My research explained to non mathematicians

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, and one thinks of shapes as "elastic", like rubber.

For instance, if you hand me a circle-shaped rubber band, I can deform it into a square-shaped rubber band. Topology would be the study of the properties of shapes that are invariant under these deformations. While a geometer might say that a circle and a square are different objects, to the eyes of a topologist, they are the same.

You may have seen the famous idea that to a topologist, a coffee mug and a donut are the same thing ! See for instance the picture below, by Henry Segerman.

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.

An example of this is the equality 1+1 = 1+1. It looks silly, I know. But there are two reasons why it is true! One is the silly one: I wrote the same thing on both sides. Another, less silly one, is that for every two numbers a and b, a+b = b+a. Now, I can decide to use this property for a = b = 1, and this is a (subtly) different reason why 1+1 = 1+1. In classical algebra, this is no cause for concern: an equality is an equality. But if one imagines numbers as living in some "space of numbers", and equalities as being paths between different points in that space of numbers, then all of a sudden this starts to matter.

"Homotopical algebra" studies the algebra one gets when one thinks of numbers this way. Just like in classical algebra, where there are many different number systems with vastly different properties (the natural numbers, or rational numbers, or real numbers, or complex numbers, where the equation x^2+1 = 0 has a solution!), there are even more different "homotopical number systems". Historically, these arose as ways of studying shapes, but they became a subject of study in their own right.

I am interested in understanding these "homotopical number systems" : what they look like, how they behave and relate to one another, how they relate to classical number systems and finally what light they shed on topology, the study of shapes.