A taste of higher mathematics! (1 Viewer)

[seanieg89]

Hi all.

Sometime over the next couple of months I intend to start a professional blog to help me organise my mathematical writing and practice my exposition. (It will also give me a relatively easy way to answer the dreaded questions from friends on the contents of my research.)

Most of this stuff will be graduate level, but I think it might be fun to include a couple of articles/videos aimed at interested high school students and early undergraduates.






Some of the topics that might be suitable for this purpose are:

-Rigour (Why proofs are necessary, tricky fallacious proofs, what is wrong with the way maths is done in high school.)

-Graph theory (The study of networks of nodes connected by line segments. Very applicable.)

-Game theory and how to use maths to beat your friends at poker.

-Number theory. (Talking about some cool things involving primes, perhaps leading into some cryptographic applications.)

-Lagrange multipliers (A general method for finding extrema of multi-variable functions, with some constraints. Eg, maximise x^2+y^2, given that x,y > 0 and xy=1.)

-What is integration? (A more rigorous discussion of integration as taught in high school, and it's various generalisations.)

-Vector calculus (the kind of maths used in understanding things like the interaction between electric and magnetic fields.)

-Calculus of variations. (Some powerful machinery for solving famous optimisation problems such as the brachistochrone. In high school, functions eat numbers and spit out numbers, but we can also study things called functionals, which eat FUNCTIONS and spit out numbers. Calculus of variations amongst other things allows us to find maxima and minima of functionals.)

-Complex analysis. (What is different about calculus in the complex plane? Miraculously, complex analysis often lets us quickly prove things that seem to have nothing to do with complex numbers.)

-Fourier analysis (The idea of breaking up a sound into its constituent frequencies can be generalised considerably and has surprising applications.)

-Differential equations (generalisation of the problem of finding a functions primitive. differential equations can model a VAST number of phenomenon in the physical and social sciences, and let us properly understand physics concepts like heat and waves.)

-Topology and the fundamental theorem of algebra. (An introduction to one of the broad spheres of mathematics that is not touched upon in high school, and an application to proving FTA. Feat. the infamous hairy ball theorem.)

-Chaos theory. (Talking about chaotic behaviour in dynamical systems.)

-Godel's incompleteness theorems (Mindbendingly counterintuitive results about the foundations of mathematics.)

-The Abel-Ruffini theorem (why is there no general formula to solve quintics and higher powers?).

-Differential geometry (the study of notions such as curvature using calculus).

-Nonmeasurability and the Banach-Tarski paradox (You can cut a pea into finitely many pieces and reassemble it into a ball the size of the sun.)

-Hilbert Space theory (The study of infinite-dimensional vector spaces in which we still have a notion of angles, like in the familiar Euclidean spaces. The theory is quite a bit more subtle than the theory of finite dimensional spaces, but still more tractable than the more general Banach Space theory.)






If any high school student can provide any feedback on what sort of things they find interesting / are curious about, that would be much appreciated!

I would also welcome additional topic suggestions, this list was just quickly cobbled together but I have a few more ideas that I am currently forgetting and I will add later.

 

[Carrotsticks]

Great idea Sean.

Something that may be useful to help guide interested HS students towards more proper Mathematics is to explain the need for more rigour in Mathematics (beyond the elementary level) and have a couple of 'brushed over' proofs versus more detailed proofs to outline the differences and assumptions made. I would have liked something like that in HS. The jump in detail when taking my first Analysis course was quite frightening.

Also, can't go wrong with Graph theory and the classical problems in it ie: Königsberg Bridge Problem, Utility Problem, Knights Tour etc, and some algorithms that we sometimes use intuitively without even learning them formally ie: Dijkstra's Algorithm, BDS etc, these are all fairly accessible topics to HS/early UGrad level.

 

[seanieg89]

Great idea Sean.

Something that may be useful to help guide interested HS students towards more proper Mathematics is to explain the need for more rigour in Mathematics (beyond the elementary level) and have a couple of 'brushed over' proofs versus more detailed proofs to outline the differences and assumptions made. I would have liked something like that in HS. The jump in detail when taking my first Analysis course was quite frightening.

Also, can't go wrong with Graph theory and the classical problems in it ie: Königsberg Bridge Problem, Utility Problem, Knights Tour etc, and some algorithms that we sometimes use intuitively without even learning them formally ie: Dijkstra's Algorithm, BDS etc, these are all fairly accessible topics to HS/early UGrad level.

Cheers, these are very good suggestions .

 

[Axio]

Hey Sean

I'm interested in learning more about Complex Analysis (which you already listed) and Hilbert Space.

 

[Kurosaki]

Sounds interesting, can't wait!

 

[seanieg89]

Hey Sean

I'm interested in learning more about Complex Analysis (which you already listed) and Hilbert Space.

Cool, these are definitely things I would be interested in talking about.

I think that Hilbert space theory might be a little hard to motivate and explain to high school students, but well-prepared early undergrads could most likely handle it. One would probably need to have a reasonable familiarity with vector spaces and metric spaces.

 

[iStudent]

Is there such thing as probability/combinatorics but at uni level? I would be very interested to know if there is one!

 

[kalstar]

all so keen.........

 

[seanieg89]

I have roughly ordered the topics by difficulty and how much you would probably need to know beforehand to properly understand the material.

 

[seanieg89]

Is there such thing as probability/combinatorics but at uni level? I would be very interested to know if there is one!

Yeah of course, and that kind of stuff leads into statistics...most of which I don't find terribly interesting but things like random walks are pretty cool.

 

[Kurosaki]

Do you plan to go through generating functions? I came across the while looking at the Fibonacci sequence and it seems quite interesting.

 

[seanieg89]

Do you plan to go through generating functions? I came across the while looking at the Fibonacci sequence and it seems quite interesting.

Potentially, but just as a pretty short self-contained article/video. There isn't all that much to them.

 

[sirable1]

Wow, very nice!

As a 1st year math prospective student intending to do honours in Pure Math or Statistics (not sure yet), I am interested in Number and Graph Theory and other theoretical areas relating to Computational and Discrete Mathematics. The whole idea of Cryptography seems very interesting like that used in the Enigma Machine in WW2 and various electronic applications used today.

I'm doing Higher Several Calculus and Higher Linear Algebra next semester, so I won't probably get to experience these kinds of areas probably until 2016 or at least 2015 S2.

 

[Trebla]

Hmm I wonder if I should do something similar for the Statistics side of things, if I find the time for it

 

[Kiraken]

Great idea Sean.

Something that may be useful to help guide interested HS students towards more proper Mathematics is to explain the need for more rigour in Mathematics (beyond the elementary level) and have a couple of 'brushed over' proofs versus more detailed proofs to outline the differences and assumptions made. I would have liked something like that in HS. The jump in detail when taking my first Analysis course was quite frightening.

Also, can't go wrong with Graph theory and the classical problems in it ie: Königsberg Bridge Problem, Utility Problem, Knights Tour etc, and some algorithms that we sometimes use intuitively without even learning them formally ie: Dijkstra's Algorithm, BDS etc, these are all fairly accessible topics to HS/early UGrad level.

knights tour is a great problem to start with for keen students imo

 

[GoldyOrNugget]

What exactly does one say about knight's tour? It's not a great motivator for graph theory because it's a contrived puzzle and doesn't make use of interesting concepts. Its solutions aren't particularly interesting and don't generalise well to other problems. AFAICT all you can really say is "here's a graph where each node is a chess board cell, and the nature of this graph permits a polynomial time Hamiltonian path divide-and-conquer algorithm and also some decent heuristics".

In terms of high school graph theory: Konigsberg and Eulerian tours is great. Bipartite graphs come up everywhere so the stable marriage problem, the alternating paths algorithm for unweighted bipartite graph matching, network flow and its relationship with bipartite matching, and the max-flow/min-cut duality are all interesting. Dijkstra's algorithm and its uselessness on paths with negative edges is interesting, and leads to the Bellman-Ford algorithm and Johnson's algorithm. Functional graphs have some interesting problems, including the one that Seanie posted ages ago about mathematicians playing a drinking game, and also constant-space cycle-finding algorithms.

 

[seanieg89]

^ this. Knight's tour doesn't really illustrate any interesting concepts, and wouldn't be a memorable example if it wasn't for the fact that you can write it as having chess squares as nodes.

My favourite parts of graph theory are actually when you get away from the combinatorial stuff and look at analysis on graphs, or random walks on graphs. Those are v. cool. (This is all personal preference though, I am an analyst.)

Goldys suggestions are excellent for the combinatorial side of things.

 

[RenegadeMx]

this has my interest

 

[FrankXie]

It seems to me what you listed all belong to pure mathematics, although part of which have found wide applications in science, engineering and econimics etc. In my opinion, applied and computational mathematics are also interesting branches. Applications can be found in HSC maths such as Newton's method, Trapezoidal formula and Simpson's formula, both of which involve approximating a function by a "simpler" function like linear or quadratic polynomial. Another example is solving linear system of equations. The Gauss elimination method (reduced to row echlon form) theoretically works perfect and is so easy. But do you ever think of solving a million by million linear system of equations (of course letting computer do it)? Due to round off errors, after massive amount of calculations, the row echlon form is merely theoretical, the resulted matrix could even be singular and thus the computer will say no solution!
Back to Newton's method. In year 12 textbook questions, you might see that Newton's method does not always give better approximation of solutions unless the initial guess is "good" enough. Even if the method gives a satisfied approximation, what is the rate of convergence? How can you improve the method?
And Simpson's formula. Some year 12 students have already known that some functions(actually most of the functions) do not a primitive ( in the sense of elementary functions), so in this situation, how can you still compute a definite integral? Simpson's rule gives a good approximation. How can you improve the approximation?
These all belong to computational mathematics.

 

[b0b101]

-game theory and how to use maths to beat your friends at poker.

-number theory. (talking about some cool things involving primes, perhaps leading into some cryptographic applications.)

uuuuhhhh, yes!