What to do after a pure math academic path? I don't know whether my question is in the appropriate place. I've studied physics, and then did a PhD in (pure) math and 2 postdocs. I definitely love math research, but I am not ready to apply all over the world hoping to find a position somewhere sometimes. Therefore I am looking for a job.
I don't have any interest in anything from the society. I only love math for its beauty. I am wondering what happened to the world. All jobs I am looking for with "math diploma" requirement seems to be in data science or finance. I hate this stuff and don't see the relation with math, at least the math that I like. I cannot see any beauty in data science and worse, in finance.
Does anyone have an idea of not-so-sad job openings? Is it our fate to change our career paths to finance if we had a pure math-physics academic background?
Sorry for these desperate questions, but I feel so lost and sad….
 A: I got a PhD in math and was on this path myself, so I understand. The number one thing you have to realize is that math for the sake of its beauty is hard to pursue even in research academia sometimes. Therefore, keep that part separated in your mind.
The next thing you should do is re-evaluate exactly what you want to do in life, regardless of whether it involved math or not. In other words, keep a blank slate. You basically have to do this because as I said, outside of academia you will never find something that will suit your ideals and in fact it's often hard inside academia. (For example, I love the beauty of math but find the mainstream of endless specialization in huge overarching fields not my thing.)
Once you decide what will really make you happy, just go for it even if it's not math-related. Why? Because you will be much happier doing math on the side than you EVER will be doing a math-related job that doesn't appeal to your ideals. Personally, it only took me a couple years of doing math that didn't appeal to me to make me lose a lot of my passion for it.
My only other advice is get a high-paying job like software development or something applied for 4-5 years. Be frugal and save up a ton of money and then just use it to pursue your passions. Make a plan to exit the traditional system and just do it and don't look back. Math does not define you as a person and I am sure that once you find your center, you will understand what you need to do.
A: Some of the recently emerged fields in machine learning have a bit of overlap with mathematics (not sure how pure they are). I'm going to name a few that comes to mind:

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*Graph theory: a recently introduced network architecture known as graph neural network can be considered a generalization of belief propagation networks on structured graph. This may in addition have some overlap with statistical physics.

*Algebra/geometry: equivariant neural networks require some sort of explicit/implicit symmetry built into each layer, and very recently people have studied this type of network with symmetry related to certain Lie algebras

*Topology/measure theory: to give a marginally related example, normalization flows are neural networks that attempts to "continuously" deform a Gaussian measure to some other non-trivial (usually multi-modal) measures. For instance, people have used the Banach fixed point theorem to show such networks are actually "trainable".

*Complexity theory: transformers are a type of network that requires quadratic memory and compute to perform inference. Recently, people have investigated ways to reduce its complexity theory methods such as hashing and kernel methods.

*Optimization theory: currently the way neural networks are trained are somewhat ad hoc, and people just use whatever optimizers (e.g. Adam, SAM) that gives the best empirical results. Recently, people have started looking into this more seriously, and neural ODE is a type of network that can be trained via the Pontryagin method.

*Random matrix theory: the neural network layer weights can be considered as a random matrix, and the heavy-tailness of such matrices have recently be studied as indicators for the "complexity" of the network (and whether it is prone to overfitting).

*Dynamical systems: a group in UWashington are looking at ways to interface machine learning with dynamical systems. For instance, one direction is to use neural networks to discover implicit low-rank structures of nonlinear dynamical systems, such as SINDY for the Navier Stokes.

*Fourier analysis: there is a line of research that tries to convert convolution networks into recurrent networks, by apply Fourier transforms (or polynomial decomposition) to the network inputs and kernels. Many theoretical problems are still open, such as the stability and convergence of such conversion.

However, similar to the case of physics lagging behind mathematics for 50 years or so, most ML fields further lags behind physics 20-30 years. So I wouldn't count on using a ML-related job (research or industry focused) as a medium to gain immediate access to novel mathematical research. Rather, I'd view it as an opportunity to apply your own mathematical knowledge (instead of advancing it).
A: I am sorry that the OP feels "desperate and sad." I agree with the comments suggesting that happiness in life is very different from achieving some specific career. I also think a lot has to do with mindset.
That said, there are zillions of jobs for mathematicians (far from data science and finance being the only options) and many of them involve working with beautiful mathematical concepts. Here are some examples, in no particular order:

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*Use math to identify cases of gerrymandering and help create maps that are fair. This involves graph theory, geometry, metric spaces, and more. It's very cool and super relevant.

*Become a senior scientist or research mathematician at a tech company, like the sort that hired Jennifer Chayes, Laszlo Lovasz, Katalin Vesztergombi, etc. There is plenty of beautiful work to do in graph theory.

*Social network analysis is a lovely blend of mathematics and sociology. I saw a great talk by Strogatz on this topic once. I imagine companies like Meta might have teams of mathematicians studying social network graphs.

*Topological data analysis (TDA) is beautiful to a lot of people, and involves mathematical concepts such as graphs, metric spaces, Betti numbers, and a whole lot more. There are government and industry research groups based on TDA, and it's a growing area. Lots of jobs.

*Work for a government intelligence service. Plenty of connections to graph theory, number theory, etc. If you like your government and believe its mission is protecting people, then this kind of work can be immensely rewarding.

*Work for a government contractor, like the IDA in the USA. I know people in jobs like that who spend most of their time thinking about elliptic curves, group laws, error correcting codes, etc.

*Be an actuary. If you like probability theory and probability models, there are really fun topics that come up in this setting.

*I push back against the idea that there is no beauty in data science. Many data mining algorithms involve beautiful mathematics, like principal component analysis (eigenvectors, change of basis), singular value decomposition and separating hyperplanes, graph clustering algorithms, etc. Many companies have realized that if they want to get their modeling right, it's beneficial to have a trained mathematician onboard rather than only people who know how to run commands and have no idea why the algorithm works. I know data scientists who spend their time tweaking these algorithms to work in new settings, which means they are constantly playing with these beautiful concepts. Additionally, there is tremendous satisfaction in feeling like you created something that has the ability to really help a large number of people in their lives, e.g., statistical models to inform government policy and help lift people out of poverty, match people to jobs they will enjoy, help people who use drugs to get out of a state of addiction, etc.

*I know a lot of people who think Fourier analysis is beautiful and there's a whole branch of data science (spectral theory, time series models) where you get to play with this every day. Same for working for companies like Sound Hound or Shazam, and probably many others that I haven't listed (Zoom? Skype? How do they denoise? Some beautiful math must be in the background.)

*I concur with comments who said secondary school teaching can be a very fulfilling job, and one full of opportunities to enjoy (and share) the beauty of math. That's especially true if you work with the IMO team, programs for gifted high school students, etc. Such students can even do cool research and there have been lots of MO questions about that topic.

*I believe certain types of engineering use fairly sophisticated tools from analysis. Sadly, I'm not an expert in this.

*Text analysis, e.g., using and developing algorithms for determining authorship, extracting summaries, etc. Imagine developing an algorithm that can use Twitter data to figure out when an emergency is happening and then dynamically allocate government resources to help.

*Mathematical art, both creating it and using math to connect people with art in new ways (e.g., Google Deep Dream)

*Using math to create improved epidemiological models, e.g., while working for a hospital system, government, etc.

Others have compiled better lists than this, e.g., the AMS has a list including the following and also a list of other lists.

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*Climate study

*Animated films

*Astronomy and space exploration

I guess the message I want to impart to the OP is that there's a lot to be excited about and a lot to look forward to. Now that you're a trained mathematician, you can go in many directions. For almost any passion, there is a way to connect it to mathematics and to bring the beauty of math into that world. Go explore and play!
A: I sympathize with you, because I have been in a similar situation. I was in mathematics because it fascinated me, although sometimes more than others. The reason I left mathematics research was not so much that I got tired from all the moving around, but that I wanted to do something "in the real world." I felt that my kind of research was hard to justify to anyone but a specialist, and that mattered to me.
But when I started to make an inventory of options that were available to me on the regular job market, I found out, exactly like you, that the vast majority of jobs for which a mathematics degree is a requirement (or even a plus) struck me as particularly "soulless." I know that might strike some as harsh, but I don't mean to be, that's simply how it felt at the time. I was in mathematics for the joy of it, and it is hard to square this with the purely utilitarian approach to math you find in finance or data science.
In the end I got a job in software development, which at first seemed just as soulless or maybe even more. However I never regretted my decision to leave mathematics. The joy that mathematics had to offer that I just mentioned was of a very elusive kind: sometimes it was there in abundance, but I could never hold on to it. It wasn't a constant stream of inspiration, and what's worse I rarely experienced it during my own "research" (if I could call it that), but almost always by reading about the exciting work done by others. And it didn't have to be cutting edge either.
So yes, quitting the academic career path was a major adjustment, and a period where I experienced loss. I was no longer allowed to devote my life to the pursuit of knowledge and understanding. Worse, I started to question whether I hadn't in fact thrown away fifteen years of my life. But there is light on the other end of the tunnel. These are after all not math problems, but life problems. If you make a decision that you know is right, then, with God's grace, it will prove to be so in the end.
A: I love mathematics, too, but I don’t expect to get paid unless I do mathematics that makes money for my employer. Nothing has “happened to the world” — it has always been this way.
If you love mathematics purely for its beauty, and you don’t care whether it provides any value to society (or your employer), then perhaps you should think of yourself as an artist, like a painter, sculptor, or musician.
To make a living as an artist, you need an audience, and the audience for modern pure mathematics is extremely small. As a musician, you can go play on street corners, and maybe make enough money to live, but it’s a hard life. The modern equivalent of the street corner is a YouTube channel. You could try that, but most potential subscribers are looking for help with calculus, and wouldn’t be interested in your research work.
Another alternative is to look for a job as an academic mathematician. As you said, this will probably involve hunting around the world for a while, and you might have to go live somewhere that’s not as pleasant as Switzerland. You said you don’t want to do this. Fair enough. Your choice.
A third alternative is to take a job that provides you with enough cash to live, and yet still allows you enough free time to pursue your art. Then you don’t need to worry about finding an audience, and you can just do things that you personally find beautiful, regardless of what anyone else thinks of them — you’re free. You said you couldn’t be a firefighter, but there are other jobs that consist mostly of sitting and waiting. Many of these jobs don’t pay very well, but I’m guessing that this might not bother you.
If none of the above sounds appealing, then maybe it’s time to re-evaluate. Do you have to do mathematics research? Could you live without it? Is it as important to you as your friends, family, mental and physical health? Could you find the same beauty in some other discipline?
The way out for me (and many others) was through software development. Learning programming is easy enough, and it’s a highly marketable skill. You won’t use much of the mathematics you learned, but well-constructed software has much of the same elegance and beauty as mathematics (in my opinion). Something to consider.
A: There are two types of career paths for a pure mathematics PhD holder who wants to continue to pursue pure mathematics:

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*Mathematics-related jobs, typically in academia: Most people only look at university jobs which involve teaching and research in roughly balanced proportions. However, one should not ignore school and college teaching positions. These can be extremely rewarding and can lead to interesting research questions in pure mathematics.


*Jobs entirely unrelated to mathematics: Here the opportunities are limited only by the current job market in your location. Note that a PhD is a higher qualification than what most people in the job market have. (In some places this could unfortunately be a dis-qualification.) It would then be incumbent on you to pursue mathematics on your own; which may be possible provided your job does not suck up your time and mind-space.
Unfortunately, many people are caught in the trap of looking for a job that "is commensurate with their qualifications" where the measurement is based external factors like remuneration and respect from society at large. Such jobs may require you to give up pure mathematics. The alternative is to look for a job which is sufficient to support the goal of pursuing one's primary interests.
Here is a tale that may be inspirational. The person who taught me music was a skilled worker in a workshop (factory-type). Even though it did not pay very well, it provided adequate support as far as he was concerned. This allowed him to develop and enjoy his music over a 30-40 year period. He never became famous, but he enjoyed and developed his music, and spread this joy to everyone he taught or otherwise came in contact with.
A: I have read your question and I am somewhat alarmed and saddened by your current state of being. I feel you need to take a broader perspective in this stage of life, zoom out; Do you really mean your only interest is in math? You also studied physics. You say "I don't have any interest in anything from the society"; are you sure?? You could contribute to medicine; how the brain might work. I'm not going to sum things up what you might do, but it vastly more than just finance or data-science. It is your quest now what else apart from pure math, you find interesting and go from there..
A: Writing a good mathematical proof is similar to writing good code. Pure math has much more in common with software engineering than data science.
