Comment by Ar-Curunir
5 days ago
Did your 7 semesters of calculus include any courses that a math major would take?
For example, a single upper-division undergrad level course on probability will expose you to measure theory.
Similarly, harmonic analysis is something that you can take as a reasonably good senior in mathematics.
I think most undergrad level course on probability expose you to measure theory _without_ telling you they are measure theory.
I took multivariate calculus (I to III), differential equations, statistics, discrete math, and linear algebra.
I didn't take those courses you're talking about, but that's my point. I have a top <1% education in math in the nation, maybe top 2-3% of all college grads, and that's not enough to even scratch the surface of the summary.
if you took an engineering course that covered fourier transform, z transform, or laplace transform, then you have seen an introduction to harmonic analysis on the circle and the real number line.
typically when I see mentions of harmonic analysis together with differential geometry I start thinking of fourier transform on more fancy shapes (on the sphere, for example, the replacements for complex exponentials are called spherical harmonics)
my PhD was EE in signal processing and I took some extra math classes at the graduate level, and continue to read some of this for fun and profit.
I still cant understand most of the terminology in the fields medal citations. that probably requires a more thorough couple of years of graduate education in mathematics.
math is by far the furthest of the sciences in terms of depth of human discovery.
Every engineering student in the US takes those courses. I did too. That doesn’t make us top 2-3% of college grads.
The stuff you mentioned is first year business for most math undergraduates. You should not expect to know most things in senior level math after taking just first-year math.
It’s like expecting to know computational complexity theory, or distributed systems theory, after taking first year CS.
Harmonic analysis is "just" working with harmonic functions which you have enough background for: f(x,y) is harmonic if f_xx+f_yy=0 (f_xx=2nd partial derivative w/r/t/ x, and so on). Higher dimensions work the same: f_{x1x1} + f_{x2x2} + f_{x3x3}+.... = 0.
Well, the snobby pure maths folks look down on applied engineering math ;)