Which field(s) of mathematics are you interested in?


EDIT 1: There really should be a +1 to the CO section. I just forgot to add it, when I first made the poll. Sorry!

EDIT 2: +1 to Topology as well. Please let me know if you guys have any more areas of mathematics that I’m missing!

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High school life, research topics, and advice for the young aspiring mathematician. (Part 3 END)

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This will be the final post in my three part saga about mathematics and high school. You can check out parts one and two here and here respectively if you haven’t already read them. In this post I want to end off by offering the final part that was mentioned in the title, research topics (I have decided to write about advice for the upcoming mathematics undergraduate in a different series some time far in the future).

If you have been reading any of my previous posts, you might remember me talking about how a critical part of learning mathematics is knowing how to explore, be creative, and think about problems outside of the classroom. It is better to sow your tiny seeds interests early than later, because as many of you find out from learning maths, many of the areas  interconnect beautifully if you just take a while at the subtleties.

Thus, below, I have provided some topics of interest that can be researched at the high school level but delve deeply into undergraduate topics; all they need is a bit of perseverance, time, and creativity to get started. You can probably even get your teacher in on this. Ask him/her if instead of completing droll textbook problems or monotonous exercises, you could start a research topic instead! No more spending one hour every day copying down exactly what the book asks you to do or sketching and drawing precise margins (at the university level, precisely drawn graphs is one of the lesser important aspects of mathematics). Instead, mathematics turns out to be a continuous process that you can do every day, without the stress.

... unless of course, your teacher wants you know the critical skills of copying and pasting. Then, it is advisable to keep at those 'exercises'. {MS Clipart}

So without further ado, here they are (with difficulty rankings relative to high school content):

Population Dynamics

Difficulty: 7/10
Mathematics Topics: Differential Equations, Chaos Theory
About: Students can examine the different  population models such as Malthusian, logistic, and island biogeography. Investigations can be about the limitations of each model, the strengths and the ideal conditions for the model to work.
Bonus: Students can talk about how current day population models are more similar to chaotic systems than deterministic ones

 Poker, Probability, and Game Theory

Difficulty: 6/10
Mathematics Topics: Game Theory, Probability, Statistics, Simulations
About: Students can investigate the best course of action to take with various poker hands, categorized by winning strategy. Approaches can be from a game theoretical, probabilistic or statistics side or all three.
Bonus: Create a computer simulation to accept or deny your theory through a null hypothesis and Monte Carlo simulations.

Rings, Fields, Groups, Vector Spaces, and Algebras

Difficulty: 9.5/10
Mathematics Topics:
Abstract Algebra, Linear Algebra
About: Students can discuss the differences and similarities between the above algebraic objects and how to construct them. Investigations can be made about various isomorphisms between algebraic objects and about interesting objects such as the power set of the real numbers being an algebra under the union and intersection operators.
Bonus: Construct your own simple algebraic object (maybe the set of all watermelons that taste like oranges; something silly) and see if you can go from a magma all the way to an algebra.

Fractals and Iteration

Difficulty: 9/10
Mathematics topics: Fractals, Chaos Theory
About: Students could take a look at the definition of a fractal and what exactly is fractal dimension. Investigations can be made about the construction of fractals under various domains, their applications (e.g. coastlines), and their self-similarity properties. Students could try to generate fractals through freeware found online.
Bonus: Students could investigate what higher dimensional fractals would look like if projected on to the 3D or 2D plane. An example of a 3D projection into the 2D plane would be a contour map.

Methods for Computing Mathematical Constants

Difficulty: 8/10
Mathematics Topics: Computer Science, Simulations, Calculus
About: Students can investigate the different methods for computing mathematical constants such as \pi, e, and \phi the golden ratio. Some investigation could be made about the complexity of the algorithms and how fast they converge to the constants.
Bonus: Come up with your own unique constant and an algorithm used to compute it. Talk about complexity and rate of convergence as well as if it can be represented by current day mathematical constants

Cardinality of Various Sets

Difficulty: 8.5/10
Mathematics Topics: Set Theory, Ordinals
About: Students can investigate the various cardinalities contained in sets such as the natural numbers, integers, irrational numbers, rational numbers, real numbers, and algebraic numbers. Students could also define their own set and try to determine the cardinality of that set.
Bonus: Students can devise a model for classifying the cardinalities of different sets based  on certain characteristics contained in each set.

If you have read Part 2, you might recall that I did write a research paper on a topic of my choice. From the above topics, my essay was actually about the last one, cardinalities, which you can access a copy, here. Some time in the future, I will be restoring that essay to a cleaner LaTeX typeset document. You may use it as a guide, but I highly encourage that readers take the time to learn about the above topics themselves. After all, to quote Georg Cantor:

 

English: Georg Cantor

"The essence of mathematics is in its freedom." {Image via Wikipedia}

 

 

I hope you all enjoyed my little mini-series rant about my high school experiences and if you did, I would love to hear about your experiences in the comments below. In the upcoming months, stay posted for even more content as I begin to dive straight into some serious mathematics this term and start to bring you my quality typsetted notes.

Happy New Year Everyone!


To all my readers and friends out there, I just want to start off by wishing everyone a happy  new year! While you may not have heard from for a quite a while now, I assure you that I have been working on quite a few other projects. Here are a few that you look forward to:

  • First off, I decided to move and develop my LaTeX templates on GitHub, as per a recommendation by one of my friends, and you can now access the latest version here. Instructions are on the page in case you are not too familiar with the layout of the pages. I will still leave the link to the uWaterloo Linux website intact, but all development on that site will cease indefinitely (new files will be put up there, but not new webpages).
  • Also on GitHub, I have put up many of my coding projects, including a Stock/Black-Scholes Option simulator and a biased random walk. You can access them via this link.
  • Today (or tomorrow), I will hopefully be finishing off my “High school life, research topics, and advice for the young aspiring mathematician.” series so you can look forward to that.
  • Course notes will start coming up soon as I continue my quest to take over Lambertw’s empire of quality notes
  • I have decided to publish the fish population article for lack of updates from the original website on that problem. The problem of the square through the sphere is still under development.
  • In the future, I will be restoring my old extended essay into a cleaner LaTeX form
  • Currently I am developing a cleaner way to view projects on the site. Suggestions are appreciated; right now I am thinking of just moving everything to GitHub.
  • As for the few projects that I will be putting off, these include my Ray Tracing project and Black-Scholes mini-series due to the six-course load that I will be taking at Waterloo in the coming Winter.
  • For those of you who would like to see some an introductory article to Black-Scholes, I will be putting that up very soon.

I hope to see old and new readers coming in for the new year because now is where the content will be really flowing in!

Future Plans and Projects.


While I am still quite busy preparing for Exam FM (completing 10 mock exams), I do have quite a few short-term and long-term plans for this blog.

For the sake of being concise, here is a short list of what I have planned; many of the projects will only be started after exam FM which is on the 10th of December. However, for some of these project, I might have some to time to start on them this week:

Short-term Projects

  • Finish my proof of the volume displaced from pushing a square through a sphere; an incomplete version can be found here.
  • Complete the third entry in the high school reflection series
  • Solve an interesting population problem about fish breeding here. I will also be re-typing the problem for better coherency (new version here)

Long-term Projects (>3 months)

  • Begin typesetting notes for Exam MLC. These notes will be extremely concise and should be considered a supplement for the recommended readings here. One of the texts is free while the other requires purchase (I acquired an eBook version). Since this is the first exam that will follow the new syllabus, I will be covering the content quite thoroughly with what is listed on the syllabus.
  • Course notes for second and third year courses at uWaterloo (specifically the ones that I will be taking next term
  • “Advice for the university freshman” series; look for it during the Winter term
  • Useful financial spreadsheets

Committed Projects (>1 year)

  • Ray Tracer project (currently on hiatus until other projects are taken care of)
  • Martingales and Black-Scholes tutorial project (essentially a 30+ series of posts intended to get readers familiar with mathematically rigourous financial concepts)

In addition, I will be posting random tidbits of interesting maths finds here and there, but just don’t expect them to come that often during these coming couple of weeks.

A gift.


For my first official content related post, I thought I would start by giving out something to you, the readers. Specifically, something that’s usually marketed on many financial websites and is often used to save money and carefully plan out investments and expenses. What I am talking about is an Excel budget workbook , and it’s free of charge, of course!

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