Where appropriate, I've listed freely available lecture notes for particular courses. However, I prefer to recommend textbooks as they tend to cover a wider set of material. They aren't "cherry picking" material in a way that a lecturer will have to do so in order to fit the material into semester-length courses. Despite this issue, there are some extremely good lecture notes available online. The rise of Massive Open Online Courses MOOCs has fundamentally changed the way students now interact with lecturers, whether they are enrolled on a particular course or not.
Some MOOCs are free, while others charge. On the whole, I've found MOOCs to be a great mechanism for learning as they are similar to how students learn at University, in a lecture setting. They provide the added benefits of being able to pause videos, rewind them, interaction with lecturers on online portals as well as easy access to supplementary materials.
Some have suggested that the quality of MOOCs is not as good as that which can be found in a University setting, but I disagree with this.
There are some extremely good MOOCs available in data science, machine learning and quantitative finance. However, I have found there to be a lack of more fundamental courses and as such you'll see me recommending textbooks for the majority of the courses listed here.
At this stage of your mathematical career you will be familiar with the basics of differential and integral calculus, trigonometric identities, perhaps some elementary linear algebra and possibly some elementary group theory, gained from highschool or through self-study. The methods for teaching mathematics at highschool level are largely mechanical in nature and do not require a deep level of thinking. At University, mathematics becomes largely about formal systems of axioms and an emphasis on formal proofs.
This means that ones thinking is shifted from mechanical solution of problems, utilising a "toolbox" of techniques, towards deep thought about disparate areas of mathematics that can be linked in order to prove results. It is the fundamental difference between highschool mathematics and undergraduate mathematics. In fact, it is this particular mode of thinking that makes mathematics such a highly sought after degree in the quantitative finance world.
Self-study of university level mathematics is not an easy task, by any means. It requires a substantial level of discipline and effort to not only make the cognitive shift into "theorem and proof" mathematics, but also to do this as a full autodidact. For those of you who are unable or unwilling to carry out formal study in a university setting and wish to tackle a full syllabus of undergraduate mathematics, I have created a comprehensive study plan below to take you from high school level mathematics to the equivalent of a four-year Masters in Mathematics undergraduate course.
I have presented it in a year-by-year, module-by-module format with plenty of further reference materials to study at your own pace. Since a degree course is often tailored to the desires of the individual in the latter two years, I have created a syllabus which broadly reflects the topics that a prospective quant should know.
However, you can obviously add your own choices for your own particular situation. To this end, I have made suggestions where appropriate. This article will concentrate on Year 1 of a degree program, with subsequent articles each covering an entire year. The courses found in a first year largely reflect this transition, whereby the following core topics are emphasised:.
Most top-tier UK undergraduate courses have a "Foundations" module of some description. The goal of the course is to provide you with a detailed overview of the nature of university mathematics, including the notions of proof such as proof by induction and proof by contradiction , the concept of a map or function , as well as the differing types such as the injection , surjection and bijection.
In addition to these topics, the concept of a set is formally outlined, as well as the induced structure on such sets by operations , leading to the concept of groups. These core topics and ideas will prepare you for the deeper topics of analysis, linear algebra and differential equations that form the remainder of a first year undergraduate syllabus. Self-study of mathematical foundations can be challenging, as it is often the first time you will have seen the concept of a proof.
It can be bewildering at the beginning to understand how proofs can be constructed, but as with everything else in life, it is possible to learn how to structure proofs through a lot of reading and practice. Perhaps the best way to learn mathematical foundations is through "bedside reading", or perhaps more rigourous study, of some of the better known textbooks. I myself learnt from the following two books listed under Study Materials below.
I can highly recommend them as they certainly provide a good taste as to what university mathematics is all about. Real Analysis is a staple course in first year undergraduate mathematics. It is an extremely important topic, especially for quants, as it forms the basis for later courses in stochastic calculus and partial differential equations. The subject is primarily about real numbers and functions between sets of real numbers.
The main topics discussed include sequences , series , convergence , limits , calculus and continuity. The primary benefit of studying real analysis is that it provides a gentle introduction to proofs, using examples that aren't too unfamiliar from A-Level highschool equivalent mathematics. In this way, real analysis courses teach not only the "mindset" of forming proofs, but also introduce more abstract concepts such as "proper" definitions of infinity , axioms such as the axiom of completeness and some good experience manipulating continuous functions and their derivatives.
In order to learn Real Analysis by yourself, I would suggest taking a look at the textbook Numbers and Functions: Steps into Analysis listed below. I used this to learn Real Analysis when I was at university and I found it extremely helpful. The book teaches you by getting you to carry out a large number of questions, rather than throwing a huge amount of text at you.
In this way you learn by doing. Excel is a good way to play around with math, since you can put the formulas in directly, without needing to do as much algebra or repeating calculations. Ultimately, the goal for learning math should be to use it, not merely pass a test. To do that, however, you need to break your understanding free of the textbook examples and apply it to real world situations.
This is more difficult than just solving a problem. When you solve a problem, you will start to memorize the pattern of the solution. This often allows you to solve problems without really understanding the principles behind how they work. This is strictly harder than solving problems, so if you want to be able to actually use what you learn, you need to practice this.
Doing all these five steps on every single thing you learn in math is going to be time consuming. Instead, think of this like a progress bar. Every math concept you learn can go from steps one through five, deepening your knowledge and increasing the usefulness of the math each time. Others will be infrequent enough that just watching the explanation is all the time you can spare.
In particular, you should try to focus on the most important concepts for each idea. Math tends to be deep, so often in a full semester class, there may only be a handful of really big ideas, with all the other ideas being simply different manifestations of that basic concept. Most first-year calculus courses, for instance, all center around the concept of a derivative, with everything being taught merely being different extensions and applications of that core idea.
If you really understand what a derivative is and how it works, those other pieces will be much easier to learn.
Learn More. The books you pick as a self-learner are also sometimes different from what you would work use if you were engaged in full-time study at a university. Personally, I lean more towards books with better exposition, motivation, and examples. In a university setting, lecturers can provide that exposition and complement missing parts of books they assign for courses, but when you're on your own those missing bits can be critical to understanding.
I recommend avoiding the Kindle copies of most books and always opting for print. Very few math books have converted to digital formats well and so typically contain many formatting and display errors. Incidentally, this is often the main source for bad reviews of some excellent books on Amazon.
I'd be remiss as well if I didn't mention the publisher Dover. Dover is a well known publisher in the math community, often publishing older books at fantastically low prices. Some of the Dover books are absolutely brilliant classics - I own many and have made sure to make note of them in my recommendations below.
If you don't have a big budget for learning, go for the Dover books first. These courses are completely free and often have full recorded video lectures, exam papers with solutions, etc. If you like learning by video instruction and find at various points that you're getting a bit lost in a book, try looking up an appropriate course on MIT OpenCourseware and seeing if that helps get you unstuck.
Pretty much all my books I recommend below focus on undergraduate level math, with an emphasis on pure vs applied. That's just because that's the level that I'm at and also the kind of maths I like the most!
And also, just a final note that the order of books I recommend below is not exactly the order I worked through them - rather, it's the order I think they should be worked through.
Sometimes I picked up a book that was too hard and had to double back and wait until I was ready. And some books have only just come out recently as well eg. In short, you get to benefit from my hindsight and missteps along the way. I'm going to assume a high-school level of maths is where you last left-off and that it's been some time since then that you've last done any maths.
To get going, there's a couple of books I recommend:. The Art of Problem Solving books are wonderful starter books.
They're oriented heavily towards exercises and problem solving and are fantastic books to get you off to a start actually doing maths and also doing it in a way that's not just repetitive and boring. Depending on your level of mathematical maturity, you may only want to work through volume 1 and come back to volume 2 after you've worked through a proofs book first though the second volume has many more questions involving writing proofs which you may not yet be comfortable enough to do at this stage.
Volume 2 has many excellent exercises though, so don't skip it! No bullshit guide to math and physics - Savov. If your calculus is a little rusty or you never really understood it in high-school, I recommend working through this book.
It's compact, free of long-winded explanations, and contains lots of exercises with solutions. This book teaches calculus in a contextually motivated way by teaching it alongside mechanics, which is how I think calculus should always be taught initially I almost recommended Kline's Calculus: An Intuitive and Physical Approach here instead, as a book I also very much like, but Kline's book is just so thick and verbose.
If you do like that additional exposition, you may want to consider this book as an alternative. Also, of course I must mention the course that started it all for me, Calculus: Single Variable.
It appears Coursera has now broken the course up into several parts and as I mentioned you can also find the full lessons on YouTube. Work through either this or Savov's book - depending on whether you prefer learning from books or online courses.
I think it's useful early on in the learning journey to have a broad map of where math has been, what has motivated its development to date, and also where it's going.
Mathematics for the Nonmathematician Dover - Kline. For a historical view, I highly recommend reading through Kline's Mathematics for the Nonmathematician. It contains a small handful of exercises, but they're not the main focus - this is one of the few math books I recommend that you can just leisurely read. Concepts of Modern Mathematics Dover - Stewart. While Kline provides the historical perspective, Stewart will provide you with the modern perspective.
This is one of the first math books I read that genuinely made me excited and deeply want to understand topology - up until then, I was only somewhat dimly aware of the subject and thought it was a bit silly.
Like the Kline book, this book also has no exercises - but for me it was a springboard and motivator to open other related books and dig in and do some hands-on math. Mathematics and Its History - Stillwell. I consider Mathematics and Its History to be somewhat optional at this point, but I want to mention it because it's so darn good.
If you read through Kline's and Stewart's books and thought "You know what, these ideas are really nice but I'd love to go more hands-on with them with some exercises" then this book is for you. Want to try to do some gentle introductory exercises from fields like noneuclidean geometry, group theory, and topology, not just idly read about them?
This book might be for you. If you prefer listening over reading, I recommend listening to the part podcast A Brief History of Mathematics that focuses on the interesting lives and personalities of some of the driving historical forces in mathematics Galois, Gauss, Cantor, Ramanujan, etc. For many, your first proof book is where everything clicks and you begin to understand that there is more to math than just calculation. For this reason, many people have very strong feelings about their favourite proofs book and there are indeed several that are quite good.
But my favourite of all of them is:. An Introduction to Mathematical Reasoning - Eccles. I think what I love most about An Introduction to Mathematical Reasoning is how it successfully pairs explanation with exercises, which is a recurring theme in books that I tend to gravitate to.
Good exercises are an extension of the teaching journey - they tell their own story and have progression and meaning. And at the time I worked through this book, the difficulty was just right. A good chunk of the book is occupied with applying the proof techniques you learn to different domains like set theory, combinatorics, and number theory, which is also something that personally resonated with me.
Book of Proof - Hammack. Book of Proof is a nice little proofs book. It's not too long and has a good number of exercises. If you're looking for a gentler introduction to proofs this is the one to go for. For the edition I used, it contained solutions for every second problem with full solutions available on the author's personal website, which I believe is still the case today. Calculus - Spivak. Spivak's Calculus is the among the best maths book I have ever worked through but don't be fooled by the name - this is an introductory book to real analysis and is very different from the Calculus books mentioned earlier which emphasize computation.
The emphasis for this book is on building up the foundations step by step for single variable calculus starting from the construction of real numbers.
It is a wonderfully coherent and realized book and what's also great about it is once again the exercises complement and expand on the content so well. Speaking of the exercises, some are seriously hard. This book took me about 6 months to work through because at the time I was still committed to solving every single exercise on my own. I have met many brilliant and generous people and been privileged to discuss math with them all these years.
If you pursue your goal but don't find a suitable college teaching job, and you like discussing math with interested kids, I suggest looking at private high schools. Some of my most rewarding teaching occurred when I volunteered one year at a good local high school. Quick answer: No, you are not too old. Yes, such people do exist. It sounds like you're off to a good start.
Don't let your age worry you. I dropped out of college when I was 21 to work as a software engineer. Admittedly my work was technical, but my background in abstract mathematics was basically nonexistent.
I returned to college to study math, basically from scratch. Like you, I'd learned some math on my own, but I feel I made faster progress in a more structured environment working with people who were also trying to learn math.
It was a great decision. I'm now a first year graduate student, well on my way to a second career in math. I turned 40 late this past year after a long and unhappy life taking care of sick family and then enduring my own illnesses that have slowed my progress considerably.
But I'm nearly finished with my Master's in mathematics after getting a subpar bachelor's in chemistry and I'm planning for my PHD. My health is the number one concern here as far as being able to do it. I know when I'm on my A game,I'm as good as anybody. The problem is that happens less and less these days. Am I scared? Especially hanging around with 19 year olds who can run rings around me because they don't need to sleep But I can't give up now.
I've suffered and lost just too much. And anyone walking the same path shouldn't have any other attitude except that. I've got a Master's degree almost, some terrific grades in some very hard graduate mathematics courses and some not so great grades and incompletes. I've learned some awesome subjects and had some fantastic teachers. I have a blog on mathematics I don't write in often enough,but I intend for that to change. I have the same philosophy on mathematics that the late great science fiction writer Fredric Brown had on writing:His wife said after his death he hated to write,but LOVED having written.
I love having written mathematics I don't even care about grades anymore. I was obsessed by it once,but you know what? After watching half your family die slowly in agony of cancer and seeing entire families live out of thier cars after falling behind on thier mortgages,it really puts a perspective on things.
Yeah,I know,in academia,what I just said is heresy. But you have to keep it real. I'll make it or I won't. I'll get a PHD in mathematics and spend a few years making contributions and teaching or I'll die of a massive heart attack trying, It's as simple as that. First of all, bitrex, of course you're not too old, and it's probably your imagination that you're slower now than you were at What's more likely is that you now have a better awareness of what you're missing, so it seems slower, even though I'll bet that you're actually learning better.
I think it's sad that anyone feels this question even needs to be asked. One of the things I find most frustrating about mathematical culture is how impressed everyone is when good mathematics is done by younger people, as though your age appeared next to your work, like in a grade-school art contest.
I really hope the Chern Medal supplants the Fields as the premier prize--I think providing something aspirational to over mathematicians will have a very salutary effect on the field as a whole, to say nothing of the salutary effect it will have on a lot of individual over psyches. But here's something odd that I've noticed: the mythology about "older mathematicians" seems to cut women more slack.
I guess sexism can cut both ways. I started college in Jan when I was just about to turn At the time I knew nothing : I didn't really remember trigonometry. I didn't know what it meant to raise a number to a negative power. I'm graduating this December at 29; I was motivated enough by feeling like I was years behind schedule that I was able to finish this degree in 3 years , in the process of applying to graduate schools, in 3 graduate classes now.
Yes, I do regret that I didn't do all this sooner. And honestly it still feels weird. Sometimes I go to class and it just seems so strange that I'm actually going to college. Still, overall, this is one of the best things I've ever done. I guess you have a couple years on me, but it doesn't matter. The unfortunate thing is that, at least at my school, you don't get to do much mathematics until your junior year, so you may be in for a long slog of general education classes you are not particularly interested in.
I actually think this is a big problem Also, you say "I'm getting into material that is going to be very difficult to learn without structure or some kind of instruction.
One of the ablest mathematicians at my grad school went to Julliard and taught music until he was He then spent 6 years at the U. Wisconson and got a PhD degree under Walter Rudin in function theory. He is a very successful professional mathematician. I have a very problematic educational impediment, I am moderately to severely disabled, and it strongly effects my ability to learn mathematics.
I'm not equating the problem of "being too old" and "being disabled", but just as a success story. I will be completing my mathematics undergraduate degree in my 6th year at UC Berkeley next year. I would not let anything stand in your way if you want to learn mathematics. It is an amazing subject and I wish I could tell more people about why it's amazing. Your enthusiasm indicates to me that no impediment could stand in your way. Pacing may be an issue since you are not working with an instructor I have friends who are your age and older that have succeeded at UC Berkeley in mathematics, and, not to exaggerate, they made it through what is for most people a rather intense curriculum from my vantage point, which is, of course, quite limited.
I never doubt the capabilities of the well motivated, so I believe you can do it! Keep with it! It only gets better :. I always feel very encouraged when I come across threads like these. I'm in a similar position. I was a math major at Georgia Tech. I lost my way and ended up in pharmacy school. I'm now in my last year and plan on going back to Tech to finish my math degree as soon as I'm financially able hopefully after one year of working as a pharmacist , after which I want to go to grad school.
It's great to hear such positive stories from others. If everything goes as planned, I'll be 30 when I enter grad school. So no, you have no reason to feel you're too old. I know of several mathematicians who obtained their PhDs in their 30s, and even a famous one who got hers at And this is coming from someone who is 47 and working on their PhD in probability. I agree with the consensus here: it is definitely not too late to be learning math. The fact that you have mastered several books worth of material on your own, without the benefit of an instructor, is proof enough.
There is nothing to stop you from continuing in the same mode if you wish. However, it sounds like you have heard the siren call of mathematics and would like to get into it in a bigger way, and more rapidly. As a mathematician I can only encourage you. Sure, there are some disadvantages to starting later in life, but there are compensating advantages as well.
My own testimony: I did complete a bachelor's degree in math straight out of high school, so I do not quite fit the profile you are looking for. I don't think my thinking is any slower than when I was young. Whether you should return to school is a personal decision, and it no doubt depends on many factors. But that is really a separate discussion, a separate thread.
0コメント