What can an algebraic geometer do outside academia? This question is inspired by this
and this. But it is not a duplicate, read on. Please don't close it: I choose to be anonymous just not to be identified.
About to be on the job market, disenchanted with academia, bored by the teaching load in grad school, I have to make ends meet (by, say, finding a job somewhere). I've seen math people finding non-math jobs. But they seem to know some math that can be ``applied'', e.g. stochastic process or combinatorics. Trained as an algebraic geometer, I don't have a strong background on those stuff or anything that helps job hunting ---- knowing things about the Weil conjecture doesn't seem to be a plus. I don't think I will enjoy a teaching job in college much either.
I would guess I'm not the only one in this situation. 
Question: Can anyone here give some career suggestions? 
(P.S. I'm in US now but not a US citizen, so National Security Agency or similar jobs appear in answers to similar posts are really not in my list.)
 A: First I wish you good luck and second I think this is appropriate question for forum.
About 3 years ago I transferred from academia to industry, so I can partly understand yours difficulties. 
Algebraic geometry is of some use in some industry. On the other hand I am not sure it that it is easy (or even possible) to find such jobs.
Example 1. Cryptography. Some private companies also research it e.g. for the purpose of cloud computing (popular topic now). Modern schemes to secure data involve Weil pairing. 
Example 2. Error correcting codes. There are Goppa codes built on algebraic-geometry ideas.
There is also current research how to use algebraic number theory in space time codes Perfect Space Time Block Codes. 
Example 3. There are applications to robotics. They are described in book  Ideals, Varieties, and Algorithms
by David A. Cox, John B. Little,  Don O'Shea, 
Well, I think the situation with research jobs in US, is better than in Russia, so if I was able find one in Russia, you have more chances there:) As another advise let me tell a bit of my story - I applied to some vacancy and was sure that they will not even interview me, cause there was big list of requirements and I knew nothing about it, except small item at the end about matrix calculations, which also was not my strong point, nevertheless to my great surprise I was hired by them. Actually I was lucky since in Russia it is difficult to find experts which they wanted to find, so they weaken the requirements.
The moral of this story - just apply everywhere, does not matter it seems very related to yours experience or not.
PS
May be it is good to allow math job advertisement on MO ? On MSE there some kind of banners at the right side of the screen, sometimes they inform about existence of arXiv. 
May be it would be good for community ? People can learn what kind of jobs are of use in industry...
I am not sure by my feelings are that there are not so many bridges between math and real world it would be great to built as more as possible...
Let me ask this in meta : http://mathoverflow.tqft.net/discussion/1433/math-job-advertisements-on-mo-/ 
A: People not originally from the US (or at least not from an English speaking country) tend to have a little more trouble with this question, because they have more difficulty understanding the following:
Traditionally in the United States, higher education is not specific training for a career.  Rather, it is training to be a thoughtful contributing citizen.  Skills necessary for being a thoughtful contributing citizen tends to be useful in all kinds of intellectual and non-intellectual work.  Companies looking for intelligent employees therefore look for intelligent, hard-working people who have done well at studying something, not necessarily anything related to what they do.
As a graduate of a liberal arts college, I saw music majors and English majors get all kinds of jobs.  What were the most common?
1) Sales.  Plenty of companies need people to sell something.  One might sell large software packages to businesses.  One might sell re-insurance to insurance companies.  One might sell medical devices to doctors.  Since salespeople generally are mostly paid on commission, hiring a salesperson is low risk.  Traditionally, salespeople know almost nothing about what they are selling; they are just good at convincing people to buy whatever they are selling.
2) Technical sales support.  Someone has to understand what they are selling.  Except technical sales support people are not actually experts in what is being sold.  They've just had a little more training so they can answer the 98% of questions that aren't actually that technical.
3) Software.  If you've learned how to program, the vast majority of jobs in the software industry don't require that much software engineering knowledge, and you've probably learned enough.  We are not talking about making a database or operating system run faster; we're talking about changing the interface of MathOverflow to move the answer box 5 pixels to the right, or just making sure this webpage is displaying the answers to this question and not some other random question.
4) Consulting.  There is both software consulting (see software) and management consulting.  This involves being in a group that is hired to look at a company's operations, gather qualitative and quantitative data, and make recommendations.  The whole point is that you are mostly not experts and therefore (at least supposedly) have a fresh view on what they are doing and can recommend the obvious stupid things they are missing.
5) Investment banking.  See Sales, except you're selling financial products to companies.  Except that in this industry, sales is frequently done in teams, so your role might be closer to that of a consultant (assembling data to support the sale) than an actual salesperson.
A: Although not an algebraic geometer, I recently went through a similar experience (I actually posted one of the questions to which you linked.) One of the reasons I became disenchanted with academia was that I struggled to learn algebraic geometry for many years but failed. So I hope you appreciate that this advice is coming from someone who is not as smart as you!
After returning to my own country following an unsatisfactory year teaching at a liberal arts college, I did some part-time teaching at my old university while I researched possible careers. Attracted by the idea of earning a lot of money, my first thought was finance and I read several books about the stock market and the history of money. By the way, it's very hard to find textbooks about finance which don't "take sides". The very best book I found, coming from zero knowledge of economics, was "Introduction to Money" by Honor Croome.
We don't have quants in this country, but the career which appealed to me most was actuary, and I contacted some actuaries via the University Careers Service. The main actuarial recruiter in my city was not interested in me, saying that I was "over-qualified" (an expression you will probably soon be hearing a lot) and would find the job boring. Most of the actuaries I talked to were very friendly but were baffled that someone who had a PhD in mathematics would want to start a new career from the beginning.
Meanwhile, I realised that any job involving maths was going to require some knowledge of programming, so I began to learn Python. This was quite fun, although I did not get very far before switching to Javascript, because I wanted to be able to share my code more easily. An excellent resource is a free book called "Eloquent Javascript". I really recommend this if you have no programming experience. I didn't get very deeply into programming, but learned enough to do simple calculations and talk about things like "APIs", "dynamic typing" and other jargon.
I attempted to get a full-time teaching job at my old university but failed. However, everyone was very impressed by my lecture. I decided that I wanted to make a positive contribution to the world and so began to consider mathematical ecology, especially fisheries. I talked to some people and they recommended that I learn Bayesian Statistics, so I got a book and started to work on this. I rapidly became starstruck by the beauty and power of the Bayesian approach. It was also a source of helpful programming exercises (Gibbs samplers etc.) and motivated me to learn R, which is the industry standard among academically-oriented statisticians.
Around this time, I failed to get a job in fisheries. It turns out that people don't really care about whether you are capable of doing the job; they want you to "demonstrate an interest" in it. It's a bit like how, when applying for a liberal arts position from a research university, you have to harp on about how committed you are to the liberal arts philosophy. Otherwise your application goes directly into the trash.
I was at a loose end and a new semester was starting, so I sat in on two courses, one on Bayesian stats and the other in data mining. The data mining was helpful for learning R, because it's a very scatty language and there are all kinds of little tricks you need to know. The Bayesian stats was an opportunity to work through more Bayesian stats, with exercises.
Around this time, I contacted an ecologist in the statistics department who happened to have a problem to work on, so I started working on this. After a couple of months I was able to amke progress on it and I'm hoping we will eventually get a publication out of it, which will certainly boost my credibility with the ecology crowd.
Anyway, at this point (about a year after my search started) I suddenly got a job in the tax department. The reason why I got this was because my boss is a mathematician. It turns out that there are lots of mathematicians in industry who were once in the same position as I was, and you now are. They like to hire mathematicians, just like how people who have emigrated to a different country like to hire people from the same country as them. Another mathematician was hired at the same time as me. I don't view the new job as a permanent thing that will go on forever, but I really like my colleagues. I do have to deal with meetings and people blathering about "going forward", "taking the first cab off the rank", "passing the ball" and all manner of similar phrases. On the other hand, I can bask in the feeling that I am helping to stick it to those nasty finance people who care about nothing but money ...
Ultimately I am hoping to return to academia (as a statistician) or become a statistical consultant if I can, and I want to move to Canada one day.
I guess the most useful pieces of advice for someone in this position are probably the following:


*

*The only way to get a job is via personal connections.

*Appearances are very important.


and (more an obervation than a piece of advice)


*

*Every real-world problem is ultimately about maximising some hideously complicated function.


I am sorry if you didn't find this too helpful, but I thought it might be useful to hear from someone who has very recently been in a similar position. It took me a year to find a job, so don't lose heart! 
A: I believe that mathematics training (of any type) is excellent preparation for work as a software engineer and, incidentally, for solving typical software engineering interview problems.
If you have a little proficiency in computer programming, or could teach yourself a little, you might land an engineering internship position.  An internship working closely with one or two highly experienced engineers would be ideal.  In any event the habits of thinking you've practiced as a mathematician would greatly accelerate your development in this field.
A: Should you be interested in applying algebraic geometry, IMA Thematic Year on Applications of Algebraic Geometry may give you some inspiration, as can searching for "algebraic geometry applications" or similar. You may also wish to take a look at the Society for Industrial and Applied Mathematics's Activity Group on Algebraic Geometry SIAM Algebraic Geometry group or visit their wiki SIAM Algebraic Geometry wiki.
Of course, you may need to take courses or do self-study in certain application areas, if they appeal to you. You may also need to develop skills in algorithms or computational techniques, assuming your background does not already include these.
All advice is perilous, but I suspect that you could increase your employment options if you know at least something of programming, statistics and optimization techniques. Perhaps there are opportunities to work for government institutions in areas such as planning or infrastructure for instance. You may find areas such as complex networks or machine learning quite accessible with your advanced background and ability to cope with abstraction.
Whatever you decide to do, the very best of luck!
A: I think first you have to answer the following question:

Do you wish to continue doing some kind of algebraic geometry or are you willing to adapt?

If you are willing to adapt, then several choices open up:


*

*Search out which jobs might interest you (ok, this question on MO is a start)

*Learn at least some of the skills needed to get those jobs (e.g., an additional degree or qualification)

*Become a consultant --- easier said than done!


If you are unwilling to adapt, or wish to only partially adapt, then:


*

*Maybe do some algebraic statistics --- you'll profitably use your AG skills there

*Some research lab might be interested

*Do a postdoc in a related yet different enough field to give you wider perspective

*Take a month off to discover yourself, and figure out what is it that you really want? The benefit of asking, and perchance answering this last question will be greater clarity, and confidence. Disillusionment might evoke a cynical response, but really, in the long-run, you should answer this question to yourself.


I wish you luck and above all clarity
A: I voted to close this question earlier, but on the principle "if you can't beat 'em, join 'em", I'll say this. I know of any number of former academic mathematicians who decided to leave academe to pursue a career as an actuary. They come from all over: algebraic geometry, number theory, logic, quantum field theory, you name it. The bit about having to have some iron in applied mathematics here just doesn't hold true: what counts to prospective employers is being able to think clearly and precisely and have good mathematical sense. Being able to pass actuarial exams (the first few of which are math-y but doable with little sweat by an ex-mathematician) goes a long way towards establishing your cred. 
The pay tends to be pretty good, and actuarial work tends to be stable, certainly compared to what struggling academics put up with. (I am not an actuary myself but I am married to one.) A lot of mathematicians I know are very happy with this career choice and have never looked back. 
Perhaps the real point is that this is merely an example. I support what Angelo said: "a PhD in mathematics is highly regarded in the 'real world', even when it is in pure mathematics." Your prospects are probably very bright.
A: The "Software Perception Engineer" position could be appropriate depending on your background: http://bostondynamics.com/bd_jobs.html
A: Your situation with algebraic geometry is probably more similar to that of humanities graduate students than to that of students of applied math or engineering.

" [...] Contrast this situation with that of academia. Professors of Literature or History or Cultural Studies in their professional life find themselves communicating principally with other professors of Literature or History or Cultural Studies. They also, of course, communicate with students, but students don't really count. Graduate students are studying to be professors themselves and so are already part of the in-crowd. Undergraduate students rarely get a chance to close the feedback loop, especially at the so called "better schools" (I once spoke with a Harvard professor who told me that it is quite easy to get a Harvard undergraduate degree without ever once encountering a tenured member of the faculty inside a classroom; I don't know if this is actually true but it's a delightful piece of slander regardless). They publish in peer reviewed journals, which are not only edited by their peers but published for and mainly read by their peers (if they are read at all). Decisions about their career advancement, tenure, promotion, and so on are made by committees of their fellows. They are supervised by deans and other academic officials who themselves used to be professors of Literature or History or Cultural Studies. They rarely have any reason to talk to anybody but themselves -- occasionally a Professor of Literature will collaborate with a Professor of History, but in academic circles this sort of interdisciplinary work is still considered sufficiently daring and risqué as to be newsworthy.
What you have is rather like birds on the Galapagos islands -- an isolated population with unique selective pressures resulting in evolutionary divergence from the mainland population. There's no reason you should be able to understand what these academics are saying because, for several generations, comprehensibility to outsiders has not been one of the selective criteria to which they've been subjected. What's more, it's not particularly important that they even be terribly comprehensible to each other, since the quality of academic work, particularly in the humanities, is judged primarily on the basis of politics and cleverness. "
