General

I’m studying for a PhD in mathematics, more specifically in a field of pure maths called Algebraic Geometry. Apart from maths I’m also interested in reading sci-fi and fantasy, watching all types of sport, ice skating and video games.

What I study

I am interested in derived categories, semiorthogonal decompositions, matrix factorisations, non-commutative or categorical resolutions and homological projective duality. I frequently use the representation theory of reductive groups to describe objects and calculate cohomology.

For non mathematicians (WORK IN PROGRESS)

I work in the general field of algebraic geometry, the fundamental idea behind this field is the fact that one can describe a line or a circle in the plane by an equation. Taking this idea much much further gives us the field of algebraic geometry which enables us to turn geometric problems into algebraic ones and vice versa. This is very helpful as we often have good geometric intiution, but it is easier to formalise and prove results using algebra.

The type of geometrical objects that I study are called varities, these are objects that locally look like Euclidean space. For example the Sphere and the Torus look like the plane if you zoom in enough, the boundary of a circle looks like a line, etc. We understand the geometry of the plane rather well, so the idea is that as we can view the sphere as pieces of the plane glued together in a specific way, we can try to understand the sphere by using the geometry of plane and the gluing information.

One other object that arises a lot in my research is a category. This is a collection of objects and maps between the objects (think of relationships between the objects). The idea behind a category is 2-fold, first considering all the objects that can be associated with another mathematical object can often tell us a lot about the object we started with. (This is basically a general principle in maths, if you can abstract away all the unnessary infomation, it is often easier to prove results as all that you have left is the important properties.) The second, and more important idea is the observation that maps between objects (i.e. relationships) are more important than the objects themselves.

As an example, consider the Sphere and a single point. We can map this point to any point in the sphere, this tells us that knowing all the maps at least gives us all the points, but maybe not how they are connected together. Now consider the boundary of the circle, knowing all the maps from this object into the sphere gives us all the paths in the Sphere that have the same start and end point, so we now know some information about how the points of the Sphere are connected. Contining this idea leads to the conclusion that just studying maps into an object gives us all the information about the original object. Now this may seem like a waste of time, but it turns out that in many cases thinking about maps is easier.

Finally, where does non-commutativity arise. First, commutativity says that order doesn’t matter, i.e. 2+3=3+2, 2x3 = 3x2, but this doesn’t always hold. It makes a difference if you put on your socks first, then your shoes, or the other way round. A more mathematical example is 2 - 3 is not the same as 3 - 2, or if you know about matricies, their multiplication isn’t commutative. It is a fact that in the algebra - geometry correspondence you always get a commutative algebra (A mathematical object where you can multiply and add). However there are non-commutative algebras that have the same category as commutative ones and more importantly there are non-commutative algebras that have the same category as geometric objects, and sometimes the non-commutative algebras have much simpler descriptions then the geometric objects. This suggesteds that if you are more interested in the associated category, it is worth considering non-commutative algebras as well.

Does it have any applications?

Maybe to theoretical physics, (there are many connections to string theory) but this work is done for the sake of pure research!

Why do you do research if it has no applications?

I view my work as a type of art that uses logic and equations instead of paint, colors and words. Admitidly this is a type of art the is interesting to almost no one.