Is it possible to start a PhD in mathematics at the age of 29? I graduated with a bachelor’s degree in mathematics. I was initially focused on branches in     analysis like operator algebra. At the third year of my undergraduate study, I experienced a financial loss in my family. It was only a slight loss and would not influence the life and regular plans of my family. But at that time I was not mentally strong enough and I could not concentrate on study. I postponed two years to graduate, in 2020.
These days I am trying to apply for a master program in mathematics. My GPA is not top, but fair enough, and I also did my graduation thesis carefully. I applied for several programs in Europe and received the conditional admission of Uni of Göttingen, but my Toefl grade did not meet the requirements. This year I have prepared all the things and I am going to apply for several master’s programs in Germany.
I am currently interested in low-dimensional topology and want to select this area as my direction. But when I apply for a PhD, I am 29 years old, is it a huge disadvantage? I also referred to several persons working on geometric topology, and the time cost seems to be high. But I am still enthusiastic about mathematics and want to get a bread.
Anyone could give me some suggestions?
 A: Just another example: I graduated late, then worked 2 years outside university, then started my PhD, defended my thesis at 37, and at 47 I became associate professor. Since you ask this frequently: my field is dynamical systems.
BUT
It wasn't easy to catch up.
I had to accept a few things which look cool in your 20s but uncomfortable in your 40s.
A: 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.
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.
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 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).
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 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.
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 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 do believe this is really an important issue to consider before getting bogged down in the academia. You really don't feel the same about those things whether you are in your late twenties or you come close to 40.
A: As a yes or no question, then certainly yes.
You are at a tiny disadvantage though, which you obviously know to be posing the question.  And tiny disadvantages sometimes gradually get bigger, but sometimes they feeble into non-importance.  Ultimately, it probably won't be the deciding factor.  But it will manifest itself: on the professional level, it might mean you get judged negatively by some (short-sighted, and hopefully not many) professors/panel members; and on the personal level, it might mean you'll attach more importance to your family and financial status than you would have done a few years ago.  No one knows how it will play out, but those will be the issues.  That's all it is.
I don't know much but I can assure you that no one will be walking about thinking "omg they're just starting their phd".  But you're right that (referring to one of your comments) it isn't that easy to overcome the social/peer pressure, but what should be easy anyway.  Just make sure you're aware of it and have your approach to cope with it (and not just ignore it - I've seen too many people fail their Ph.D.'s not because they're mathematically incapable but because they don't know how to deal with the pressures involved).  Anyway, pretty soon you'll talk to enough people to realise few people care about your age, they just care what maths you do.
Also, as a small personal supporting note cause no one likes getting rejected and it was that bit that made me catch your post: good that you avoided Goettingen, my supervisor there literally told me I was too old to go for a postdoc (at 28).  So look at it as a bullet dodged, if it helps.  (By the way, Mihailescu is at Goettingen and he didn't start his Ph.D. until he was aroud 40!  (He's great by the way, incase this last paragraph is otherwise too negative.))
A: I completed my PhD at the age of 32 which is not uncommon to Israelis. We often lose several years due to military service. So starting at 29 might be a bit late, but it is not a disaster. It is more about talent and commitment. Good luck.
A: This is/was certainly possible.
Proof : Reuben Hersh started a PhD after 30 (born in 1927, he defended his thesis in 1962 at the age of 35) after having been working a decade as a machinist. He eventually became a successful professor at the University of New Mexico. His scientific work ranges from hyperbolic PDEs to Probability and Philosophy.
A: I started my PhD at age 30, and don't feel my age was a significant obstacle. However, I believe almost no one should do a PhD, regardless of age.
A: May be I live in another part of the world? I never asked this question to myself when I started my PhD from Mathematics Department and I was 29 at that time :)
I thought I was too young to do a PhD :D
A: I think the bottom line is that some things are easier if you start earlier, but that talent and quality will ultimately find a way and that these things are not determined by your physical age.  If you have something original to say, you should still be able to say it regardless of your age.
Note that obtaining a PhD at a relatively old age is quite common for Israeli mathematicians and physicists.  For example, if you need inspiration, Yuval Ne'eman started his PhD in physics aged 33.  His main contribution (age 36) was his discovery of the classification of hadrons using $SU(3)$ flavour symmetry (known colloquially as the ''eightfold way'').  This is a major achievement in twentieth-century physics.
