Welcome to regulous geometry

Interview ---------

What field of research does your thesis fall within?

My field is real algebraic geometry, a major branch of mathematics. I chose the broader field of geometry and algebra because it is full of the mathematical questions and methods I enjoy most, along with logic.

Why did you choose this topic?

I settled on this particular field after meeting my supervisor in the second year of my master’s degree, when I was looking for a thesis topic: I went around my laboratory talking to the various researchers. He had posted a PhD opening, and I applied. The specific topic was chosen during the first year of my PhD. We had a lead for another topic, but after four months it proved fruitless, so we changed direction and took up a question he had encountered on several occasions and that was not too far removed from our previous lead.

How does this thesis fit into your plans?

For quite some time now, my plan has been to do a PhD in mathematics. It is more or less an end in itself. Since I work in a theoretical field, there are not many career paths: the private sector is not necessarily interested in these topics, while the number of public-sector research positions has been plummeting for some twenty years.

In 1637, René Descartes published the Discours de la méthode. Alongside the famous "I think, therefore I am" was a manuscript, La Géométrie, describing one of the major mathematical advances of the period. It first showed how, by using a coordinate system, every geometric object could be given an algebraic counterpart. For example, every line can be represented by an equation of a line: an equality satisfied by the coordinates of the points on the line, and only those points. Every straightedge-and-compass construction can then be associated with an algebraic operation. Finding the intersection of two lines, for example, amounts algebraically to solving a system of two equations in two unknowns. All that remains is to apply the theory of equations in order to solve the various geometric problems that arise.

If we consider only equations expressed using addition, subtraction and multiplication—so no powers, sines, cosines and so on—we obtain the field of algebraic geometry.

Dividing by zero! -------------------

The framework of algebraic geometry works extremely well with complex numbers. It struggles, however, to capture certain phenomena over the real numbers, for which it is useful to add division to our toolkit. But division can itself be problematic, since there is always a risk of dividing by zero. To avoid this, we divide only by quantities that never vanish, such as 1 + x².