What is on the cutting edge of mathematics?

The 20th century had many advances. Claude Shannon created information theory, Alan Turing created theoretical computer science before there were computers, John von Neumann created game theory and John Nash made significant additions to game theory. As to whether all the low lying fruit has been taken, this is my favorite story.

The work of Shinichi Mochizuki might be the sort of thing you have in mind. He has been examining the simplest of mathematical relationships eg a + b = c and coming up with new ideas about the numbers involved. His ideas are pretty deep and even mathematicians don’t yet understand them.

Srinivasa Ramanujan did pioneering work in mathematics in the early twentieth century but I wouldn’t call it low hanging fruit. He was a poor Indian clerk and a mathematical genius who was brought to Oxford. There’s a very good film about him The Man Who Knew Infinity that you might enjoy..

Vera Rubin died on Christmas day, 2016. Her research on the existence of black matter changed science. Not bad for someone many in the science and mathematical community would not recognize because she was a woman.