A lot of my research work is somewhat technical/abstract. There is a connection between the theory and the application but the questions are usually defined from the perspective of theory. A question I often deal with is whether something is well-defined. I will explain this within the framework of singularities and regularizations. There is a cute photo of my cat at the end :D
A large part of applied math is concerned with making the reasoning in experimental subjects rigorous. A subpart is concerned with developing theory for singularities. Singularities here mean that somewhere you are doing an illegal operation like dividing by zero. The corresponding math is then concerned with extending the system with the singularity to a system without singularity.
These singularity problems are in a sense very similar to the limit problems which you come across in a fundamental calculus course. So let's stary with an example. Below you can see the graph of the function sin(x)/x:
It does not exist at x =0, but I can make it defined at zero by saying that at x=0 it is equal to 1 since if I approach zero from positive or negative x it appears to go to 1. This idea results in the definition of the limit. So for this example x=0 is a singularity. Specifically, it is called a removable singularities since we can extend the definition of sin(x)/x by fixing sin(x)/x at x=0. This yields a regularized equation for sin(x)/x.
This concept can be extended to more complicated systems. Systems which play an important role in the science subjects are differential equations. Informally, these are systems where governing equations can also depend on infinitesimal change of temporal or spatial variables. For example, for planetary motions the force between two bodies is proportional to 1/distance^2 (up to scaling constants). This means that if planets collide the governing equations are not defined. Technicalities on regularizations for the n-body problem can be found here
So we need some kind of limit which regularizes the system. There are two ways to proceed with these type of problems:
You can prove that solutions of these singular systems have a regularized form, just as with the sin(x)/x case,
or you can regularize the governing equations themselves and then apply standard theory to show the existence of the desired solution.
Usually you don't only want to prove that there exists a regularized solutions. You want additional properties of the underlying solutions. So I find that (1) generally doesn't allow you to apply standard theory when you have regularized the system. This is because standard theory usually only is available in a setting where the governing equations do not contain singularities. So generally, I opt for method (2). If you expect that there is no theory for the regularized form you can opt for (1).
Singularities and their regularizations have been a somewhat common theme in my research work. For most of my applications they are the result of a global transformation which gives the governing equations an overall nice form but at some points singularties pop up which you have to deal with. Only after having regularized the system I can apply standard techniques to prove the existence of the desired solutions. Hence, this regularization step is just a subproblem which kicks off the main problem.
Hope you enjoyed my somewhat cryptical mathematics post. Will try to pick something with pretty pictures next time :P
Anyway here is a cute picture of my cat:
Cat tax
