Concerns about a very old PhD advisor?

My PhD advisor was already retired when I started my PhD. He is in his 80s now. One of my collaborators is due to retire soon; the other is in his 90s. All I can say is that everyone of them has had a tremendous impact on me. They have been the most brilliant people I know.

So I don't see any problem at all with very old PhD advisors. In fact, my experience suggests that you should consider yourself privileged to have an old PhD advisor. A few reasons stand out, at least from my experience:

1. They have a wealth of experience.
2. Old professors are usually kind, generous, and so eager to share with you what they know.
3. They have nothing to lose. They don't have issues that younger professors face such as getting promoted, tenured, or the pressure to publish.
4. Retired professors don't have as many responsibilities as they used to, and so can devote more time advising you. Of course, they probably have other interests they would like to pursue now that they are retired, but research will always be their main interest.

All I wanted to say is that age doesn't matter when it comes to choosing an advisor. Passion is more important. Some professors manage research; some others do research. Choose the latter.

One minor counterpoint to @adipro's answer: rather than age of the professor, consider where they stand in their professional career. Age aside, you want to make sure that they're still highly active in the research community. I made the mistake of being what turned out to be my advisor's last graduate student; not a good position to be in. To present the case more generally: While all the positive points listed by @adipro were are true of someone close to retirement, they may simply might not care as much about things you care about. Specifically:

• They have no pressure to publish, and consequently, your need to publish doesn't scratch any itch of theirs
• Since they're close to retiring, they won't want to enter extensive collaborations, as they will likely be gone before the work is done. (This is referring to long, multi-year projects, not smaller stuff.)
• Their interest in grant-writing will be far less than yours, for the same reason as the preceding point; multi-year research efforts will likely complete after they're gone.

When researching potential advisors, talk to their graduate students about the number of grants they've applied for in the past two years, the number of new students they've taken on, the number of new collaborations they've started, and the like. You want to make sure they're not winding things down, as that means they'll be less interested in things you care about, such as creating new relationships, writing more papers, and finding new grants.

You seem to be thinking only about the downsides, but having an experienced supervisor has many benefits as well:

1. Decades of experience as a research mathematician.
2. Decades of experience as an advisor.
3. Connections to lots of other experienced mathematicians.
4. A developed mathematical taste.

I would gauge your decision accordingly, but if he is an accomplished mathematician, you will probably get to know many of his contacts. If, God forbid, something were to happen to him, both the department and his colleagues/friends should be able help you. Looking after orphaned PhD students sometimes doesn't go very well, but I think the risk of being abandoned is much higher as a student of a young and inexperienced advisor (who quit/moved/whatever).

My biggest worry would actually be #4: If he hasn't been keeping up with new developments in mathematics, you might be learning classical theory, learning to think about problems in an old-fashioned way. This can go two ways: (1) gaining solid foundations in an area of mathematics that's still relevant today or (2) learning mathematics that's not relevant in modern mathematics. There are many old mathematicians who lead research in modern/new developments. There are also old mathematicians, who haven't substantially changed their focus of attention for 40 years and keep on working on old problems that your future colleagues might not find interesting any longer.

If your potential advisor still publishes papers in a range of journals and with younger collaborators from well-known universities, you probably don't have to worry that your knowledge will become obsolete and/or unfashionable any time soon.