Chasing varieties in infinite dimensions

Our world is permeated by geometric objects. These objects have associated curves and surfaces, otherwise known as varieties, which in turn have a dimension. Lines have dimension one, while surfaces, such as the screen you are reading this text on, have dimension two. And there’s volumes, which have dimension three.

In mathematics, there is no limit in terms of the number of dimensions that a geometric object can have. However, it is impossible for the human mind to envisage a geometric object with, for example, 15 or more dimensions. The primary reason is that we live in a three-dimensional world, a finite-dimensional world.

But is it possible to think about a variety in infinite dimensions? For his PhD research, Alessandro Danelon set out to address this question.

Analogue with numbers

The analogue with numbers expressed in decimals led Danelon’s intuition throughout his research. Consider the numbers 1/7 and √2.  The former is rational while the latter is not, and this is reflected in their decimal expressions.

Indeed, 1/7 = 0.142857142857… is a number where the digits ‘142857’ keep appearing infinitely many times, and this is called the period. On the other side, √2 = 1.4142135623730950488016887242096980785696718753769480731766797379… and there is no period: the numbers after the decimal point never repeat, they do not show any recurrence.

The point here is that even if both of these numbers have infinitely many nonzero digits, the first number is somehow easier as it ‘stabilizes’. It’s possible to record a finite number of digits (142857, the period), and then reconstruct the remainder from those (by translating the period over six digits again and again). It is not possible to do this with √2.

Infinite dimensional varieties

Infinite dimensional variety can be thought of as a limit of finite dimensional ones. Danelon and his colleagues constructed, or approximated their variety step-by-step, by first shaping it in one dimension one, then in two dimensions, then three, and so on. This is the exactly how the numbers 1/7 and √2 are approximated by considering at each step more and more digits.

In the same way as 1/7 is intrinsically different than √2, not all the infinite dimensional varieties are equal. Therefore, Danelon looked at the varieties that stabilize when the dimension tends to infinity - sort of what happens for the rational numbers with the period.

However, in this case, Danelon needed to exploit the geometric symmetries to let stabilization appear. A geometric object can be moved around - in a mathematical sense - with the movements sending each point of his chosen variety to other points in that same variety which are called symmetries.

For example, if a square clockwise is rotated more than 90 degrees around its center (intersection of the diagonals), all points will have hanged their positions but the square occupies the same spot as before.

Danelon’s research focused on infinite dimensional varieties that stabilize up to symmetry, in the sense that the infinite data of the variety can be recovered from a finite amount of data and the associated knowledge of the symmetries.

Amazingly, this can be done, and finding criteria for stabilization depending on the varieties and the number of symmetries is the first and main step in his research.

Application

Danelon’s work might be theoretical but it is rooted in applications, particularly in the fields of theoretical computer science and algebraic statistics, where many algebraic models come in infinite families. Stabilization of the limit of such a family implies theorems that hold for all members of the family.

In a nutshell, taking symmetries into account reduces the complexity of an object and makes it (more) manageable both for the human mind and for a computer.

Title of PhD thesis: Stabilisation for varieties in polynomial functors. Supervisors: Jan Draisma and Rob Eggermont.