How is Pure Maths utilized in Computer Science research?
Let's do these in the order you listed. I will not be comprehensive, I will just try to impress you by pointing out cool applications of mathematics in CS. Others may know of even cooler applications.


Linear algebra is used to compute the Page rank of web pages, which is the basis of Google engine ordering pages from more relevant to less relevant.
Abstract algebra is used to model computational effects, such as exceptions, input/output, memory, parallel computation etc.
Functional analysis is used in the JPEG image format which uses Fourier analysis to achieve lossy compression.
Topology is used in domain theory to give mathematical models of programming languages. Algebraic topology is used in topological data analysis, and homotopy theory is used in homotopy type theory to study equivalence of datatypes (I am oversimplifying) and how to convert between them.
Number theory is used in cryptography to provide public key protocols, such as the RSA algorithm. These are used every time you buy anything on the Internet.


So, is that convincing enough?