Areas of contemporary Mathematical Physics
Here is the 2010 Mathematics Subject Classification list. It has 6500 entries, working out the mathematical physics projection operator is left as an exercise to the reader.
There was a time when the Mathematics and Physics departments weren't separate to begin with - one just studied "Natural Philosophy". During the 20th century, these disciplines separated and become quite specialized, obscuring the relations between them.
The links between mathematics and physics are very broad as was championed by Atiyah, Witten, Verlinde, Dijkgraaf and many others in the 1980's. The later cohorts in the 90's and 00's are too many to list, but as a sampler: Ashoke Sen, Rajesh Gopakumar, Michael Douglas, Andrew Neitzke, Masahito Yamazaki, Tudor Dimofte and various others.
These days there are institutes devoted to establishing the relationships between the two fields.
- Simons Center for Geometry of Physics
- Kavli Institute for Theoretical Physics
- Institue for Physics and Mathematics of the Universe
- Institute for Advanced Study
- Harish-Chandra Research Institute
My main criticism of mathematical physics is that study tends to concentrate in connecting a few very specific areas of math and physics. However, the consequences are still very far-reaching.
Instead of writing a complete list of "mathematical physics topics" I recommend reading through these and similar pages too see who is doing what these days.
"Data Science" as gimmicky as it sounds, but it may be looked at a restructuring of applied mathematics to address commonalities in many disciplines considered "out of reach" by traditional mathematics.
Well, where I study in Sao Paulo in the theoretical institute of physics the researcher Andrei Mikhailov is working in mathematical physics with relation to the pure spinors in string theory (My interest in mathematical physics is more related with statistical mechanics, so I don't know the details of this area). The open problems you can consult in the papers http://arxiv.org/find/hep-th/1/au:+Mikhailov_A/0/1/0/all/0/1. The line is much more geometric. Other area and problems that I me remember is associated with higher spins and twistor theory, works important in this direction are of Alexey Rosly http://inspirehep.net/search?p=find+a+rosly. Too geometric.