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I know someone who, as a PhD student, answered a question of Serre (you might call it a conjecture) on cohomological invariants of some finite groups. Instead of congratulating him for his results, Serre became mad at this guy, accused him of stealing his ideas, saying that "the main steps of the proof were already known to him, and that he was going to publish very soon a paper answering his own question." The guy was forced by Serre's affiliates to rewrite his paper and explicitly mention that his work contains no original contribution as "everything was already known to Serre" (but of course not published).

I know someone who, as a PhD student, answered an implicit \footnote{added as per suggested by Dan Petersen} question of Serre (you might call it a conjecture) on cohomological invariants of some finite groups. Instead of congratulating him for his results, Serre became mad at this guy, accused him of stealing his ideas, saying that "the main steps of the proof were already known to him, and that he was going to publish very soon a paper answering his own question." The guy was forced by Serre's affiliates to rewrite his paper and explicitly mention that his work contains no original contribution as "everything was already known to Serre" (but of course not published).

I don't think being 29 could ever be considered a disavantage on the intellectual or creative level to start a PhD. The comments below your question give you links to lists of famous mathematicians which were late starters. One famous example is Stephan Banach who wrote the equivalent of his master thesis at 28 and got the equivalent of what we call today a PhD at age 30.

If you plan to get a PhD and then move to the industry or/and work for private compagnies, I guess (though I don't have a formal proof, only examples from friends and colleagues) that your professional life won't be any significantly different or harder than that of your colleagues which got their PhD a few years younger than you. On the other hand, you have to know that there is an extremely fierce competition for jobs in the academia.

It has now become standard to be on very unstable positions (called post-doc positions) for at least 3-4 years after the completion of your PhD. And sometimes up to 10 years! (I have seen that among younger colleagues). During these years, you need to gain recognition from the bigwigs in your field, so that they can support your application for the next stage of your career  : the tenure track position (which I will describe below). And that might be extremely difficult, even if you prove some big results.

I know someone who, as a PhD studend, answered a question of Serre (you might call it a conjecture) on cohomological invariants of some finite groups. Instead of congratulating him for his results, Serre became mad at this guy, accused him of stealing his ideas, saying that "the main steps of the proof were already known to him, and that he was going to publish very soon a paper answering his own question." The guy was forced by Serre's affiliates to rewrite his paper and explicitely mention that his work contains no original contribution as "everything was already known to Serre" (but of course not published).

But that is a single example, and obviously, most PhD don't end up like this. On the other hand, even if you succeed in having your pairs acknowledge your work in a positive way and find some good post-doc positions, you still are in the middle of the jungle. Indeed, if you gain enough support from the bigwigs in your field, you can only upgrade from post-doc positions to a tenure track position.

Then, finally, after 10 to 15 years of such a life (where you might have to move out places every 2 or 3 years), you may hope for a stable and permanent position. Which means that if you start your PhD at 29 and plan to work in the academia, you might secure a permanent position at 40 the earliest. Granting the fact that you have been able to give plain satisfaction to the numerous bigwigs you will encounter during this 10 to 15 years period of time.

I don't think being 29 could ever be considered a disavantage on the intellectual or creative level to start a PhD. The comments below your question give you links to lists of famous mathematicians who were late starters. One famous example is Stephan Banach who wrote the equivalent of his master thesis at 28 and got the equivalent of what we call today a PhD at age 30.

If you plan to get a PhD and then move to industry and/or work for private companies, I guess (though I don't have a formal proof, only examples from friends and colleagues) that your professional life won't be any significantly different or harder than that of your colleagues who got their PhD a few years earlier than you. On the other hand, you have to know that there is an extremely fierce competition for jobs in academia.

It has now become standard to be on very unstable positions (called post-doc positions) for at least 3–4 years after the completion of your PhD. And sometimes up to 10 years! (I have seen that among younger colleagues.) During these years, you need to gain recognition from the bigwigs in your field, so that they can support your application for the next stage of your career: the tenure track position (which I will describe below). And that might be extremely difficult, even if you prove some big results.

I know someone who, as a PhD student, answered a question of Serre (you might call it a conjecture) on cohomological invariants of some finite groups. Instead of congratulating him for his results, Serre became mad at this guy, accused him of stealing his ideas, saying that "the main steps of the proof were already known to him, and that he was going to publish very soon a paper answering his own question." The guy was forced by Serre's affiliates to rewrite his paper and explicitly mention that his work contains no original contribution as "everything was already known to Serre" (but of course not published).

But that is a single example, and obviously, most PhD don't end up like this. On the other hand, even if you succeed in having your peers acknowledge your work in a positive way and find some good post-doc positions, you still are in the middle of the jungle. Indeed, if you gain enough support from the bigwigs in your field, you can only upgrade from post-doc positions to a tenure track position.

Then, finally, after 10 to 15 years of such a life (where you might have to move out places every 2 or 3 years), you may hope for a stable and permanent position. Which means that if you start your PhD at 29 and plan to work in the academia, you might secure a permanent position at 40 at the earliest. Granting the fact that you have been able to give plain satisfaction to the numerous bigwigs you will encounter during this 10 to 15 years period of time.

I don't think being 29 could ever be considered a disavantage on the intellectual or creative level to start a PhD. The comments below your question give you links to lists of famous mathematicians which were late starters. One famous example is Stephan Banach who wrote the equivalent of his master thesis at 28 and got the equivalent of what we today call a PhD at age 30.

I would nevertheless advise to be very careful on other aspects of a mathematician's carreer. Unfortunately such a career is rarely based solely on talent and genius (unless you solve the equivalent of the Riemann hypohtesis in your field). Financial and sociological issues are very important and might become more and more difficult to tolerate as you grow older.

It has now become standard to be on very unstable positions (called post-doc positions) for at least 3-4 years after the completion of your phD. And sometimes up to 10 years! (I have seen that among younger colleagues). During these years, you need to gain recognition from the bigwigs in your field, so that they can support your application for the next stage of your career : the tenure track position (which I will describe below). And that might be extremely difficult, even if you prove some big results.

His career in abstact algebra (which should have certainly flourished in the best possible way, considering his brilliant debut) brutally stopped there. This guy was only 25 or 26 at the time, and was strong enough to start a new career in another field. On the other hand, I can't imagine him doing the same if he was 36 (instead of 26).

But that is a single example, and obviously, most PhD don't end up like this. On the other hand, even if you succeed in having your pairs acknowledge your work in a positive way and find some good post-doc positions, you are still in the middle of the jungle. Indeed, if you gain enough support from the bigwigs in your field, you can upgrade from post-doc positions to a tenure track position.

While tenure track positions are certainly less insecure than post-doc positions, they still aren't permanent positions. They last between 5 to 10 years, and the same game has to be played again with the bigwigs : publish (a lot and frequently) on the subjects which are considered by them interesting, gain their recognition and ask them to support your application.

Then, finally, after 10 to 15 years of such a life (where you might have to move out places every 2 or 3 years), you may hope for a stable and permanent position. Which means that if you start your PhD at 29 and plan to work in the academia, you might secure a permanent position at 40 the earliest. Obviously granting the fact that you have been able to give plain staisfaction the numerous bigwigs you will encounter during this 10 to 15 years period of time.

I don't think being 29 could ever be considered a disavantage on the intellectual or creative level to start a PhD. The comments below your question give you links to lists of famous mathematicians which were late starters. One famous example is Stephan Banach who wrote the equivalent of his master thesis at 28 and got the equivalent of what we call today a PhD at age 30.

I would nevertheless advise to be very careful on other aspects of a mathematician's carreer. Unfortunately, such a career is rarely based solely on talent and genius, unless you solve the equivalent of the Riemann hypothesis in your field. Financial and sociological issues are very important and might become more and more difficult to tolerate as you grow older.

It has now become standard to be on very unstable positions (called post-doc positions) for at least 3-4 years after the completion of your PhD. And sometimes up to 10 years! (I have seen that among younger colleagues). During these years, you need to gain recognition from the bigwigs in your field, so that they can support your application for the next stage of your career : the tenure track position (which I will describe below). And that might be extremely difficult, even if you prove some big results.

His career in abstact algebra, which should have certainly flourished in the best possible way, considering his brilliant debut, brutally stopped there. This guy was only 25 or 26 at the time, and was strong enough to start a new career in another field. I can't however imagine him doing the same if he was 36 (instead of 26).

But that is a single example, and obviously, most PhD don't end up like this. On the other hand, even if you succeed in having your pairs acknowledge your work in a positive way and find some good post-doc positions, you still are in the middle of the jungle. Indeed, if you gain enough support from the bigwigs in your field, you can only upgrade from post-doc positions to a tenure track position.

While tenure track positions are certainly less insecure than post-doc positions, they still aren't permanent positions. They last between 5 to 10 years, and the same game has to be played again with the bigwigs : publish (a lot and frequently) on the subjects which they consider to be interesting, gain their recognition and ask them to support your application.

Then, finally, after 10 to 15 years of such a life (where you might have to move out places every 2 or 3 years), you may hope for a stable and permanent position. Which means that if you start your PhD at 29 and plan to work in the academia, you might secure a permanent position at 40 the earliest. Granting the fact that you have been able to give plain satisfaction to the numerous bigwigs you will encounter during this 10 to 15 years period of time.