large dimension. And properties of differential equations (DEs) may well depend on the dimension of the ambient space.’

Recently, there has been a growth of interest in the study of particular geometric settings where there are properties other than dimension or curvature to take into consideration.

‘At each point of the space, your particle is able to move only in certain directions. Some are forbidden. For example, think of an ice skate on a surface: If you want to model it, you think of it as having a position on the surface, which is 2D. But there is also the direction the skate is pointing in, which gives a third parameter – the orientation of the skate – to be considered. Moving the skate sideways is forbidden: One must turn the skate first. This is the kind of geometry – sub-Riemannian – that I’ve been interested in with DEs.’

One reason for the growing interest in this area of mathematics is because of its potential real-life applications – in particular, modelling the way human perception works.

‘Each 2D cell in the retina somehow may record some orientation and then you get a similar parameter space, so when you look at a picture, the cell orients so you follow shapes.’

The question, says Alessio, is how do you define dimension in this context?

‘If you just think of the configuration space, and forget about the forbidden direction, it’s easy: the answer is 3. This is what we call “topological dimension”. But if you want to define a different notion that takes the forbidden direction into consideration, then it’s more difficult.’

You could ‘count it twice and get 4 instead of 3’. This alternative notion is known as ‘homogeneous dimension’ and better captures the special geometric features of sub-Riemannian spaces. However, if you are interested in PDEs, that wouldn’t be the right answer. Trying to work out what is the right answer is the subject of Alessio’s paper.

‘It’s about trying to understand how certain DEs in those sub-Riemannian settings depend on the geometry,’ he explains. ‘It turns out there is a clear relation between solutions to certain DEs and the dimension. What is not so clear is the right dimension we should be considering.’

In fact, trying to understand how these solutions behave and what is the right dimensional parameter has baffled mathematicians for decades. And, until now, no progress had been made on solving the problem more definitively since the Heisenberg group (the group of unipotent 3×3 upper triangular matrices) in the 1990s.

‘For a particular 3D example, this problem was solved then,’ says Alessio, who came to Birmingham in 2014. ‘Despite the forbidden directions, the original 3 (topological dimension) is the right answer, not 4. But for more general examples of higher dimensions the answer is still widely open. What our paper does is to give a substantial contribution in this direction.’

That contribution is that for a much larger class of examples, this parameter is at least the topological dimension and at most the homogeneous dimension. In particular, Alessio and his co-researcher have shown that it is always strictly less than the homogeneous dimension.

‘It was always known that it was not more than the homogeneous dimension, but now we have shown that it can’t be exactly that. So it’s at least topological but not homogeneous. It was conjecture before; now we’ve proved it. This is a major step forward and it comes at an exciting time for harmonic analysis internationally.’

For more information on research in mathematics, visit the School of Mathematics research pages.