Nature of Advanced Mathematics Major (1 Viewer)

[leehuan]

So just two broad questions:

1. What kind of concepts do you learn in this course? From what I heard the broad categorisation was between algebra and calculus. So briefly, what kinds of new concepts are introduced in calculus? (e.g. multidimensional etc.) And what exactly is in algebra anyway? I never fully understood exactly how much further algebra could be. All I could really guess were matrices.

Random: How much geometry do you still see at uni?

Also, I'm looking into a pure maths major, if not quantitative risk (to supplement actuarial, also assuming I make it in). Any brief clarification on the concepts here?

2. For my other degree, be it actuarial, commerce, or computer engineering, it's quite clear the job prospects for them. Where would a maths degree take me? Especially if I chose pure maths, because I have no clue what pure mathematicians actually do these days. The idea of "inventing new maths" seems so difficult and surely can't be the only thing in existence.

(P.S. Title clarification - I meant under the umbrella of a BSc)

 

[Paradoxica]

So just two broad questions:

1. What kind of concepts do you learn in this course? From what I heard the broad categorisation was between algebra and calculus. So briefly, what kinds of new concepts are introduced in calculus? (e.g. multidimensional etc.) And what exactly is in algebra anyway? I never fully understood exactly how much further algebra could be. All I could really guess were matrices.

Random: How much geometry do you still see at uni?

Also, I'm looking into a pure maths major, if not quantitative risk (to supplement actuarial, also assuming I make it in). Any brief clarification on the concepts here?

2. For my other degree, be it actuarial, commerce, or computer engineering, it's quite clear the job prospects for them. Where would a maths degree take me? Especially if I chose pure maths, because I have no clue what pure mathematicians actually do these days. The idea of "inventing new maths" seems so difficult and surely can't be the only thing in existence.

(P.S. Title clarification - I meant under the umbrella of a BSc)

TBH the scope of mathematics is pretty huge. Modern mathematics is like a field of mountains, and you can only climb one mountain at a time.
The easier stuff are like a bunch of hills. You can scale them pretty quickly, moving over to other hills and returning after a short while to rescale it again.

Of course, some people (like Hilbert) use quantum uncertainty to allow themselves to scale multiple mountains at the same time (provided they are unobserved). Okay maybe that analogy went over my head but you get the point.

 

[VBN2470]

Currently have exams atm, but I'll write up a more detailed response covering what you do in first-year maths and beyond (for UNSW at least). As Paradox said, maths as a discipline is immensely huge and everything you do will be done step-by-step and with an increased level of rigour (esp. from 2nd year onwards). Calculus in first-year builds upon Calculus you see in 3U/4U so it will be more familiar to you than Algebra, but you will still be learning more rigorous concepts. Algebra will be completely new, only Algebra 'concept' you would have come across is Vectors in 4U Complex Numbers but even that's not close. It is more abstract in nature and often has tight connections between the different ideas you learn as part of it. Challenging, but still fun

 

[RealiseNothing]

This post is for USYD, but I guess it would be similar for UNSW?

You'll do heaps of different topics such as:

- Linear Algebra (algebra)
- Group Theory (algebra)
- Analysis
- Topology

First year:

Semester 1 you do differential calculus and linear algebra. In differential calculus you learn limits properly using the epsilon-delta definition of a limit (https://en.wikipedia.org/wiki/(ε,_δ)-definition_of_limit)
If I remember correctly you also deal with functions of several variables and stuff like grad(f) (basically gradient for several variables - https://en.wikipedia.org/wiki/Gradient)

In linear algebra you will learn about vectors and linear independence (https://en.wikipedia.org/wiki/Linear_independence). Then you will go on to learn about matrices and the determinant, and then finish off the course learning about eigenvalues, eigenvectors, and diagonalisation (https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors).

Semester 2 you have a choice between integral calculus, discrete maths, and statistics. In integral calculus you learn about Riemann sums (https://en.wikipedia.org/wiki/Riemann_sum) and ODE's (https://en.wikipedia.org/wiki/Ordinary_differential_equation).
In discrete maths you learn multinomial theorem (https://en.wikipedia.org/wiki/Multinomial_theorem), generating functions (https://en.wikipedia.org/wiki/Generating_function), and Boolean algebra (https://en.wikipedia.org/wiki/Boolean_algebra).
In statistics you learn about hypothesis testing (https://en.wikipedia.org/wiki/Statistical_hypothesis_testing) and probability distributions (https://en.wikipedia.org/wiki/Probability_distribution).

Second year:

There are 7 maths subjects to choose from this year, they are:

Linear algebra and vector calculus: You learn about double and triple integrals, surface and line integrals, a few different theorems regarding vector calculus like Green's theorem (https://en.wikipedia.org/wiki/Green's_theorem), then a whole heap of linear algebra like what is a basis, the image and kernel of linear maps (https://en.wikipedia.org/wiki/Kernel_(linear_algebra)) etc.

Real and complex analysis: This is a HARD course. You learn analysis PROPERLY. How do we define the real numbers? What are all the axioms we use? (completeness axiom etc) You start off with limits of sequences and series, then move on to a little bit of topology (like what is a ball https://en.wikipedia.org/wiki/Ball_(mathematics) and how to we define open/closed sets etc). Then you study analytic functions and their properties (https://en.wikipedia.org/wiki/Analytic_function), and finish off the course with some complex analysis where you learn about Cauchy's integral theorem (https://en.wikipedia.org/wiki/Cauchy's_integral_theorem). You learn about poles and stuff here.

Discrete maths and graph theory: One of my personal faves. The first half of the course is stuff you learn in 3U/4U/1st year mostly. However you do learn about stirling numbers (https://en.wikipedia.org/wiki/Stirling_number).
The second half is graph theory which is brand new unless you've done Olympiad stuff (https://en.wikipedia.org/wiki/Graph_theory). You learn about Eulerian and Hamiltonian paths (https://en.wikipedia.org/wiki/Eulerian_path https://en.wikipedia.org/wiki/Hamiltonian_path), chromatic polynomials (https://en.wikipedia.org/wiki/Chromatic_polynomial), and chromatic numbers of graphs (https://en.wikipedia.org/wiki/Graph_coloring).

In second sem you have the choice of:

Financial Maths: Easy subject, you learn about linear programming and the simplex algorithm (https://en.wikipedia.org/wiki/Simplex_algorithm). You then go on to learn about Lagrange multipliers (https://en.wikipedia.org/wiki/Lagrange_multiplier). You finish the course off by learning about utility theory and portfolio theory, then CAPM (https://en.wikipedia.org/wiki/Modern_portfolio_theory https://en.wikipedia.org/wiki/Capital_asset_pricing_model).

Number theory and cryptography: You learn about cryptosystems (main one is RSA). You learn heaps of number theory starting with the Euclidean algorithm and Fermat's Little Theorem (https://en.wikipedia.org/wiki/Fermat's_little_theorem). You do a whole lot of cool things such as the Mobius inversion formula (https://en.wikipedia.org/wiki/Möbius_inversion_formula).

Partial Differential Equations: Basically starts off learning about ODE's then moves onto PDE's like the wave, heat, and Laplace equations (https://en.wikipedia.org/wiki/Partial_differential_equation). You learn fourier series (https://en.wikipedia.org/wiki/Fourier_series) and Laplace/Fourier Transforms (https://en.wikipedia.org/wiki/Laplace_transform).

Algebra: This is another HARD course. This course starts off with group theory. You learn what a group is, about the order of a group, the orbit and stabiliser, what is a coset, etc (https://en.wikipedia.org/wiki/Group_theory). You learn Lagrange's Theorem (https://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory)) and the three isomorphism theorems (https://en.wikipedia.org/wiki/Isomorphism_theorem). You learn about homomorphism and automorphisms too (https://en.wikipedia.org/wiki/Homomorphism https://en.wikipedia.org/wiki/Automorphism). Then you learn about the centre and centraliser (https://en.wikipedia.org/wiki/Centralizer_and_normalizer), and finish off group theory with Sylow's Theorems (https://en.wikipedia.org/wiki/Sylow_theorems). Then this course goes on to study linear algebra again, learning about minimal polynomials (https://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra)) and Jordan Normal Forms/Blocks (https://en.wikipedia.org/wiki/Jordan_normal_form).

I have only done two 3rd year subjects so far so I won't comment on them too much. But I studied some formal logic such as propositional calculus and predicate logic (https://en.wikipedia.org/wiki/Propositional_calculus). I also did some more topology but from a combinatorial perspective, such as knot theory (https://en.wikipedia.org/wiki/Knot_theory) and looking at surfaces and stuff.

Next year I'll be doing the rest of my third year subjects, which will include metric spaces (https://en.wikipedia.org/wiki/Metric_space), galois theory (https://en.wikipedia.org/wiki/Galois_theory), measure theory (https://en.wikipedia.org/wiki/Measure_(mathematics)), and differential geometry (https://en.wikipedia.org/wiki/Differential_geometry).

If you go on to do Honours, you do stuff like functional analysis (https://en.wikipedia.org/wiki/Functional_analysis), Riemannian geometry (https://en.wikipedia.org/wiki/Riemannian_geometry), and commutative algebra (https://en.wikipedia.org/wiki/Commutative_algebra).

In regards to your question about what is in algebra, I recommend you look up group theory. Very interesting and shows how deep algebra really goes, yet is probably understandable by a very smart 4U student.

You won't see much geometry that you are used to in uni, you will see it in a different way mostly.

Where can you go with pure maths? Almost anywhere really. You could do research in academia, work for a quantitative trading company (or just finance in general really), work in meteorology and oceanology, become a teacher (a masters in education required though), heaps of options.