Representation theory is the art of understanding mathematical concepts by looking at them in the right way.
Consider the complex numbers. William Rowan Hamilton describes them as ordered pairs of real numbers, thus making calculations easier. He then tries to give a similar structure to tuples of arbitrary size. That led to his discovery of the quaternions. They have later proved highly useful, for example in 3D graphics (using quaternions circumvents the gimbal-lock problem encountered when using traditional coordinates).
Over time, representation theory has grown to be a large subfield of algebra. We use it to study algebras, which are mathematical structures with addition, multiplication and scalar multiplication. Instead of studying a certain algebra, we study the modules over that algebra. These give us all the information we need. However, there will typically be infinitely many modules over a given algebra and we have to come up with some way of organizing them.
One approach is homological algebra. It is concerned with short exact sequences, which are groupings of three modules with morphisms between them that fit together in a particular manner. Another approach is tilting theory, which tells us how algebras relate to each other. For instance, it can describe when different categories have similar homological structures.
Now that we have these nice descriptions, the same question arises as it did for Hamilton. What appears if we make the structures larger?
Higher homological algebra is a recent generalisation of homological algebra. Here, the short exact sequences are replaced with n-exact sequences, which contain n+2 objects, instead of three objects. The idea is that we can better understand algebras by using these higher structures.
Thus far in the project we have followed two main lines of investigation. One is to find more examples to work with, to facilitate further research. Existing examples are either very complicated (and thus hard to work with) or very simple (which makes it hard to draw the right conclusions). Our work has excluded a large class of algebras, the so-called gentle algebras, from the possible sources of examples. We are currently investigating wether representations of continuous graphs can be a good source of examples.
The other main line of investigation is to study higher torsion classes. Our results will extend the possibilities for use of tilting theory in higher homological algebra.
The ambition of the project is to implement tilting theory in higher homological algebra.
Homological algebra is a set of powerful tools that provide structure to big, complicated mathematical systems. It often concerns short exact sequences, that is sequences of one injective and one surjective map, which match so that the former is the kernel of the latter. We can also look at it as a sequence of three objects, where the middle object should be larger that the two others.
Higher homological algebra takes the sequences of homological algebra and asks what would happen if one were to put more than one object in the middle, so that we have sequences of a longer but still fixed length. This lets us understand higher-dimensional phenomena and structures of systems.
Tilting theory has been important to understanding rings and algebras by creating equivalence classes of objects that behave in similar ways. That way, an object that looks complicated may turn out to act almost like an object we understand well. We can even tell where the differences are! Tilting theory relies on the tools of homological algebra to work, but also provides tools to use with homological algebra. Outside of algebra, it has also been used in quantum field theory in physics, and in theoretical computer science.
There has already been some effort to implement tilting theory in higher homological algebra. However, tilting theory is a rich field, and there is much work left to be done. We hope to do that, and thus understand still more complicated objects and structures.
A lot of the work we do uses category theory. We can think of category theory as the universal language of mathematics. Thus something that for us looks like a solution to a problem in representation theory may also help someone in geometry, topology or even further afield
