mathematicians work on problems which are interesting to other people, usually mathematicians. Sone people dont seem to be aware of this, but this is a graph with a general direction trend from the pure to the applied. Show me any math paper and I can tell you how this could potentially help solve real world problems.
I am not an expert on this so I can obviously not give a good judgement but this is about Cohomology and p-adic numbers. Whatever it is they may prove, they are clearly furthering our understanding of these subjects. When you want to apply a theory it really helps if the theory is well-developed, people know different approaches to define things, what are the standard results, what things are equivalent, etc. These kinda of theoretical results work in the background to allow people to effectively and comfortably apply their linear algebra or calculus, for example.
So now I will argue that Cohomology and p-adic numbers are interesting and useful.
Cohomology and hodge theory are about geometry and partial differential equations. This can have applications in AI for example, since data lies on manifolds. I saw some paper a while ago that layers in neural networks fold the data manifold onto itself to reduce its topological complexity and this can be measured by computing some "Betti numbers", which are related to homology and thereby also cohomology. Now is this really true or useful? I don't know but having a mathematical theory makes it possible to even start thinking about such ideas. Also, partial differential equations have obvious applications. By the way, most theories have finite-dimensional/discrete analogues, for example discrete Hodge theory exists, and usually when you understand something about the smooth version of a theory then there are some equivalents for the discrete theory. So if you have data as a graph then you may want to investigate some discrete forms of homology/cohomology on that and may wonder how different types are equivalent etc.
P-adic numbers are related to modular arithmetic and therefore pretty useful just because of computers and cryptography and these things.
