How is Pure Maths utilized in Computer Science research? How can People with a background in Mathematics pursue PhD in Computer Science"

Although I have a Postgraduate degree in Mathematics with basic programming knowledge,it sounded quite interesting how knowledge in Mathematics can help to solve Computer Science Research Problems.

Since this site is for researchers in Computer Science,Can someone kindly tell:


What type of problems can a Mathematics postgraduate take up in Computer Science?
What is the current scenario of research in Computer Science where Mathematicians like us can make a significant impact?
What are the other skills needed apart from a sound Programming knowlegde to work here?


Any links/references would suffice too.

I have done courses on Linear Algebra,Abstract Algebra,Functional Analysis,Topology,Number Theory and have enjoyed them.

I really dont know how to use the concepts learnt in these courses can be used to solve research questions on Computer Science.

If someone can kindly comment how concepts in the above Mathematical facts can be used to do research in CS,I can understand why a reputed instititution in our country is asking for postgraduate in Mathematics for a PhD in CS.
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?