To extend Pete Clark's very nice work, I imitated his methods for the decades before and after the period that he covered. It is exhausting to keep making one decision after another, and I didn't bother separating people into main list and honorable mention. Maybe Pete or someone else can help with the latter — feel free to edit this answer!
1850: Sonia Kovalevskaya
1852: William Burnside
1853: Heinrich Maschke, Gregorio Ricci-Curbastro
1854: Henri Poincaré
1856: Andrei Markov, Emile Picard
1858: Edouard Goursat, Giuseppe Peano
1859: Otto Hölder, Adolf Hurwitz
1911: Shiing-Shen Chern, Garrett Birkhoff
1912: Alan Turing, Hans Zassenhaus
1913: Samuel Eilenberg, Paul Erdős, Israel Gelfand, Paul Teichmüller
1915: Kunihiro Kunihiko Kodaira, Yuri Linnik, Laurent Schwartz
1916: George Mackey, Claude Shannon
1917: Irving Kaplansky, Atle Selberg
1918: Richard Feynman
It's clearly time to stop around 1920, not just because of the Fields Medal, but also because more and more mathematicians born after then are still alive. Also, I understand that Feynman is a debatable choice as a mathematician, but I think that he should at least be considered.
