By way of background, I have worked in a business finance related field for many years and found aspects of it becoming tedious and unpleasant so chose to make a shift into GIS/spatial data analysis, as I enjoy working with numbers. However, until the last couple years, I had not cracked a math text book since college (many years ago)and basically never went beyond first year calculus,some linear algebra, and some fairly basic stats.
I am hunkering down on calculus again and more advanced linear algebra now with the goal of getting into optimization and more advanced stats. But I feel like I need a better "math overview" that explains more broadly fundamental but advanced concepts of math and how the subjects I am learning relate and fit into a "bigger picture".
For example, I have read that algebraic topology and the geometry of manifolds are highly relevant to mathematical decision-making technologies. But I have no idea how, and I have no idea how such advanced topics relate to the more traditional equation-solving type of math I have always studied.
I am hoping you might be able to point me to a book or books that present, in a very readable manner to a non-expert, a survey or overview of an array of math topic areas and fundamental concepts (e.g., duality, operators, etc.), how they relate, and how they are applied individually or in combination to solve various families of problems.
Can you recommend anything?
I'm afraid I can't answer this question directly because there is no good direct answer here. Without a significant amount of additional information, it would be very hard to act as academic-counselor-via-internet for you.
I will say this -- you have barely seen any introductory math, but you want to talk about advanced concepts that are not
digestible to a layperson, at least not at the level you are hoping to find. There is no "graduate level advanced algebraic topology and manifold theory for the layperson" because if you've never done multivariable calculus (and really
mastered it) you have very little chance to just skim through such a book and "get it."
That being said, you can start with one of these two ideas:
1. Check out a basic book that introduces advanced mathematics. Based on what you seem to be interested in, this means a "basic proofs" book that covers topics like sets, proofs, etc. as well as some other stuff (often number theory or the analytical theory of calculus). Work your way towards a basic book in abstract algebra. There are more books that I'd care to mention of either the introductory type
or for basic abstract algebra
2. Check out a "survey" book like the Princeton Companion
or Mathematics 1001
. These books will give you insights into these fields written (roughly) at a generic mathematically-educated person. While much of the texts may be "over your head," they are very comprehensive and broad and might give you some idea of what these concepts are. Tim Gowers also has a tiny little book that's about $10 that you could pick up, although it is very quick and shallow (it's tiny, after all -- but it is well written).
I was unsure whether I should consider this an "answer" or a "rejection," but since it may be useful for others looking for texts to read, I will call this an "answer" and hopefully it will be of use to you.