How to Learn Advanced Mathematics Without Heading to University - Part 3

How to Learn Advanced Mathematics Without Heading to University - Part 3

In the first and second articles in the series we looked at the courses that are taken in the first half of a four-year undergraduate mathematics degree - and how to learn these modules on your own.

In the first year we discussed the basics - Linear Algebra, Ordinary Differential Equations, Real Analysis and Probability. In the second year we built on those basics, studying Metric Spaces, the Riemann Integral, Group Theory and calculus on Vector Spaces.

In the third year of a four-year masters-level course, especially one with an applied focus that will be of interest to quants, we need to begin thinking about more abstract concepts that will prepare us for study of Stochastic Calculus, Probabilistic Machine Learning and Bayesian Econometrics.

With that in mind it is essential that we study topics such as Measure Theory and Linear Functional Analysis.

Both of these courses contain ideas that underlie Probability Theory, Time Series Analysis and some aspects of Machine Learning. Measure Theory teaches us about generalising the Riemann Integral to the Lebesgue Integral, while Linear Functional Analysis discusses function spaces, many of which are necessary for solutions to certain Partial Differential Equations.

At this level there are less video lectures available, since the content becomes quite complex. However, there are still plenty of accessible textbooks and lecture notes, many of which contain questions and solutions to test your knowledge.

## Year 3

Here is the course list for Year 3:

#### Complex Analysis

Complex numbers are a generalisation of real numbers motivated by the need to define the concept of $\mathbb{i}=\sqrt{-1}$. This comes about because of the solution to particular equations, such as the familiar quadratic equation from highschool algebra, which possesses complex roots when $b^2 - 4ac \lt 0$ in the solution to the equation $ax^2 + bx + c = 0$.

The set of such complex numbers is denoted by $\mathbb{C}$. One of the key results involving complex analysis is the Fundamental Theorem of Algebra, which states that every complex-valued polynomial or order $n$ with coefficients in $\mathbb{C}$ contains $n$ roots within $\mathbb{C}$.

Complex analysis is concerned with the concepts of sequences, series, differentiation and integration, as in real analysis. However the fact that $\mathbb{C}$ is two-dimensional (each element $z \in \mathbb{C}$ has a 'Real' and 'Imaginary' component) and possesses a different geometry compared to $\mathbb{R}$ means that certain familiar results are not quite the same.

Complex analysis may be seen to be quite abstract and not immediately applicable to "real world" situations that are often modelled to take place in $\mathbb{R}^n$. However it is a highly applicable subject in various areas including quantum mechanics (the Schrodinger equation), fluid dynamics (via conformal mappings) and electrical engineering (signals analysis/Fourier transform, control theory etc).

For quant trading applications in particular, signals analysis is a large part of some funds (Renaissance Technology being the oft-quoted example!) and hence having a good understanding of how complex numbers behave will help you grasp the tools you are using in a much more structured way.

##### Study Material

I learnt complex analysis primarily from university lecture notes as well as the course text from Ian Stewart and David Tall.

The Springer Undergraduate Mathematics Series (SUMS) book by John Howie is also great for self-study and is pitched at a slightly simpler level (maybe second year undergraduate).

An excellent addition to the more "algebraic" approaches is the geometric approach of Tristan Needham in his book Visual Complex Analysis. I myself tend to understand concepts better when I see them visually represented and this book really helped me grasp the main concepts, particularly around complex differentiation.

If you prefer to learn via lecture notes or video courses then both Coursera and MIT offer free alternatives in a MOOC format:

#### Topology

Topology is a subject of study in mathematics that attempts to determine particular properties of abstract spaces that are preserved under continuous transformations. The "continuous" aspect of these transformations means that only concepts such as "stretching" or "bending" are considered. If holes are teared or parts are "glued" together, these are not considered topological transformations.

An old mathematical joke involves a topologist who cannot distinguish between a teacup and a doughnut (donut for those in the US!), as both have a single hole (the hole in the doughnut and the hole in the handle of the teacup).

The main subjects of study are topological spaces, continuous transformations between them (homeomorphisms and diffeomorphisms) as well as the concepts of compactness and connectedness.

Elementary topology leads on to some fascinating areas of mathematics including differential topology and algebraic topology. The former touches on the study of manifolds and differential geometry, a deep area of mathematics that has close connection with theoretical physics. The latter makes use of abstract algebraic tools in an attempt to classify topological spaces via determination of invariants.

Admittedly, Topology is not a subject that easily admits a direct relevance to quantitative finance techniques. However, that is not to say it isn't useful. In particular a startup known as Ayasdi makes direct use of topological methods in order to carry out regime detection.

Hence if you want to be one of the top quants and work on the most cutting edge problems, as with other more abstract areas of mathematics presented in this article series, it helps to really learn as much as possible, as it is often initially unclear how some areas of mathematics might eventually be applied.

##### Study Material

I learnt introductory Topology from a lecture course given at the University of Warwick while I was an undergraduate. The core textbook for the course is by Sutherland (below). This is a great book and gently moves from real analysis to metric spaces to full abstract topological concepts.

However, I later decided to casually read through some introductory topological concepts to refamiliarise myself with the area and I decided to try Crossley's Springer Undergraduate Mathematics Series book. Arguably this is better for self-study. It goes quite far, covering homotopy as well as singular and simplicial homology.

Unfortunately Topology is not an easy subject to find video lectures for and so the main path to learning it is via textbook self-study. That being said, given that it is not often a prerequisite for quantitative finance, it can easily be "skipped" if your sole focus is to learn sufficient mathematics to be a quant.

#### Ring Theory

In the first and second year of a traditional undergraduate degree it is common place to study the abstract algebraic concept of a group in some depth. Groups have a huge range of mathematical and physical applications and are one of the most fundamental mathematical structures.

A ring, however, is similar to a group except that rather than simply being a set with a single associated binary set operation it is a set with two binary operations, which attempt to generalise the concepts of addition and multiplication. The motivation for such an object is that it allow arithmetic theorems, which apply to numbers, to be extended to more sophisticated sutrctures such as polynomials and matrices.

One of the most important aspects of ring theory is whether the ring is commutative or not (that is, whether the order of multiplication operation has an impact on the result). Ring theory is closely related to algebraic number theory and algebraic geometry and results in those fields overlap significantly with ring theoretic results.

Ring theory is certainly an abstract area of mathematics. It is unlikely in most quant finance roles that you will ever need to use many results from theory directly. However, as with study in topology, if you are aware of these concepts and become well-versed in the material it is likely to open up opportunities in some of the more sophisticated commercial quant finance research firms who utilise non-traditional, but highly advanced, methods.

##### Study Material

Once again the Springer Undergraduate Mathematics Series provides two useful self-study texts for abstract algebra and ring theory. Another recommended text is an older work (1970) by Burton.

#### Fluid Dynamics

Fluid Dynamics may seem an odd course choice for a prospective quant to learn. Despite the seemingly different areas of research the subject is highly applicable to quants who wish to become expert at derivatives pricing.

Fluid dynamics is formulated via the principle of conservation laws taken from theoretical physics. By assuming continuity of mass, momentum and energy, it is possible construct the compressible Navier-Stokes equations, which are the main equations used within fluid dynamics research.

The equations are written in the language of vector calculus since fluids are a three-dimensional field-based phenomena. The equations themselves (at least in certain forms) appear as Partial Differential Equations for density, velocity and enthalpy.

Many solution techniques for fluid dynamics problems involve numerical approximation and make use of Finite Difference, Finite Element or Finite Volume Methods, which attempt to approximate partial derivatives or integrals at certain points or volumes within a space.

Why is this relevant for quants who wish to become derivatives pricers? Stated simply, the Navier-Stokes equations are very similar to the Black-Scholes PDE used for pricing options. Not only that but once exotic option information is added it is often necessary to use numerical approximation techniques as analytic techniques are difficult or impossible to apply.

Hence an expert in numerical analysis of PDE is likely to be highly sought after for pricing securities that have a stochastic component associated with them.

For full disclosure, my PhD was actually in compressible fluid dynamics and I utilised a lot of the knowledge gained on the course to transition to the quant finance world. Hence, in my opinion, fluid dynamics is not only highly relevant for pricing options, but it is also an extremely fascinating area of study in its own right and is one well worth pursuing.

##### Study Material

Fluid Dynamics is a vast subject stretching across applied mathematics, theoretical physics, aeronautical and chemical engineering as well as more recently the biosciences. Hence it is difficult to suggest one or two textbooks to "learn the subject".

I myself gained insight into many of the underlying ideas from a book by Acheson, called Elementary Fluid Dynamics, as well as my university lecture notes. I picked up more theory in textbooks on aerodynamics and computational fluid dynamics. Anderson is particularly good for self-study and covers a vast amount of aerodynamics theory (and application!) in a conversational style. I made significant use of his "Compressible Flow" textbook for my PhD, although that is somewhat more specific than the texts I've listed here:

#### Measure Theory

Measure Theory is usually considered a difficult course by many undergraduates. However it is an absolutely essential prerequisite for a quant who wishes to be an expert at derivatives pricing.

Measure Theory is motivated by the fact that the "traditional" Riemann integral, familiar from high school calculus, is unable to be applied to certain classes of functions. The canonical example is the indicator function on the rational numbers, $\mathbb{1}_{\mathbb{Q}}$. This function is equal to unity for elements $x \in \mathbb{Q}$ and equal to zero for elements $x \in \mathbb{R} \setminus \mathbb{Q}$. The Riemann integral is not defined here as it is impossible to create upper Riemann sums and lower Riemann sums that equate.

This motivates the concepts of the Lebesgue Measure and the Lebesgue Integral, which serve to generalise "distance" and integration to a wider class of sets and functions. While these may seem like abstract concepts, they are absolutely crucial in the mathematical definition of probability and how numbers can be chosen at random. These are key concepts that underlie stochastic calculus and the mathematics behind derivative pricing.

Hence, having a solid grasp of Measure Theory will pay dividends when studying stochastic calculus at the fourth year undergraduate level or on a Masters in Financial Engineering (MFE) course.

##### Study Material

My favourite resource for learning Measure Theory is yet another Springer Undergraduate Mathematics Series (SUMS) book by Marek Capinski and Ekkehard Kopp called Measure, Integral and Probability. The book is perfect for self-study. It spends a good deal of time initially on building up the Lebesgue Measure and Lebesgue Integral, as well as how they lead to spaces of integrable functions (such as Hilbert Spaces). In particular it frames all of the theoretical aspects in the language of probability at the end of each chapter, providing significant motivation for the quant who wishes to become a derivatives pricing expert or sign up to an MFE program.

In addition to Capinski and Kopp, MIT have produced a series of shorter PDF lecture notes which are free to download, although they are quite dense for self-study purposes. If you are more of a visual learner then there is also a YouTube channel by MathematicalMonk on Measure Theory and Probability.

#### Linear Functional Analysis

Linear Functional Analysis is primarily concerned with extending the ideas from finite-dimensional vector spaces, learned about in Year 1, to infinite-dimensional spaces, often with some form of "structural" addition, such as an inner product, a norm or a topology. Such spaces are motivated by the need to identify a setting for solutions of differential and integral equations. An infinite-dimensional vector space over the real or complex numbers often provides such a setting.

Quantum mechanics, time series analysis, extensions of measure theory and stochastic calculus all make heavy use of ideas from linear functional analysis and so quant analysts and quant traders can all benefit from having a rigourous grounding in the subject.

The main concepts discussed include norms and normed spaces, inner products and inner product spaces. Complete normed vector spaces are known as Banach spaces, while complete inner product spaces are known as Hilbert spaces. A key area of study in linear functional analysis is that of linear transformations between these sorts of infinite-dimensional vector spaces, which in some cases the set of such transformations themselves define another infinite-dimensional vector space.

As a self-studying prospective quant your time to study will inevitably be limited due to career, family and even social commitments. A key question to ask in the "third year" of your study is whether to make the effort to study each course if you are to pursue further study directly relevant to becoming a quant.

While Linear Functional Analysis is not a direct prerequisite for further quant finance study, having a grasp of the main results and objects will certainly make graduate-level mathematical finance research a lot more straightforward. Hence, it is up to you whether to "dive in" to linear analysis or whether to simply view the main theorems and results, without detailed study of the proofs.

##### Study Material

There are a wealth of textbooks - both introductory and advanced - that cover elements of functional analysis. The SUMS book by Bryan P. Rynne and Martin A. Youngson, Linear Functional Analysis, is highly readable and great for self-study. Also, Kreyszig is the usual recommended book that only requires calculus and basic linear algebra as prerequisites.

There is a set of freely-available lecture notes given by Professor Melrose at the MIT Open Courseware site for those on a budget:

#### Elementary Differential Geometry

Elementary differential geometry is predominantly concerned with curves and surfaces lying in three-dimensional space, that is $\mathbb{R}^3$. For curves the notions of "curvature" and "torsion" allow us to determine how a curve can twist in $\mathbb{R}^3$. However, surfaces in $\mathbb{R}^3$ require additional notions in order to specify their behaviour, namely the Guassian and Mean curvatures.

These geometric notions have relations to topology via the Gauss-Bonnet Theorem, which relates the average of a surface's Gaussian curvature to that of its topological Euler number. Such connections make differential geometry an extremely exciting area of modern research.

Elementary differential geometry leads naturally on to manifolds, which are generalisations of curves and surfaces in $\mathbb{R}^3$. In addition, General Relativity, one of theoretical physics' greatest theories makes heavy use of concepts from Riemannian Geometry, which is a subfield of differential geometry.

As with many courses in a typical third year mathematics undergraduate degree, elementary differential geometry is not obviously related to many of the common quant finance topics. However, I will once again state that Jim Simons was originally a geometer prior to founding the Renaissance Technologies quant fund and it is highly likely that their huge success is due in part to thinking about financial markets in a geometric and topological vein.

##### Study Material

Elementary differential geometry is usually restricted to the geometry of curves and surfaces in $\mathbb{R}^3$. Courses on the topic do not often delve into the deeper aspects of higher-dimensional manifolds or Riemannian Geometry until the fourth year.

Hence the following books, two of which are from the Springer Undergraduate Mathematics Series (and hence ideal for self-study) and the third from Cambridge University Press, are pitched at the right level of difficulty for a third-year course.

#### Partial Differential Equations

At university Partial Differential Equations (PDE) were my favourite area of study and were one of the original reasons that I eventually became a quant, namely through numerical solution of Black-Scholes type models. They are one of the most diverse and fascinating areas of applied mathematics and touch on a huge range of other mathematical topics. I cannot emphasise enough how important they are in mathematics and quantitative finance in general.

PDE are equations that relate spatio-temporal field values to their derivatives in various dimensions. For instance, the diffusion of heat in time, through an insulated rod, is a second-order linear PDE with specific boundary conditions. They encompass a huge range of phenomena in mathematical physics, including continuum mechanics, fluid dynamics, electromagnetism, quantum mechanics, general relativity and many other field theories.

Solutions of PDE (which are themselves functions of space and time) can be sought either through analytical or numerical methods. The former allow direct specification in space and time of a field property, once initial and boundary conditions are set, while the latter must approximate the solution discretely on a lattice, or mesh, using a theory such as Finite Elements, Fininte Differences or Finite Volumes.

Classification of PDE is quite a challenge. For second-order linear PDE, the main approach is to group them into Parabolic, Elliptic and Hyperbolic type. These three classifications have an associated physical interpretation, especially in fluid dynamics.

Deterministic PDE leads naturally on to the study of Stochastic PDE (SPDE), which is a highly active area of research. Indeed, SPDE are highly applicable to quantitative finance within the field of stochastic calculus, hence it is essential for the quant wishing to be involved with derivatives pricing, model validation or risk management, to be aware of PDE and thus eventually SPDE.

##### Study Material

There is no shortage of introductory textbooks on PDE. The field is so vast that it can be difficult to know where to begin. I found the book by Strauss to be particularly engaging at the undergraduate level.

The Springer Undergraduate Mathematics Series book on analytical solutions of PDE is also a great self-study resource.

#### Numerical Linear Algebra

Numerical Linear Algebra is a more specialised subject for a mathematics degree, but I have included it since it was a module offered on my own undergraduate course, as well as being extremely relevant for computational finance.

The subject concerns the use of computers to efficiently solve problems that occur in finite-dimensional linear algebra. Rather than being solely of academic interest, these problems usually form the computational "bottleneck" for solutions of applied problems.

Finite Element and Finite Difference equations for the solution of fluid dynamics PDEs are often formulated as a matrix inversion problem. Efficient inversion of these "structured" matrices can significantly improve computational efficiency and thus allow a more detailed analysis. In addition, machine learning algorithms often make use of efficient implementations of LU decomposition, QR decomposition or Singular Value Decomposition (SVD).

The same is true for PDE models in quantitative finance, which also rely on similar numerical techniques for efficient matrix computation.

Despite the availability of "off the shelf" software for carrying these tasks out, I always believe that it is extremely worthwhile for the quant to be aware of the underlying principles of how these techniques work, in order that when developing new models, it is straightforward to extend more elementary techniques.

##### Study Material

The classic text in the field is "Golub and van Loan", now in its 4th edition. In addition, I made heavy use of a text by Trefethen for my undergraduate dissertation, which I also recommend.

## Next Steps

The third year of an undergraduate syllabus is mainly about preparation for the more advanced courses in the fourth year. For the autodidact who wishes to become a quant trader, quant developer, quant/risk analyst or data scientist, it is necessary to choose modules that make sense for each respective career path.

As the courses become more complex it is harder to state that they are an absolute necessity for quant finance work. Many of the courses will feel rather abstract and will seem quite distant to the day-to-day work of many quants. However, a vast amount of quantitative finance theory, including probability, time series and machine learning, is based on the underlying principles of these courses. By learning these principles you will give yourself a distinct advantage when applying to the top quant jobs, or researching for your own strategies.

In the next article covering Year 4 we will take a look at areas that begin to overlap with graduate-level applied mathematics, including applications to mathematical physics and stochastic processes. All of these modules will provide a solid grounding in material that will be necessary to learn when applying for Masters in Financial Engineering courses.