> regarded by many as the world’s greatest living mathematician.
Oh?
Perelman comes to mind (as the only person who has been eligible to claim one of the Millennium prizes), although he is no longer actively practicing math AFAIK. Of Abel prize winners, Wiles proved Fermat's last theorem, and Szemeredi has a number of number-theoretic and combinatorial contributions.
Recently deceased (past ~10 years) include figures such as John Nash, Grothendieck, and Conway.
Tao is definitely one of the most well-known mathematicians, and he's still got several more decades of accomplishments ahead of him, but I don't know that he rises to "greatest living mathematician" at this point.
That said, I do appreciate that he indicates that even his papers get rejected from time to time.