I'm aware that PDEs, linear algebra, numerical analysis and such are staples of most applied mathematical endeavors. I want to know if there are other areas of "higher" mathematics that have useful applications in climate science (like topology, measure theory, Riemannian geometry, etc) or whether the things I mentioned first are the only really important ones. Thanks.
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The question may generate primarily opinion based answers but while it is open one would like to present the following information.
At the outset most of the people I know in the field work in NWP and the reason people avoid higher mathematics(by that I mean looking for analytical solutions or pure math based projects such as you mention topology or Riemannian geometry) is because of the issue of jobs after completion of their respective Ph.ds and post docs. More of the recent advances use approaches from statistics and Big Data and that is because of the massive amounts of data that is available on a daily basis.
Presuming one wants to move on a "path" in academia then there is this application of topology in meteorology - Hairy ball theorem and this book should provide more information on the topic An Introduction to the Geometry and Topology of Fluid Flows but applications of the material in the book to actual problems in meteorology may be few in so far as I am aware and others can perhaps provide better information. This workshop Complex Analysis And Geometry does provide some applications of differential geometry to meteorology but these I reckon are most of the outlier groups and most of the work is on improving skill of NWP models.
A colleague of mine in academia works on Potential Vorticity and Slow manifolds which is a part of Stability theory and fast manifolds/inertial manifolds. Essentially one is looking for a point after which the flow becomes unstable and why is that ? That is because unstable flows can be ultimately linked to weather forecasting and NWP models. There is the famous paper by Edward Lorenz that reviews that topic - The slow manifold what is it ?
While it is numerical work I and many others use Vector Spherical Harmonics because we all deal with global circulation models(spectral method) and others use wavelets to analyze equatorial waves
Finally there is this paper by Tapio Schneider that mentions Gauge transformations