General request for a book on mathematical history, for a VERY advanced reader.

I suppose that a very good choice is A History of Mathematics, by Victor J. Katz.

I have not read it myself, but I have heard excellent things about The Princeton Companion to Mathematics. It is not specifically a history book, but apparently has a decent amount of history in it, with many pages devoted to mini-biographies of mathematicians. It is written by, and primarily for, mathematicians.

In the comments, Kevin Long suggested

Hopefully this isn't getting too off topic, but I've heard that when Stan Ulam was in the hospital for encephalitis, Paul Erdos went to meet him when he was discharged and spent a few weeks at his house, plying him with math questions and playing chess with him. At the time, Ulam was afraid that the incident would have affected his mathematical ability, but Erdos helped him build his confidence back up. So if the professor in question is feeling suboptimal, some (small) math problems might be good.

Personally, when I had to stay for longer periods at the hospital, the most difficult part for me was to overcome the boredom. There aren't that many books I would enjoy reading for 8 hours a day, 7 days a week... from that perspective, math problems might make sense, because mathematicians like to spend sheer endless amounts of time on problems that they find engaging. In that spirit, I'd like to suggest The Art of Mathematics: Coffee Time in Memphis by Béla Bollobás. It contains many interesting problems, all with elegant solutions, some due to famous mathematicians such as Erdos.

Littlewoods’s Miscellany

• It is a classic.
• Littlewood does not write for the general public.
• There are wonderful anecdotes about the british academic life in the 1st half of the 20th Century.
• There is plenty of hard-core mathematical content.

The World of Mathematics by James R. Newman, 1956, 4 volumes. This is a collection of 133 essays written by the pantheon of mathematical thinkers: Descartes, Archimedes, Newton, Euler, Galileo, Bernoulli, Malthus, Laplace, Poincare, Mach, Einstein, Boole, Turing and dozens more.

Chapters average about 20 pages and need not be read in order, so the book is ideal to pick up for a brief diversion and then put aside for later.

For more detail, see the review by David E.H. Jones in Nature, 337 (February 2 1989), p. 420. (Link here.)