How to Learn Advanced Mathematics Without Heading to University - Part 1

I am often asked in emails how to go about learning the necessary mathematics for getting a job in quantitative finance or data science if it isn't possible to head to university. This article is a response to such emails. I want to discuss how you can become a mathematical autodidact using nothing but a range of relatively reasonably priced textbooks and resources on the internet. While it is far from easy to sustain the necessary effort to achieve such a task outside of a formal setting, it is possible with the resources (both paid and free) that are now available.

We'll begin by discussing the reasons for wanting to learn advanced mathematics, be it career-driven, to gain entrance into formal education or even as a hobby. We'll then outline the time commitment required for each stage of the process, from junior highschool (UK GCSE equivalent) through to postgraduate/research level work. I will then present the different study materials available for the equivalent of an undergraduate course, how to access them and how to make the best use of them. Finally, I will describe a mathematical syllabus that takes you all the way through a modern four-year Masters-level UK-style undergraduate course in mathematics, as applicable mainly to quantitative finance, data science or scientific software development.

In this particular article we will consider the first year of an undergraduate course. The remaining articles will each discuss subsequent years.

The first question to ask yourself is why you want to learn mathematics in the first place. It is an extremely serious undertaking and requires substantial long-term commitment over a number of years, so it is absolutely imperative that there is a strong underlying motivation, otherwise it is unlikely that you will stick with self-study over the long term.

For the majority of you on this site, it is because you wish to gain employment and/or further formal study in the field of quantitative finance, data science or scientific software development.

You might be an individual at the beginning of your educational career, deciding whether to take a formal university program in mathematics. You might have worked in a technical industry for 10-15 years, but seek a new role and wish to understand the necessary prerequisite material for the career change. You might also enjoy studying in your own time but lack a structured approach and want a reasonably linear path to follow.

One of the primary reasons for wanting to learn advanced mathematics is to become a "quant". However, if your sole reason for wanting to learn these topics is to get a job in the sector, particularly in an investment bank or quantitative hedge fund, I would strongly advise you to carry out mathematics in a formal setting (i.e. at university). This is not because self-study will be any less valuable or teach you less than in a formal setting, but because the *credential* from a top university is, unfortunately, what often counts in getting interviews, at least for those early in their career.

An alternative reason for learning mathematics is because you wish to gain a deeper understanding of how the universe works. Mathematics is ultimately about formalising systems and understanding space, shape and structure. It is the "language of nature" and is utilised heavily in all of the quantitative sciences. It is also fascinating in its own right. If you are heavily interested in learning more about deeper areas of mathematics, but lack the ability to carry it out in a formal setting, this article series will help you gain the necessary mathematical maturity, if you are willing to put in the effort.

I want to emphasise that studying mathematics from the level of a junior highschooler to postgraduate level (if desired) will require a huge commitment in time, likely on the order of 10-15 *years*. Clearly this is a staggering commitment to undertake and, without a strong study-plan, will likely not be completed due to the simple fact that "life often gets in the way".

However, 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!). In this instance it is likely that you might be able to begin learning at the start of the undergraduate level, or possibly at the level of an advanced highschool student.

Even if you have the equivalent qualifications in A-Level Mathematics or A-Level Further Mathematics, you will still have a long road ahead of you. I estimate that it will take approximately 3-4 years of full-time study or 6-8 years of part-time study, in order to have an equivalent knowledge base gained by an individual who has carried out formal study in a UK undergraduate mathematics program to masters level.

While I don't think it is necessary to have postgraduate qualifications to become a quant, it is useful and can certainly put you ahead of the competition. However, do not be put off by the time commitment for postgraduate study. It isn't absolutely necessary and is likely to be carried out in a formal, full-time setting regardless.

If you are happy with this overall level of commitment, then the broad path that you will follow should look something like this:

- GCSE Mathematics or equivalent - 1-2 years part-time
- A-Level Mathematics/Further Mathematics or equivalent - 1-3 years part-time
- Masters of Mathematics (UK) equivalent - 3-4 years full-time or 6-8 years part-time
- Postgraduate Study/Certification/Research - 1-4 years full-time or 1-8 years part-time (depending on qualifications/research project)

As you can see, a mathematics education to a high level can take anywhere from 3 years to approximately 15 years (or more!) depending on your chosen path. Hence this is not something to be undertaken lightly. You must give it serious consideration and make sure that the payoff (financial or otherwise) from study will be worth the serious effort required.

These days it is possible to study from a mixture of freely available video lectures, lecture notes and textbooks. There are those who learn better from watching videos and making notes, while others enjoy working methodically through a textbook. I've listed what I consider to be the most useful resources below.

At the undergraduate level, I am a big fan of the Springer Undergraduate Mathematics Series of textbooks, which cover pretty much every major course you will find on a top-tier mathematics undergraduate degree in the UK. I will go into detail regarding choices of books for specific modules below.

I've also found the Schaum's Outlines series of books to be extremely helpful, particularly for those who like to learn by answering questions. While they don't go into the detail that others might (particularly the SUMS books above), they do help consolidate the basics by working through a lot of questions. I highly recommend them if you've not seen any of the material before.

Many Universities provide publicly accessible course pages that contain freely available lecture notes, often in PDF format, typeset in LaTeX or similar. 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. Leaders in the field include MIT Open Courseware, Coursera and Udacity. 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. On the whole, most MOOCs are actually *lectures filmed in University settings*, so I feel this point is somewhat moot.

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. As the focus turns to quantiative finance (in Year 3 and 4, as well as at the MFE level), I will be able to recommend more MOOCs in addition to traditional textbooks.

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.

However, there is a substantial shift in mindset when moving from A-Level/highschool mathematics to that studied in a typical UK undergraduate program. 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 first year in an undergraduate mathematics education is primarily about shifting your mindset from the "mechanical" approach taught at highschool/A-Level into the "formal systems" approach that is studied at university. Hence, there is a much more rigourous emphasis on mathematical foundations, including formal descriptions of sets, maps/functions, continuity and symmetry, as well as theorems and proofs.

The courses found in a first year largely reflect this transition, whereby the following core topics are emphasised:

**Quantity**- Number theory, Cardinality**Structure**- Groups, Linear Algebra**Space/Geometry**- Trigonometry, Non-Euclidean Geometry**Change**- Calculus, Real Analysis, Differential Equations

Here is the course list for Year 1:

- Foundations
- Real Analysis - Sequences and Series
- Linear Algebra
- Ordinary Differential Equations
- Geometry - Euclidean
- Algebra/Group Theory
- Probability
- Mathematical Computing

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.

*Textbook/~$24*- The Foundations of Mathematics by Ian Stewart and David Tall*Textbook/~$35*- Introductory Mathematics: Algebra and Analysis (Springer Undergraduate Mathematics Series) by Geoff Smith*MOOC/Free*- Introduction to Mathematical Thinking by Keith Devlin

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*. In addition to that book, I have listed a few others that are helpful. Finally, I've listed a YouTube playlist series from Harvey Mudd College, by Professor Francis Su. The video quality is not great, but the content is extremely good.

*Textbook/~$62*- Numbers and Functions: Steps into Analysis by R.P. Burn*Textbook/~$43*- Real Analysis by John Howie*Textbook/~$35*- Principles of Mathematical Analysis by Walter Rudin*YouTube/Free*- Real Analysis: Lectures by Professor Francis Su

Linear Algebra is one of the most important, if not the most important, subjects to learn for a prospective quant or data scientist.

In an abstract sense Linear Algebra is about the study of linear maps between vector spaces. It teaches us that in certain cases linear maps and matrices are actually equivalent. This latter result makes it extremely useful when dealing with matrix equations, of which there are many within quantitative finance and data science.

The majority of statistical machine learning methods are based on the principles of linear algebra and calculus, as are many quantitative finance theories, such as the covariance matrix and the capital asset pricing model. Hence, it is imperative for prospective quants to learn it well.

Thankfully, Linear Algebra has such a wide applicability in mathematics, physics, engineering and science in general, that there are many great resources available for learning it. One of the best books to learn about it is by Gilbert Strang, a professor at MIT. In addition to his textbook, you can also find a set of video lectures presented by him on MIT Open Courseware.

*Textbook/~$94*- Introduction to Linear Algebra, Fourth Edition by Gilbert Strang*Textbook/~$50*- Basic Linear Algebra by T.S. Blyth and E.F. Robertson*MOOC/Free*- MIT Open Courseware Video Lectures - Linear Algebra by Gilbert Strang

The subject of differential equations permeates wide areas of quantitative finance. They are an extremely important subject for a prospective quant to learn, as stochastic differential equations play a large part in options pricing theory.

Formally, a differential equation is a relationship between a function and its derivatives. Informally, they are equations, which describe how rates of change of the function, with respect to some other quantity, affect the function itself.

Ordinary differential equations (ODE) are the first type considered at university (as well as A-Level/Highschool). An ODE is a differential equation where the underlying function has one independent variable. For instance, an ODE could represent the *rate of change* of population growth as a function of the population level itself.

As a quant, it is necessary to understand the basics of ODEs and how to solve them. Since the more complicated partial differential equations (PDE) and stochastic differential, equations (SDE) are widely found in quantitative analysis and trading, understanding the solution of the more simpler ODEs helps with understanding solutions of these problems.

Some ODEs can be solved *analytically*, that is, with a closed-form solution, using elementary functions. However, the solution to many ODEs can only be written as a series or integral relationship. ODEs can be solved "numerically", on the computer, using approximate methods. A large part of quantitative finance involves numerically solving differential equations in this manner.

There is no lack of study materials available for ODEs as they are a staple of the first year undergraduate mathematics program. I used the book written by my lecturer at University, and I found it to be approachable for a first year undergraduate (see Robinson, below). In addition, there is the famous "Boyce & DiPrima" (now in its 10th edition!), which is the staple of many ODE courses. In addition there is a free video lecture series on MIT Open CourseWare:

*Textbook/~$74*- An Introduction to Ordinary Differential Equations by James Robinson*Textbook/~$150 (although you'll get it much cheaper second-hand)*- Elementary Differential Equations and Boundary Value Problems by William Boyce and Richard DiPrima*MOOC/Free*- MIT Open CourseWare Video Lectures - Differential Equations by Arthur Mattuck

Geometry is one of the most fundamental areas of mathematics. It is absolutely essential for many areas of deeper mathematics, including those related to quantitative finance. Many undergraduate courses introduce Euclidean geometry to students in their first year, and it is also an appropriate place to start for the autodidact.

The primary setting is often Euclidean Geometry in three-dimensions, namely the geometry of "everyday life". You will learn a lot about constructing proofs from studying geometry, particularly with regards to projective geometry in the plane and geometry of the sphere.

In highschool (or at GCSE!) students are often taught about triangular geometry, and an introductory university module in Geometry will formalise these concepts, ultimately with the idea of gaining practice understanding and writing geometric proofs.

Euclidean Geometry eventually leads on to more general geometries such as Spherical Geometry or Hyperbolic Geometry, where familiar results from Euclidean Geometry are shown not to hold. In addition, and perhaps more relevant to the quant, having a good understanding of trigonometry is essential for later courses such as Fourier Analysis, which plays a substantial role in signals analysis and time series analysis.

Geometry is a tricky subject to introduce as it is extremely broad and covers such a diverse area of mathematics. However, I have found the following book, part of the Springer Undergraduate Mathematics Series, to be very helpful:

*Textbook/~$32*- Geometry by Roger Fenn

Groups are one of the most important algebraic structures found in mathematics. They provide the basis for studying more complex structures such as rings, fields, vector spaces (which we mentioned above in Linear Algebra). They are also strongly related to the idea of mathematical symmetry.

While it might be considered that groups are more of a "pure mathematics" topic, and thus are less applied, this is actually not the case. Groups find applications in chemistry (crystallisation), physics (symmetry and conservation laws) as well as in cryptography.

However, are they relevant to the quantitative analyst? This is a tricky question to answer. While it isn't clear how a direct study of groups and symmetry might be applied on a day-to-day basis in the world of a quant, the study of groups does form the basis of many more advanced mathematical topics, particularly advanced Linear Algebra.

For the autodidact who is short on time, I would state that it is worth studying them at an introductory level in order to "be aware of their existence", as many advanced quantitative techniques will indirectly refer to them.

*Note however that one of the most successful quant hedge funds in history, Renaissance Technologies, was founded by Jim Simons, a notable mathematician who carried out a substantial amount of work on manifolds (which requires a solid understanding of group theory). Read into this what you will!*

There is no shortage of elementary textbooks on group theory. Since it it such a common topic for first year undergraduates, many authors have tried to write introductory books. I've found the following to be useful:

*Textbook/~$39*- Groups by Camilla Jordan and David Jordan*Study Book/~$15*- Schaum's Outline of Group Theory by Benjamin Baumslag and Bruce Chandler*Textbook/~$38*- Groups, Rings and Fields (Springer Undergraduate Mathematics Series) by David Wallace

Along with Linear Algebra and Real Analysis (Calculus), introductory Probability is the most important first year course for a quant to know. This applies for quantitative traders, quantitative analysts (derivatives pricers), risk managers (VaR, CVA etc) and data scientists. I cannot stress enough how important it is for a practising quant to have an intuitive grasp of probabilistic concepts. Time spent studying here will pay dividends over a quant career.

Undergraduate introductory probability courses usually begin by discussing the laws of probability, including Bayes' Theorem, probability distributions, discrete random variables, expectation, covariance and continuous random variables. These are *all* necessary topics for the quantitative analyst.

Probability courses naturally lead into more advanced courses on (classical) Statistics, Bayesian Statistics, Stochastic Processes, Stochastic Analysis, Econometrics and Time Series Analysis.

As with Groups, there are no shortage of textbooks on Probability for the undergraduate student, nor MOOCs for that matter. I learnt probability primarily from Ross, below, as well as the Schaum's Guide (I prefer to learn by doing!). There is also a Coursera course on Probability, given by the University of Pennsylvania:

*Textbook/~$162*- A First Course in Probability by Sheldon Ross*Textbook/~$13*- Schaum's Outline of Probability and Statistics by Murray Spiegel, Jason Schiller and Alu Srinivasan*MOOC/Free*- Coursera - Probability by Dr. Santosh Venkatesh of the University of Pennsylvania

What is "Mathematical Computing"? Broadly, it is carrying out mathematical analysis using computer programs. This is essentially the definition of a quant! Hence, it is absolutely essential that you gain a grounding in programming algorithms at the earliest possible stage.

For the autodidact, such a course may seem a little unnecessary, as it is straightforward enough to learn how to program from the various sources on the internet, along with a large array of textbooks. However, I will state that "learning how to program" and understanding how to take a mathematical algorithm and turn it into efficient computer code are completely different skillsets.

One of the key benefits for a quant of carrying out a PhD in a scientific computing discipline is that it teaches you how to take complex algorithms, found in papers that often leave out the essential details, and write them into fully working pieces of software in a reasonable time frame. Undergraduate courses such as Mathematical Computing are often the first steps in learning how to carry out scientific computing.

What do you actually learn though? Usually, a mixture of MATLAB, Mathematica, Maple, Python, Java or C++ is taught, along with simpler algorithms such as basic numerical integration of Ordinary Differential Equations, symbolic manipulation, root-finding, optimisation etc. These are all key skills for a quant.

It is difficult to suggest study materials for a course such as Mathematical Computing as the syllabus can vary substantially between universities. An introduction to MATLAB or Mathematica is often a good first step, and the following books reflect this:

*Textbook/~$36*- Hands-On Start to Wolfram Mathematica by Cliff Hastings*Textbook/~$47*- Matlab - A Practical Introduction to Programming and Problem Solving, 3rd Edition by Stormy Attaway

The first year in an undergraduate syllabus is all about introducing the student to new ideas, as well as formalising old ones. It is usually a "make or break" situation for those in formal study, and often students will transition to other courses such as physics, computer science or economics. It is a substantial step up from highschool mathematics and is not to be underestimated.

However, the autodidact has a lot more flexibility, in that the "course" and "modules" can be tailored to the particular career path or hobby-learning desire. For prospective quants, it is easy to "cherry pick" courses such as Linear Algebra, Differential Equations, Probability and Real Analysis (Calculus) to suit more specific quantitative finance topics.

In the next article, covering Year 2, we will look at more advanced topics in the subject areas outlined above, including the Riemann Integral in Real Analysis, more complicated topics in Group Theory, an introduction to Metric Spaces (a precursor to Topology), Vector Calculus and Statistics (an absolutely essential subject for the practising quant trader or risk manager). We will also gain our first taste of Stochastic Processes, as a precursor to more fundamental study of Stochastics in Stochastic Analysis.

*Read the next article in the series: How to Learn Advanced Mathematics Without Heading to University - Part 2*

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