To cheer you up in difficult times, here is a wonderful post by Peter Cameron about doing research. And see also this entertaining post by Peter on mathematics and religion.
Probably every research mathematician has been asked the question, “How do you do mathematical research?” Some lay people think we simply figure out ways of doing bigger and bigger long multiplications. Many more people think that all the mathematics must have been discovered by now, so what are we doing?
These misunderstandings are about what we do research on rather than how we do it, the question I want to focus on here.
In 1945, Jacques Hadamard published a remarkable book, The Psychology of Invention in the Mathematical Field. He was a great mathematician himself, one of whose achievements was the proof of the Prime Number Theorem fifty years earlier. He examined his own mental processes, searched the writings of his predecessors, and sent questionnaires to many leading mathematicians and scientists of his time, including Albert Einstein.
Hadamard’s conclusion was that the process of mathematical discovery can be broken…
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Combinatorics and more 