What is the role of mathematics in programming? I am almost totally uneducated in programming and computer science in general, though I am really fascinated by these topics. I read that mathematics is important in programming and that nowadays many mathematicians specialise in applied mathematics in this field. So, I wonder how these two branches are linked with each other. What are the fields of mathematics that are useful (maybe unavoidable?) in writing a computer program?
You don't need any math to write a Hello World or a very simple website.

You will need to know some discrete mathematics and algorithm analysis to write a program that finds a route between two cities.

You will need to know matrix transformations and quaternions to write a game engine.

You will need to know a lot about all kinds of mathematical fields to write the next Wolfram Mathematica.

What I'm trying to say, what fields of mathematics you will need to know and to what degree will depend a lot on what kind of programs do you want to write. But the ones that tend to be most widely useful are data structures, algorithms and their analysis.