Søren Gammelgaard

DPhil in Mathematics

About my research:

I work in the branch of mathematics known as algebraic geometry. This is a really interesting (and fun!) field, with connections to many other parts of maths and physics.

The basic object that we study in algebraic geometry is a variety.

A circle is a basic example of a variety – a circle lying in the plane, say with radius 1 and centred at the origin, is the set of points where the polynomial (x^2 + y^2 = 1) vanishes. An affine variety is the set of points in n–dimensional space where a bunch of polynomial equations in n variables all vanish – and a variety is then a geometric object glued together from affine varieties.

For my master’s thesis, I looked at cubic hypersurfaces nd their Fano schemes. A cubic hypersurface is a variety defined by the vanishing of a single degree-3-polynomial. Especially, such varieties contain lines, which we can parametrise with another variety, namely the Fano scheme associated to the cubic hypersurface.

Currently, I’m more interested in varieties that are quotients of the plane by certain group of symmetries; these are sometimes called Kleinian singularities. Especially, I’m interested in various ways of parametrising tuples of points on these singularities. If you want to parametrise tuples of points on a variety, there are different ways to do it, each slightly different. Together with A. Craw, Á. Gyenge, and B. Szendrői, we found out that many of these different parametrisations for the Kleinian singularities could all be described in a fairly simple way, essentially with just a diagram of dots, lines between them, and two numbers for each dot.

This research forms most of my DPhil work.

Since then, I have extended this work in various ways. Currently, I am trying to investigate three-dimensional singularities.

A more technical explanation of my current research

can be found here.

My completed DPhil thesis can be found here.

Unfinished projects, notes &c:

In 2018, I found a few results involving cubic hypersurfaces, Grothendieck rings, and Hodge numbers. Here’s a short note with some of the ideas.