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Comment by helmett | Hacker News Reader

congratulation...I see a hedge fund career for many

I hope not. When I did math Olympiads (including the IMO) I was presented with a false dichotomy of pure math or finance. This is really unfortunate because finance in general does not use very deep math. A tiny number of people might use SDEs but by now the techniques are standard and boring anyway. Furthermore, even mainstream economists doubt that this sort of finance has positive externalities. The amount of resources that go into finance is just way out of proportion to what seems necessary for price discovery.

In contrast, all of the science and engineering disciplines can make use of very interesting math. Not deep compared to research math, but used in a much more interesting way than in finance. E.g when you study the statistics of markets, you are just playing a game, and don't care that much about external reality per se. On the other hand if you study the statistics of DNA or gene expression, you are doing real science.

I think the best advice to a young person studying math is what was given to me at the age when I was doing the IMO (and interestingly, after I graduated by someone else): Don't neglect statistics.

As someone who neglected statistics as a student ( topology was a lot funner) , would you have any recommendations for self-learning tools for statistics?

Casella & Berger's "Statistical Inference" is a nice introduction to basic probability theory and statistics. I found it pretty readable, and it's used for many 1st year graduate stat programs.

Duda & Hart's "Pattern Classification" is one of the best introductions to machine learning IMO. It assumes very little in the way prerequisites, which is nice for first time exposure.

Hastie & Tibshirani's "Elements of Statistical Learning" can be a little intimidating without having been exposed to the ideas of the previous two texts. Afterwards, however, it is a gem.

I would suggest "elements of statistical learning". If possible I would also try to study some econometrics which gives unparalleled insight into the correlation vs causation issue. You can think of econometrics as a branch of statistics that remained separate from the mainstream for historical reasons.

I also neglected statistics, it seems there's no avoiding it these days. What cured me was a MOOC from edx/MITx called 6.041x. It literally had me close to tears a couple of times. There was carnage, whining and general malaise. I couldn't imagine a better course for persistent programmers who don't know when to quit.

It's been offered during the spring term for the past two years, so maybe Feb 2016 will see the next run.

https://www.edx.org/course/introduction-probability-science-...

I haven't finished it yet, but what I've read of Wasserman's All of Statistics I've liked. The chapters are a bit terse, so I'd plan on doing a bunch of the exercises. The good news is that there are lots of exercises and most of them feel well chosen.

Probably doesn't need to be said here, but I would also add in: Don't neglect computer science.

the transition from pure math to the application of statistics to the real world (or science) requires a philosophical adjustment.

In math, the model and axioms are sound (by definition) within the mathematician's world.

However, in order to make judgments about reality by using statistics, one has to come up with reasonable models and assumptions, otherwise the resulting deductions can be worthless. The leads to a lot of subjectivity and grey areas for debate that a mathematician may not be accustomed to.

A good proportion of maths olympians go into mathematics in academia rather than into industry. Industry does need problem solvers, but maths olympians develop creativity and a sense of elegance and beauty as well. Industry requires efficient solutions to technical problems, more than elegant or beautiful ones, at least relative to what is possible with full academic freedom.

As a minor nitpick, some mathematicians see elegance in efficient solutions. It's interesting to get your head around a specific problem and come up with the solution that is optimal regarding some given metric.

However, I agree that industry and mathematics are not best friends. That's because mathematicians righteously demand and require a level of freedom and support the industry is not always willing to give because of the social problem it creates with other employees and because the value of the work of a mathematician can be too hard to judge.

From my experience, the fight to get the working conditions you need is not worth it. My advise to fellow mathematicians is that — when you want go into the industry — to go where there are already mathematicians.

Just to clarify, I meant efficient from the perspective of business, not efficient from the perspective of mathematics. In industry, often a solution which "works", but is neither mathematically "efficient" or "elegant" is "good enough". By and large, industry is trying to generate revenue, not scientific knowledge.

In software engineering, elegance is prized and is often related to efficiency in several layers.

At the human layer, one expression of this idea is the principle of least astonishment: "People are part of the system. The design should match the user's experience, expectations, and mental models." (Wikipedia). A system involving people is not efficient if the people are often surprised by its design, implementation, or behavior.

Efficient software solutions also tend to involve elegance. Take Git for example as an improvement over other source control systems. The conceptual primitives that Git is built upon are elegant and recognizing them as the correct basis for source control (along with a strong implementation) resulted in Git's efficiency and power.

Granted, software is different from mathematics, but I find the parallel interesting. I suspect that a mathematician's desire to find an elegant formulation, and appreciation of it, is very similar to the software engineer's.

My anecdotal experience is that most may start with math for their first degree, but then (or during) pivot to some related but more applied area like CS, Econ, etc.

There was an interesting article that looked at what the USA Mathematical olympiad trainee candidates in 1980 were doing now (This batch included the famous Noam Elkies) -

http://andrewgelman.com/2015/03/17/1980-math-olympiad-progra...

HN Discussion: https://news.ycombinator.com/item?id=9225683

A bunch of them went into academia, though not necessarily pure math. Some became engineers and only one ended up in finance.

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