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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.

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    $\begingroup$ I'm afraid that this question is far too broad to be able to have a clear and concise answer. While some of these kinds of questions are tolerated as a 'soft question' on e.g. Mathoverflow or Theoretical Computer Science, here on Computer Science, we generally do not have such questions. I can provide some quick guidance, however. CS is a very diverse field, and a lot of subfields in theoretical computer science could have been subfields of pure math. While elementary knowledge of programming is assumed to be mastered by experts in the field, a lot of CS work can be done without any knowledge of programming at all. $\endgroup$ – Discrete lizard♦ Mar 2 '19 at 17:11
  • $\begingroup$ This is far too broad. But, essentially, the entirety of theoretical computer science is mathematics, just like theoretical physics (except mostly different parts of mathematics). $\endgroup$ – David Richerby Mar 2 '19 at 18:52
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    $\begingroup$ Mathematics is about as useful in computer science as it is useful in physics. If you tell us what branch of mathematics you are most familiar with, we will tell you where computer scientists use it. I bet they do. $\endgroup$ – Andrej Bauer Mar 2 '19 at 20:19
  • $\begingroup$ @AndrejBauer,I have edited my question as much as I could.In case you require more insights do ask me $\endgroup$ – user101150 Mar 3 '19 at 4:00
  • $\begingroup$ @DavidRicherby,Can you kindly comment how knowledge in Topology serves as a base for CS research $\endgroup$ – user101150 Mar 3 '19 at 4:01
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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.

  1. 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.

  2. Abstract algebra is used to model computational effects, such as exceptions, input/output, memory, parallel computation etc.

  3. Functional analysis is used in the JPEG image format which uses Fourier analysis to achieve lossy compression.

  4. 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.

  5. 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?

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  • $\begingroup$ Its really a wonderful answer.Thank you very much $\endgroup$ – user101150 Mar 4 '19 at 3:35

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