Except for John Conway, I believe that these "modern mathematicians" are all dead. Most were dead by the time the book was published, so that was not the author's criterion for "modern". It is not clear what was.

This book is a bunch of capsule biographies of mathematicians, starting with Ada Lovelace and Charles Babbage and ending with John Conway. Several of the dozen or so mathematicians made significant contributions to computer science, that circumstance may give them a claim to be modern. Mathematicians covered:

Charles Babbage and Ada Lovelace - the analytical engine
George Boole - the laws of thought
Georg Cantor - classifying infinities
Sofia Kovalevskaia - Of aristocratic descent and married a principled revolutionary so she could travel. Worked in calculus and theories of planetary motion.
Emmy Noether - Because her mathematical contributions had such an importance in modern physics her invention of Noetherian induction is not even mentioned. It is not clear to me, from the general description, what her contributions to physics precisely were. We all know what an invariant is.
Srinivasa Ramanujan - Self-taught number theorist.
Stanislav Ulam - UW-Madison - Began work with von Neumann on automata.
Shing-Chen Chern - A geometer whose work influenced modern physics.
Alan Turing - Favorite dead computer scientist
Julia Bowman Robinson - Another number theorist and the first woman in this book who was not absolutely stamped all over by the establishment.
Benoit Mandelbrot - Fractals, chaos, and economic markets
John Conway - Cellular Automata ( )