I wrote a lengthy reply to this, but unfortunately the server lost it for me... grr!
It is interesting to hear that diff. geometry is almost unrelated to high school geometry, and is closer to calculus of manifolds and analysis.
I am very ignorant of analysis, and so I would love to hear what it is all about (all I know is that the subject has a lot of relevance to theoretical physics). As for this fundamental problem of calculus on manifolds, that sounds interesting. I was certainly not aware that this is where tensors are from.
Now, regarding tensors, are they as scary as many people say? Or is this a percieved image with no realism to it?
Mathwonk states in his paper on algebraic geometry (that I briefly skimmed through) that this subject is related to the study of geometry of polynomials, or as he now likes to think, the study of geometry of rings (or was it groups?). This sounds interesting, but as with a large chunk of higher mathematics, I am again very ignorant of the subject. I am aware that groups and rings are important concepts in abstract algebra, but that is all.
This shows my ignorance of mathematics!
I will have a closer look at solid state physics, as it is an area of theoretical physics that I have not looked at. I think, at my age, it would be better for me to see the "big picture", perhaps, something that I haven't been doing.
I am guessing, because higher-level geometry and analysis are "pure" branches of mathematics, they are on a completely different level, than say, partial diff. eqn's?
Is it not Fast Fourier Transforms (or FFTs) that govern the rendering of computer graphics from the graphics pipeline? Is it correct that Fourier transforms are essentially methods to solve differential equations?
Now, as for my mathematics courses, I wrote a huge list of the topics covered in the post that I lost, so if you are after further details (as I wrote in my earlier post) then just reply to me. I do two courses, my top-course year ten mathematics course, and my "mathematics methods 4b" course, which I will refer to as "extended maths", is a course on pure mathematics, one step below pre-tertiary level (the 4 indicates it is one step below, as 5 is for pre-tertiary, and the "b" refers to the length [100 hours], whereas a standard pre-tertiary subject here is denoted as 5C [150 hours long]).
In my regular mathematics course, these are the subjects (that I have studied, or in the process of studying, so far):
Trigonometry covered non-right-angled triangles and problems dealing with bearings. Financial Mathematics is simply simple and compound interest.
Geometry covered three-dimensional shapes, with concepts such as volume and total surface area (it also included questions such as: "Given these dimensions of a wine glass [cylinderical, with a hemispherical bottom], find to what height, you must pour the wine, to fill the wineglass to exactly half of it's volume"). It was these types of questions that I liked solving, and as a sidenote, geometry is my favourite branch of mathematics so far.
As for my extended maths class, the following is a list of subjects that I have covered, or yet to cover:
Division of Polynomials
Exponential and Logarithmic Functions
Theoretical, and Experimental Probability
I am currently working with logarithmic functions, and so calculus is up next! I have found I am doing much better currently, with logarithms, than I did with cubics (probably my weakest mathematics topic).
I really hope to master the mathematics I am taught, as often I find, say the algebra topics (like cubics and simultaneous equations) to be less intuitive to my thinking than geometry, for example.
So, at the moment, I am dealing with pre-calculus topics, but it won't be for too much longer (I'm going to do pre-tertiary Mathematics Methods next year). This is essentially a calculus course (this is all that's required for all of the maths/physics units at the University I will be attending, excepting for the advanced calculus and applications units offered in first-year of an undergrad programme).
If I do well enough with this, I will do the pre-tertiary Mathematics Specialised course in Year 12 (the hardest high school mathematics course available). This course covers calculus, series, sequences, convergence and divergence of functions and linear algebra, and so this course would give me a huge advantage for my degree.
Take note that these courses are generally taught by people who have a really solid mathematics grounding (ie a degree in it, most commonly), as opposed to what I've heard about the American AP Calculus courses (I have heard rumours that they are often taught by incompetent teachers. Is this true?).
You mention that it generally takes more effort to visualise concepts in algebra than it does in calculus and geometry. I have (clearly) no experience with calculus yet, and so I can't talk about that, but I feel that, as I stated earlier in the post, I find geometry concepts more intuitive to my style of thinking than algebra concepts. I always like to visualise concepts, and this is far more difficult with algebra.
What do people on this forum find about their intuition when dealing with higher mathematics? Do some of you grasp algebraic concepts better than calculus concepts? Or geometry concepts? Do you have no real, *deep* intuition regarding the topic you spend the most time with? To succeed in an area of higher mathematics, how good must your level of intuition be? I would like to hear about this.
I am looking forward to the journey through this material, and particularly looking forward to becoming an undergraduate (in a little more than 2 years)!
Also, a few questions regarding analysis, before I finish writing this post. How intuitive is this field? Does it throw you in the deep end? And how elegant is the field? At my university, as an undergraduate I can study these two courses: "Analysis 3" and "Topics in Advanced Mathematics".
The "Topics in Advanced Mathematics" course covers various topics such as geometry, set theory, number theory, history of mathematics and a few others. If I'm not mistaken, you are able to choose the area of the list of topics provided that you wish to study. The course itself only requires that you have studied a second-year mathematics unit. Perhaps if I can choose, I wold be wise to choose to study algebraic and differential geometry?
As background, I will have had grounding in second-year abstract algebra, second year calculus, second year differential equations, and second-year linear algebra. Differential equations and linear algebra are covered in the same course, so I would be guessing that it is structured as half/half. The university/college I plan to attend is the University of Tasmania (aka UTAS).
I am particularly looking forward to hearing about a mathematician's intuition, and would love to hear my other questions answered! Thanks for reading my post, and for your time!