What is mathematical research like? I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail exactly? For other research like in economics, or biology one collects data and analyzes it and draws conclusions. But what do you do in math? It seems like you would sit at a desk and then just think about things that have never been thought about before. I appologize if this isn't the correct website for this question, but I think the best answers will come from here.

For me as an independent mathematical researcher, it includes:
1) Trying to find new, more efficient algorithms.
2) Studying data sets as projected visually through different means to see if new patterns can be made visible, and how to describe them mathematically.
3) Developing new mathematical language and improving on existing language.
It is not so different from how you describe biological or economical research, only that you try to find patterns linked to mathematical laws rather than biological or economical laws. 
