What are current trends in Mathematical Analysis?

I was curious what active researchers believe are current trends in mathematical analysis. A lot of mathoverflow seems focused on algebraic geometry and topology etc., so it’s hard to get a good idea of what’s big right now. I checked MSRI and the AMS Notices, and recently I have found there has been a lot of discussion in:

  • Uniqueness and existence of weak solutions to nonlinear PDEs: especially in fluid dynamics, and trying to build an understanding of the phenomenological models of turbulence. Examples include work by Isett, De Lellis, Titi, Székelyhidi, Buckmaster, Vicol, etc.

  • Geometric measure theory: Ambrosio has been publishing a lot on this, Toro is also very big and publishes quite a lot

  • Microlocal analysis: there was recent seminars at msri on this area of analysis

  • Optimal transport and calculus of variation: Figalli recently won the Fields medal for his work in this area and Ambrosio will be leading (I believe) a hot topics seminar on the intersection of optimal transport and machine learning at MSRI this year, this seems to be a fast growing area, there is also the intersection of optimal transport and Riemannian geometry which seems quite new, Villani and McCann are also big names I know of who work in this area

  • Geometric analysis: of course after Perelman’s proof this area has been steadily active i.e. studying the Ricci flow and other PDEs governing the solution of Riemannian metrics, Uhlenbeck recently won the Abel prize and her work is within this area

Of course there are other areas like functional analysis, operator algebras etc., but I was curious what other people think are the current trends in analysis.

Edit: moved Uhlenbeck to geometric analysis Edit: Corrected Szekelyhidi’s name

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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs https://nevinmanimala.com

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