This page is still very much under construction.

Topology is, to a certain extent, the study of shapes, such as the circle, or a plane, or a coffee mug - but also far more exotic shapes, in higher dimensions, or still in low dimensions but that cannot be represented graphically faithfully, such as the so-called Klein bottle. It differs from what school students learn as "geometry" in that in topology, one does not care about "rigid" notions such as distances, or angles.

This is not to say that they are literally the same thing, just that a topologist will only study properties that are common to both objects. More generally, a topologist will only study properties of shapes that remain unchanged when the object is deformed "continuously".

This is like putting on a blindfold, and not seeing some properties of the objects - but it can sometimes be very helpful and simplify their study.

This idea, of focusing on specific features of some objects, permeates through modern mathematics. It is in fact related to other fields of research I am interested in : homotopy theory, and higher category theory. Indeed, while the above description of topology is fairly benign to a mathematician, there are other operations where one focuses on a specific features of objects that are less so. I cannot go into too much detail, but let me still try to make this statement slightly more precise :

In focusing on properties of shapes that are invariant under continuous deformations as above, one can stay with no problem in the world of "classical mathematics", where equality is a reasonable notion.

However there are other types of properties of certain objects, where the only reasonable way to really make sense of "putting on the blindfold that only sees those properties" is to go into a different setting. This means, one must leave those classical mathematics into a world where "equality" is a much more subtle notion. In this new world, two objects can be equal in many ways, and the ways in which they can be equal can themselves be compared : an equality between two objects becomes an object of the study itself. You can for instance check out this Quanta article.

Let me also say a few words about representation theory. In mathematics, a lot of objects are rich in symmetries : for instance the square is symmetric along its diagonal, or a horizontal line, or a vertical line, etc. These symmetries can be encoded by some objects, which are called groups.

Groups can be found by taking an object and looking at its symmetries, such as the square from above, but they can also be found "abstractly", with no a priori object whose symmetries they encode. One of the goals of representation theory is to understand how those groups can encode symmetries of various objects - this can help in understanding the groups themselves, as well as the objects whose symmetries they encode. If a group encodes some of the symmetries of a given object, we say that this encoding is a representation of the group.

More specifically, the representation theory I am interested in studies symmetries of "linear" objects. Here, "linear" means that we are in a context where it makes sense to add and subtract things - the square is not such a context, but for instance the real numbers would be. In fact, some tools allow one to pass from "geometric" or "topological" contexts, such as a representation on a square or a circle, to "linear" contexts - so there is a rich interplay between representation theory and geometry and topology.

This has been studied for a very long time, but I am also interested in some more modern version of these ideas, where we change what "linear" means. The way it changes is that we allow equality to become itself an object of study. Where classical algebraists write things like x+y = 3z, or 2+2 = 4 , a homotopy theorist will want to say not only that 2+2 = 4, but also why. This may seem a bit weird, but an example of this is the following : there are two reasons why 2+2 = 2+2: one of them is that x is always equal to x, so x = x, and you can apply this to x = 2+2. Another reason is that x+y is always equal to y+x, so x+y = y+x, and you can apply this to x = 2, y = 2. These two reasons feel very distinct, and in homotopy theory this is somehow reflected by saying that there are two equalities (or "homotopies") between 2+2 and itself.

This complicates matters a whole lot, but also makes the whole thing very fun !