# Dear Math Students…

**Dear Math Students**

Article By: Vanny Khon

You, as an undergraduate in Cambodia, might want to read this article if you want to learn mathematics so that you can contribute something significant to the field. Studying mathematics in our country can be very challenging. If you want to learn mathematics so that you can do something meaningful for the field, you have a lot to do and a lot to know. Not only do you have to be self-disciplined to learn the subject all by your own, but you also have to know what you have to know and what you don’t know. Here I am trying to give you my idea on what a mathematic student in our country should learn and do. Hopefully, this article can help you to know what you should know in math (and possibly what you might never know) so that you can do well when you are an undergraduate, and also, you can prepare yourself to go on for higher education.

First of all, I believe it is important I mention that mathematics has been being developed by many mathematicians for more than three thousand years. And since the time that Newton and Leibniz invented calculus in the seventeen century, mathematics has been growing rapidly; and at this present time, it is a very huge, very broad, and very active subject that has many subareas. Now researches are being done are at very high levels that master students are not supposed to be able to understand. So, it is very unlikely that you can contribute anything important to mathematics if you don’t know enough about it. Now I am only in my fourth years in university, I don’t really know enough about the subject that I can tell you how much it is to be enough, but I think I can tell you how much it is not enough.

To my belief, it is not enough to go on doing mathematics in somewhere else outside of Cambodia if you just learn what you are taught at your university, and, here, I am not even talking about doing research in mathematics but just to go on and pursue higher education somewhere else that has good education system. Let me illustrate my point by telling you my experience when I went to a summer school in Vietnam in March 2016. That summer school was about algebraic geometry, and it was designed for advanced undergraduates and master students who knew some algebra, as it was said in the course’s introduction. When I first heard about it, knowing that I knew undergraduate algebra reasonably well, I thought I would get benefits from it, so I applied for it. As it turned out, I couldn’t follow the courses since the middle of the third lecture. And I endured through it until the course finish, and at that time I was lucky because it was only two weeks. The course required the understanding of commutative algebra, which I hadn’t learned. Commutative algebra is something that you can learn after you have done abstract algebra and generally it is not taught to undergraduates. So you see, if you go for a course, and if you don’t know its prerequisite well enough, you are most likely to get lost. In higher education, it is almost impossible for anyone to teach you all the way back from the first point that you need to understand his\her class. Lecturers (or speakers) assume that you understand mathematics enough for their courses, and they start from somewhere where they think their courses (or talks) should begin. And generally the place where they think they should begin is quite advanced compare to what are given to us in our country.

So, to be able to take a course in mathematics in higher education, you have to possess a certain amount of knowledge in mathematics. Expectation can be varied from place to place, depending on what you do and where you do it. The only thing that I am sure of is that you are required to understand some core subjects in mathematics. They are single and multi-variable calculus, linear algebra, abstract algebra, real analysis and complex analysis. If you don’t reasonably understand these subjects, you are more likely get lost when you sit in master courses somewhere else outside our country. And Professor Rod Halburd, professor of mathematics at University College London, believes that this list contains only the absolute minimum that you need to know.

There are lots of things that you need to learn and can learn in mathematics. In the following paragraph, I am going to tell you some of the subfields in mathematics that I know. As I have mentioned above, I am only a fourth year undergraduate, so what I am going to say here is likely to be disagreed by experts in the field. But I hope to give some ideas of possible directions that you can go on in mathematics, and make you feel the grandeur of mathematics.

Here I am trying to give some ideas of subfields in three of the main fields in mathematics, namely analysis, algebra and topology.

**Analysis**

- a)
**Elementary Real Analysis**: It is the first subject you encounter when you do analysis. It includes both single and multivariable analysis. A part of it is about defining precisely the concepts of limit, derivative and integral and series. You get to know what a limit of a function really is, what an integral of a function is, the concepts of different types of integrals–namely Cauchy integral, Darboux integral and Riemann integral–and the relationship between them etc. Throughout analysis, you learn to prove some important theorems about these concepts. - b)
**Complex Analysis**: After doing real analysis, some people choose to do complex analysis. From the name it bears, you might think it is the extension of analysis that you do for real variable function. Of course, we define limits, derivative and integral for function of complex variable. But the interesting thing about complex analysis is that its theory is much richer than that in real analysis. Things in complex analysis behave remarkably well, and there are lots of interesting situations in the theory of complex analysis compared to real analysis. For example, in real analysis you can do integral only on an interval or union of intervals, but in complex analysis, you can do integral along various curves in the complex plan. So, complex analysis has its own flavor different from real analysis - c)
**Fourier Analysis**: Broadly speaking, Fourier analysis is about representation of a function by a cosine and sine series, and some application of it in physics, namely heat diffusion problems and vibrated strings. The problems that lead to the development of the subject are from physics. One of it is to understand the behavior of strings, which attached to two fixed poles, when you apply force on it. As it turns out, this question in physics is related to the representation of a function by a series of trigonometric function. - d)
**Measure Theory**: What we really do in integration is we give a numerical value to a set. For instance, integration of a function along an interval is equal to some numerical value. Measure theory is the generalization of integration theory. And when we develop the theory well enough, we come to developing theory of integral that allows us to use integral wider class of functions than that Riemann integral can do. - e)
**Hilbert Spaces Theory**: It is the study of infinite dimensional version of spaces like R^n or C^n. When we do real analysis, we only do analysis on those spaces whose dimensions are finite. But in Hilbert Spaces theory, you come to do analysis on infinite dimensional spaces.

Besides all these, in analysis, there are things like**analytic number theory, p-adic analysis, harmonic analysis, Nevannlina theory, partial differential equation, ordinary differential equation, differential geometry and differential calculus**and I believe there are more.

**2. Algebra: **

In elementary set theory, when you have a set, all you can do is to form subsets of that set and count the number of elements of its subsets or to perform other set theoretic operations such as union and intersection. One of the first things in abstract algebra is the study of the way to combine two elements of sets, called operation, as we want to see how elements of some sets interact with each other as the whole while considering a certain operation. The subjects of algebra are mostly concerned with the structures of a collection of objects rather the object itself. The study of geometry usually boils down to the study of groups or other algebraic objects that we associate to the geometric object.

- a)
**Abstract Algebra**: It is the first thing you see when you do algebra. It talks about operation, property of operations, and basic properties of some very important algebraic objects called Group, Ring and Field. - b)
**Galois’s Theory**: It is a very beautiful subject in mathematics with very interesting history. The subject itself deals with the question of the possibility of whether one can find roots of polynomials in traditional sense. It is a heavily-used tools from group theory and field theory. - c) I don’t know how to give the ideas of the following subjects of algebra, so I just mention them here.
**Representation theory, field theory, ring theory, commutative algebra, category theory, abstract linear algebra, algebraic number theory, algebraic geometry, homological algebra, sheaf theory, quantum group, Lie algebra, differential Galois theory**and there are more that I don’t know.

**3. Topology **

is geometric in nature, but it is just that we concern in different feature from what we do when we learn Euclidean geometry of other geometry. I list here some subareas of topology. They are **algebraic topology** (the study of topology by using tools from algebra), **geometric topology **(I never have a taste of it), **differential topology** (I think it uses the tools from differential geometry), **Knot theory** and **surgery theory** (cut and clue topological spaces).

Each of those subfields of mathematics I have mentioned above is very big itself. Some people are doing a research in a particular part of algebraic topology or representation theory. For something like representation theory, a standard text book for introduction of the subject is generally contain more and 500 pages and things are developed in a very intensive pace.

I myself have only learned a fairly small amount of mathematics, but I am not discouraged by it. I am learning it and improving myself every day. I know I am going to do a research in math when I am ready, and I know that I am going to learn it all the rest of my life. Doing mathematics is a very sustainable way to achieve happiness, for me. The truth in mathematics is eternal and that is very satisfactory things to know. Math gives me an aspiration to live. Finally, I hope you feel the same for math.