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.