How to Learn Advanced Mathematics Without Heading to University | Hacker News Reader

How to Learn Advanced Mathematics Without Heading to University

quantstart.com

178 pointsshogunmike74 comments

The main thing to remember is that nobody learns mathematics to any significant depth by reading. The only way to learn it is by doing it. Doing it carefully, and in full detail, not falling into the trap of "and I understand it from there".

So the biggest barriers to doing it on your own aren't source material (there is lots of that) it's a good source of correction. There is also the usual problem of self study, in that you don't have a roadmap and can waste time easily.

That said though, one advantage if you are diligent is that you probably by necessity learn techniques of checking your work (formally and informally) earlier than typical students, which is a good thing.

This is probably the reason why there are so many more autodidact programmers than mathematicians. When programming, you know if you made an error as soon as you try to run your code. The program, by definition, must be "correct" in order to execute. The "source of correction" is the error detection built into any programming language runtime or compiler.

No such "source of correction" exists for mathematics, and that makes it an inherently more difficult subject to teach yourself, because any errors you make will "fail silently" unless you are capable of detecting them yourself, which by definition you cannot do without experience. This is why a mentor/professor makes learning mathematics so much easier; he/she plays the role of mathematical compiler.

Good point and great analogy, though I think EWD might be turning over in his grave. While the error detection built into compilers and the testing process are, in general, indispensable tools for developing stable software, a program that executes without any evidence of error is a great distance from a program that has been shown to be correct.

It's still easier to build up an understanding of computers under that environment. You may not be proving your programs correct, but you are probably proving them useful.

I find that after doing some proofs work you start to get a sort of sixth sense feeling for when something isn't rigorously correct. In the words of one of my professors, "something smells wrong".

> "The program, by definition, must be "correct" in order to execute"

This definition of 'correct' is why software may have a poor user interface and many security holes.

Your observation also assumes that the language is completely specified. In C, for example, certain constructs lead to undefined behavior. As a modification of the old warning, running your program may unexpectedly cause demons to fly out of your nose tomorrow. Execution therefore does not imply correctness.

Proof verification, btw, offer a "source of correction" for mathematics -https://en.wikipedia.org/wiki/Automated_theorem_proving#Rela... .

>There is also the usual problem of self study, in that you don't have a roadmap and can waste time easily.

Can you expand on what you mean by "roadmap"?

To me, a roadmap is very easy to get and can be arrived at from different angles.

method 1) Pick up an "Advanced Mathematics" book and start at page 1. The sequential chapters of that book would start a roadmap. If page 1 looks incomprehensible, look at the preface/introduction to see what the author lists as prerequisites. Seek out the book(s) on the prerequisites and start on page 1 of that book. If that prerequisite looks like gibberish, then look at that book's prequisite. And so on.

method 2) Google "advanced mathematics study roadmap" and look at various answers from math.stackexchange.com, reddit.com, blogs, etc.

method 3) Look at the undergrad curriculum of math courses for degree requirements (e.g. Bachelor of Mathematics, Bsc Electrical Engineering, etc) published by universities. (e.g. go to http://mit.edu).

It seems like a "roadmap" for self-study is readily accessible for anyone curious.

The usual problem in self study (of any subject) is that you go towards what is easily accessible from where you are right now, which means you might miss out on useful avenues visible to someone who already has a good view of that "lay of the land", so to speak. You may also lack the discipline to push through something that is difficult for you when you can't see they payoff - an outside influence could convince you it will be worthwhile. Depending on the type of student you are, you can also spend too much time inefficiently on things that are comfortable.

You can try and do something like recapitulate the core curriculum of a math degree, say, but there are constraints there you aren't aware of and it won't be the right path for everyone.

method 2 expanded to "actually ask questions at some of these locations" is actually a good way to inject some outside direction. So it's a mitigation for this problem.

>That said though, one advantage if you are diligent is that you probably by necessity learn techniques of checking your work (formally and informally) earlier than typical students, which is a good thing.

Or you can post them on math stackexchange and get feedback for free from bored? mathematicians :)

http://math.stackexchange.com/

The same could be said of programming, or anything really.

To a degree. It's particularly true of mathematics.

The CS parts (which you can view as a branch of mathematics) of programming certainly have this feature.

If anyone knows a good way to check proofs by yourself, let me know. I love it, but you pretty much have to have other mathematicians tell you when you're doing something horribly wrong.

By the way, did anyone else have to take engineering maths instead and feel they were somewhat lacking?

Just check everything carefully. If you've written your proof properly every step should follow logically from previous steps, and you should be able to check if yourself without issues. The problems happen when you're not critical enough of yourself, when the proof is very large and complex, or when you don't know your base axioms and the assumptions of your theorems well enough.

Proving the security of cryptographic protocols is a counterexample to this. Sometimes basic assumptions are very difficult to catch.

> So the biggest barriers to doing it on your own aren't source material (there is lots of that) it's a good source of correction.

Does anyone have any recommendations for where I can find a "good source of correction" (private tutor) online or locally for discrete math/algorithms/college-level mathematics? I looked at WyzAnt which gave me no results for my area -- most of the tutors seem to be for SAT/high school prep.

The only way to learn it is by doing it. Doing it carefully, and in full detail, not falling into the trap of "and I understand it from there".

This. So much this. A lot of people think there is some big difference between mastery from a physical versus intellectual level. No one who is serious about learning to play a musical instrument will only play a song up to a point and then stop, saying "... no need to go further, I already know how to play the rest." I think the way the brain consolidates high level learning into long term memory is essentially the same as for "muscle memory". This is like when you learn to drive. At first, you have to think consciously about every little detail but with practice, your unconscious mind takes those over and your conscious mind is left to operate on higher and higher level concepts. Mathematics is hard and the sooner you can offload the details to your unconscious mind the better. And the only way to do that is by practice.

I consider myself to be a smart but sometimes intellectually lazy person and had to struggle to develop the habit of working problems out rigorously and completely when learning new material. The belief that you can learn material by just reading is seductive because it feels like you're saving time and getting to the interesting topics faster. But in all likelihood, this belief is false and will be proven as false the moment your understanding is put to the test, either in a real test, or when subsequent material requires a solid understanding of older material. Sure, you may indeed be understanding it at the moment you're reading and in the flow of the material. But what makes your knowledge solid and reliable even under adverse circumstances (e.g., when you're distracted or learning a difficult new subject that builds on that knowledge) is practice and repetition. This applies not just to mathematics but to learning a new framework or programming language.

I've been getting into math recently and found https://www.youtube.com/user/professorleonard57 Professor Leonard's videos are excellent, clear and he injects just enough humour to keep it from been too dry.

Learn it in a group. Mathematics is a social activity. Meeting even once a week to discuss the material you're trying to learn will vastly improve the learning rate for everyone involved.

Brilliant suggestion!

There is no reason to shun university.

If you want to learn mathematics but do not have the time for a full time degree do it part time.

You are most definitely not actually going to reach the same level of mastery of mathematics as even the most mediocre graduate by studying alone. It requires a huge amount of commitment and staying focused and discipled enough is very hard when you don't have a set goal and good continuous guidance and motivation.

It is also very boring and much less productive to study alone. I'd estimate that 70% of my learning comes from the tutorial sessions. Reading the literature is just the starting point. It only counts for 10% if that. Homeworks and exams are also very useful because they make sure you have the prerequisites to move on and start learning something new.

I disagree, at least at the undergraduate level.

There is no shortage of problems and solutions to check your knowledge for much of undergraduate mathematics.

As to the value of tutorial sessions for you, it is important to remember that people learn differently. Speaking with an expert is valuable, but not required in my opinion.

To really learn a subject, you have to go beyond the minimum required to get a good grade on homework and tests. You can absolutely do this in university, but I found that I was rarely willing to do so. Studying on my own, I find it easier to reach a deeper level of understanding.

>it is much less productive

My annecdotal evidence differs. I miss a lot of university class because I find either the pace or the content lacking, in which case I just read the book and go to exams. It takes me way less time and I usually learn a lot more.

An early place where the OP dropped the ball:

> and some good experience manipulating continuous functions and their derivatives.

Nope. If a function is differentiable, then it is continuous, but continuity is not sufficient for differentiability. So, we can't talk in general about the derivatives of continuous functions.

E.g., each sample path of Brownian motion is almost surely differentiable nowhere.

Just f(x) = |x| is continuous but not differentiable at x = 0.

Maybe the OP advice is okay in England, but here in the US I would advise people wanting to learn to f'get about the OP and get better advice.

For job opportunities in quantitative trading, I tried that on Wall Street here in NY, and got nowhere. I came with a Ph.D. from a world-class US research university with my dissertation research on stochastic optimal control, which should have put my resume near the top of any stack. My favorite prof was a star student of E. Cinlar at Princeton and, thus, about the best there is for mathematical finance. I came with a long, solid background in software, peer-reviewed publications in mathematical statistics, optimization, and artificial intelligence.

I had a good background in second order stationary stochastic processes, power spectral estimation, and the fast Fourier transform -- no interest.

Got nowhere. That was before I heard about James Simons.

One interview was by a guy who recruited for Goldman Sachs, and he didn't have a clue about my background.

Another interview was at Morgan Stanley: The interview was in their computer group, but I indicated that I'd like to get into quantitative trading -- they acted like they had never heard of any such thing.

I got the impression that only a very tiny fraction of the people on Wall Street had good backgrounds in measure theory and stochastic processes based on measure theory, that my resume never got in front of any such person, and that the other people didn't know measure theory, anything about Brownian motion, power spectra, time series analysis, etc. and were looking to hire people like themselves.

Candidate Lesson: Study all the math you want, but don't expect Wall Street to be interested.

Thank you for pointing out my mistake! Indeed, that was a slip of mine to write about continuous functions, assuming they had derivatives.

>> and some good experience manipulating continuous functions and their derivatives.

> Nope

1. "good experience manipulating differentiable functions and their derivatives" sounds weird in prose.

2. Some continuous functions are differentiable. Those ones have derivatives you can manipulate. In fact knowing when a function is not differentiable is a pretty useful skill.

> The interview was in their computer group, but I indicated that I'd like to get into quantitative trading -- they acted like they had never heard of any such thing.

Sounds like you interviewed for position X and talked about wanting position Y, and were rightly rejected as "not a good fit; likely to leave at first opportunity".

> only a very tiny fraction of the people on Wall Street had good backgrounds in measure theory and stochastic processes based on measure theory

Probably true. Think of this as "calculus for engineers vs. analysis", and imagine how well a civil engineering interview would go if you talked about different types of integrals instead of talking about how to use the basic stuff to build good bridges. Fact is most people in industry are looking for "calculus for engineers" levels of formal understanding. Enough to be useful and make money while avoiding expensive mistakes.

> and were looking to hire people like themselves.

Also probably true. This is a good assumption not just on Wall St but everywhere.

> Think of this as "calculus for engineers vs. analysis", and imagine how well a civil engineering interview would go if you talked about different types of integrals instead of talking about how to use the basic stuff to build good bridges.

That analogy shouldn't apply to quantitative trading on Wall Street: That challenge needs more than just engineering math approaches if only to read the literature.

E.g., apparently broadly the first cut way to evaluate exotic options is to use the Brownian motion solution to the Dirichlet problem, that is, the subject of Markov processes and potential theory. The subject is awash in measure theory, e.g., stopping times, the strong Markov property, regular conditional probabilities, of course conditioning and the Radon-Nikodym theorem. This isn't advanced calculus for engineers. E.g., the work of Marco Avellaneda at NYU Courant, Steve Shreve at CMU, no doubt the work of E. Cinlar at Princeton.

> Sounds like you interviewed for position X and talked about wanting position Y, and were rightly rejected as "not a good fit; likely to leave at first opportunity".

My point is not that I was rejected but just that they had no hint that anyone at Morgan Stanley was doing applied math for automatic trading.

If you do study (continuous) mathematics, one of the things you build up is a little stable of strange functions to help you test your intuition.

A very common "right of passage" exercise in introductory analysis is to come up with the Weierstrass function or something similar (usually with a little coaching). This is an everywhere continuous and nowhere differentiable function that then goes into your little toolkit.

Ah, a favorite is a function that is differentiable but its derivative is not Riemann integrable!

Recall, a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0. So, the derivative has to be discontinuous on a set of positive measure. Now, to construct one of those!

Here's another favorite: For positive integer n and the set R of real numbers, suppose C is a closed subset of R^n with the usual topology. Then there exists a function f: R^n --> R that is 0 on C, strictly positive otherwise, and infinitely differentiable. So, for C, take, say, a sample path of Brownian motion, the Mandelbrot set, a Cantor set of positive measure, etc. Can use that function to settle an old question in constraint qualifications for the Kuhn-Tucker conditions in optimization.

Or, any closed set can be the level set of an infinitely differentiable function.

Sure, really fun reading for such things is:

Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis, Holden-Day, San Francisco, 1964.

Fun book indeed!

Your first example you won't typically run into until an introductory measure theory course.

No, it's doable at the level of Rudin's Principles: He shows that a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0.

For the definition of measure 0, he gives that quickly, and don't really need a course in measure theory. Besides a course in measure theory likes just to f'get about Riemann integration, and thankfully so.

Agree, it's do-able with baby Rudin. But as I recall it's a typical example used (early) in measure theory, not so much intro analysis. Hence "typical".

Ymmv, of course.

A single book is enough to learn mathematics:

Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide

It has a whopping 1300 pages, but it has everything you need. And if that is not enough for you get

Cahill: Physical Mathematics

This will give you advanced topics like differential forms, path integrals, renormalization group, chaos and string theory.

Thanks for the suggestion. Like you mentioned though getting to the end of 1300 pages will take ages. Hope I can find that sort of time.

I know that many learn programming themselves but I'm curious if anyone have learn advanced mathematics this way.

What are the best ways for a non-genius, normal person to learn advanced mathematics?

That depends primarily on your favoured learning style. MOOCs are certainly changing the available resources and lecturers are publishing freely available notes for particular courses on their home pages.

Unfortunately some textbooks can be expensive, but some are more reasonably priced. Unfortunately, the more "niche" the mathematical area becomes, the harder it becomes to find freely available sources.

I personally prefer a mix of video lectures and textbooks. Being able to watch video lectures, with the ability to pause and rewind, is a very useful feature that is not available in live lectures!

It is very easy to get most mathematics textbooks online, through slightly unsavoury means. It is far more difficult to figure out which textbooks are worth reading---this is a process that requires trial and error, and browsing through recommendations (math.stackexchange and mathoverflow.net have many good textbook recommendation questions, with many excellent answers).

Also, it is very easy to audit courses at universities! Get out there, ask the professor if you can audit the courses (make friends with them too!), and enjoy yourself a stress-free, and money-free, quality education.

"Unsavoury", as in?

Piracy.

Though, between price fixing and booksellers going under and not being able to guarantee that you will retain access to the books you bought, I'd say that this form of piracy is morally ambiguous.

I wrote this answer to a similar question in /r/math last year. You might enjoy it:

https://www.reddit.com/r/math/comments/3ixyw3/how_can_i_begi...

Notes from the intro math course at my University are pretty good.

https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/Math117...

First off, "genius" talent is by no means required.

Second, it should be something you really, really like doing. Like music is to a musician (or an audiophile), cooking (and watching people get off on your creations) is to a chef, sports training is to an athlete, etc.

And third, like anything else of true value in this life -- it will take a significant amount of time; in particular devoted to practice (and very importantly, play), especially solving (often obscure-seeming) problems on your own, just to scratch an itch, or to know that you can.

Easily a few thousand hours to attain what's called "mathematical maturity"[1], and probably somewhere on the order of the fabled 10,000 to obtain what might be called true expertise in the field. Which should (by itself) be no obstacle, if it's something you're really, really, really into.

Just like any other field, basically.

[1] https://en.wikipedia.org/wiki/Mathematical_maturity

I did books recommended from here, and lots of math stackexchange searching/questions.

I started with Basic Mathematics by Lang, Eccles book on Mathematical Reasoning, A Course of Pure Mathematics by Hardy combined with the lecture notes of MITs honors single variable calc and Polya's How To Solve It, currently doing Advanced Calculus by Loomis & Shlomo.

I would imagine if you're interested in advanced math you would go to free university seminars from visiting professors and network with whoever is there as a self learner.

I get up 3hrs before work everyday and read a chapter then do as many exercises as I can. Repeat until done, or I get stumped and skip that exercise then come back to it later.

One famous, although admittedly not particularly recent, example of a mathematical autodidact is Srinivasa Ramanujan (https://en.wikipedia.org/wiki/Srinivasa_Ramanujan).

How does bringing a genius into the conversation in anyway answer the question?

Hopefully, they do a good story with the most recent movie on Ramanujan's life:

http://deadline.com/2015/09/the-man-who-knew-infinity-exclus...

I did. It wasn't easy and I had the support of a large team of people to fall back on.

10 years ago I was an early employee of a risk analytics firm. I got pushed into a quant roll because I had an engineering background and could solve PDE's.

My first meeting I remember the head of the quant team hand-waved over a bunch of math that took me two weeks to go through on my own.

I had to spend 2-3 nights a week for 1 year going over my old statistics texts, Differential equations text and Knuth's Concrete Mathematics before I moved onto Stochastic Calculus.

It would have been almost impossible without having a team of people who were willing to help me get through sticking points. That was 8-10 years ago. I still spend atleast 1 night a week learning new math, though now its mostly non-linear regression and chaos theory.

And to be honest I still get alot of quants looking down on me, "A guy who dropped out of his masters, trying to claim he's our equal?". That does bother me a bit, and it means I probably couldn't get a quant job at Goldman, but I'm very proud of where I am now and how I got there!

TL/DR It can be done, but it gets done over the course of years, not months or weeks. You won't get there without applying what you learned to real world applications, reading the text's aren't enough.

Thanks for sharing this. It's good to hear from others who have been through similar experiences!

I fully agree with you that one needs to think in terms of years and not days, weeks or even months. I do make this point in the article, as well.

I took a Masters in Mathematics with the Open University. It's not quite as brutal as simply telling you which textbook to read and see you in nine months for the exam (repeat five times), but it's not far off.

It's not quite entirely by yourself, as in the webpage linked, but it worked like this:

  1) Get sent problem sheets and a list of what chapters
     in a textbook to work through.
  2) Read textbook.
  3) Solve the problems in the textbook.
  4) Watch a one-hour live webcast to you (and a dozen 
     other people) four times.
  5) Send off completed problem sheets; if you make the 
     pass mark, your prize is a seat in the exam.
  6) Optionally pay for a weekend of direct instruction to 
     you (and a dozen others) in person somewhere (I did
     this 3 times out of six, I think).
  7) Sit three hour exam in supervised exam hall.
  8) Go home, open next textbook.
  9) Repeat four more times, and then write a dissertation.

As in the linked webpage; it's just you and the textbook, for nine months. Not the same as in the linked webpage; there is a tutor you can eMail. This was often not as useful as you might think and some years I eMailed my tutor fewer than five times.

Also not the same; a formal, full-on three hours exam in a formal exam hall. This sets the bar pretty high and is a really good indicator to yourself. I think this makes a real difference; someone else validating that yes, you really do know what you're talking about (or not, as the case may be - I think that of everyone who starts this, less than fifty percent pass the first exam, and if I had to guess, I'd say that it's not so much that the maths is hard; making yourself study it to the level required month after month is hard).

I estimated that the time I spent on this was roughly equivalent to working full-time for 6-8 weeks per year. Interestingly, if I'd been given an actual 8 week block, I don't think I could have done it. I couldn't spend 8 hours a day reading a maths textbook and get the same from it as if I'd done it in four two-hour blocks spread over a week.

Coming back from an exam, sitting down at the table, putting all the notes and papers and books from the previous subject to one side and then sliding the virginal textbook and brand new notes and problem sheets onto the table was brutal. It can be done, but you have to want it. I found that in terms of taking in new knowledge, I couldn't do more than a couple of hours with the textbook at a time. For practising and solving problems, I could sometimes sit down at lunchtime on Saturday to just have a go at one quickly and stand up again eight hours later, table covered in notes and the problem at hand having been thoroughly explored and answered (typically, sadly, spread over several pieces of paper which I would come back to in the future and condense into a smaller paper space).

The study habit it forced into me is enormously valuable in itself. I can now pick up a high-level maths text book and simply start learning. I won't understand much of it, but I know that if I keep at it, I will. Grind, grind, grind. Not just mathematics, either. Someone gave me a Learn Japanese book for Christmas and now it's in me. I ground through it, and now I've got another one and I'm going through that. It's as if the subject doesn't really matter, so much as the studying.

Can you explain what steps you would take to learn a new subject? For instance, lets say you wanted to learn quantum mechanics - Where would you start? How would you study and check your work? Etc.

I don't know if my journey can be called self study. I keep taking classes at community college when I can. I am not a fan of the emphasis on technique over understanding concepts but staying on track is hard when I try to do it all alone!

I guess I sort of did. I never took mathematics seriously, have a GED instead of a HS diploma, and subsequently started college in remedial HS-level algebra. But now I have a PhD in applied mathematics and work in the overlapping gray area of research that exists between the mathematics, computer science, and systems engineering disciplines.

I did have some very good (and very patient) instructors early on. But at some point when I had the chance to read and understand the basics of discrete math and intermediate calculus in-between my semesters (undergraduate was a bit broken up for me, I was a deployed military reservist), I guess I found it interesting enough at that point to go deeper and change my major.

What books would you recommend? I'm currently in university, and I've had some exposure to discrete mathematics. But I'm definitely not comfortable with discrete math.

To start from absolute zero, check out Suzanna Epp's Discrete Math[0]. I believe even a motivated high school student could get started with it and even finish it. If your proof-writing is shaky, the book provides a very good workout. From there it will be easy to choose the areas of discrete math to specialize.

[0] http://www.amazon.com/gp/product/0495391328?keywords=susanna...

Thanks, I'll check it out. My proof-writing is definitely shaky. I can clearly see the relationship between programming on writing proofs, but I can't get immediate feedback on the validity of my mathematical proofs like I can with code.

Questions in Epp are by no means unique. If you search MSE, you'll see that every question in Epp has probably been asked and reasked about a thousand times each. That goes for subjects like Real Analysis, Abstract Algebra, Topology as well.

If you're looking for historical examples:

Stefan Banach, Israel Gelfand, Edward Witten...

Amazing how thorough the Math curriculum is in the UK! I got a minor at my undergrad just by doing stats, linear algebra and calc through multivariate. I think I was just 2-3 courses shy of a BA in the subject.

I'm curious how many people have met self-trained mathematicians in real life. I know there are storied examples, but is it feasible? (Compared to say - self-trained programmers or writers)

Where did you go to school? Those were just the prerequisites for my engineering degree.

> I got a minor at my undergrad just by doing stats, linear algebra and calc through multivariate.

I wouldn't call this a good thing; this just seems like a sane basis for any maths work (including Comp Sci).

That's my point. It was pretty much what was required for the CS degree. The UK standard is much higher.

I don't think it's just the UK. At Berkeley, for example, the Math minor requires 4 lower division courses and 5 upper div.

https://math.berkeley.edu/programs/undergraduate/minoring-ma...

Yes - 9 versus 5. Is Berkeley quarterly or semester? Either way, it looks like a much more rigorous program. (I've generally been very impressed with Berkeley Math and CS grads)

Any one know any books or resources that concentrate on invariant programming? Translating recursive code into properly tail recursive or iterative code can be pretty difficult.

I recall that Concrete Mathematics has a pretty section on solving recurrence relations. That's probably as close as I can get you to this.

`Pearls of Algorithmic Design` by Bird - Beautiful little book. It's a series of problems that are solved by first writing the naive program and then transforming it rigorously to make it more efficient.

`Algebra of Programming` by Bird and De Moor - This a treatment of the theory that is implicit in the methods of the previous book.

It's cute that quants think what they do is advanced, isn't it?

It's cute that the article is littered with Amazon referral links to textbooks.

I'm curious why people care about articles with referral links. It's not like you're paying more if you end up buying something, is it?

It is always good to know when someone is profiting financially from a recommendation they're giving you.

Sure, I'll buy that in principle. But wouldn't you agree that it mostly matters if someone is recommending some shady financial product or otherwise high-ticket item? What's the worse that could happen when someone makes (text)book recommendations?

Affiliate links are also a good way to compensate people for curating valuable information. So the sort of "it goes without saying affiliate links are bad" attitude is kind of a mystery to me.

I don't know any successful quants that claim what they do is "advanced" in any fashion.

A year or so ago a chum asked me to take a look at a paper that some of her quants were basing their work on. When I dug through it and pulled out all the plagiarism in it (and I do mean that; it had literally been hacked together out of other papers by some chancer in a middle-eastern university) it turned out to be a Crank–Nicolson method, and then when I saw what they were doing with it, I reckoned they could have just used a crayon and drawn over the graph they were interested in to get their smoothed graph. I do know that soon after that she got a job somewhere else. You can find hacks and chancers in every walk of life.

[1] In the above text, "hack" and "hacker" is used in the pejorative sense, similar to a "hack journalist".

The word "advanced" is relative to the intended audience and it's obvious how he meant that word to be applied. The author wrote:

", chances are if you are considering studying advanced mathematics you will already have formal qualifications in the basics, particularly the mathematics learnt in junior and senior highschool (GCSE and A-Level for those of us in the UK!). "

The author is not a Fields Medalist writing to other PhD mathematics graduate students. In such a case, the adjective "advanced" would have a different meaning.

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