Areas of contemporary Mathematical Physics I have often heard that some developments in Physics such as Gauge Theory, String Theory, Twistor Theory, Loop Quantum Gravity etc. have had a significant impact on pure mathematics especially geometry and conversely. 
I am interested in knowing a list of areas of Mathematical Physics which have important and interesting open research problems. Please mention such areas and some references where one can get started in each of them. 
Just to give an idea of what I have in mind following areas come to my mind  for example when I say Mathematical Physics : Knot Theory, Mirror Symmetry, Atiyah-Singer Index Theorem & Dirac Operators, Topological Quantum Field Theory etc. I believe that such listing could be useful to other members of the m.se community as well. 
Is there an article/website/blog where I can find such listing ? 
I had earlier asked a question about the existence of a website similar to string wiki, but unfortunately it does not seem to exist. Unfortunately this does not have a very systematic classification of sub areas of Mathematical Physics though it does provide some references. Another very useful website exists for Physics but I am unaware of a similar one for Maths. 
Please note that my question deals with interactions between Pure Mathematics and Fundamental Theoretical Physics. There are interesting and valuable aspects like applications of mathematics in statistical mechanics or fluid mechanics but for the purposes of this question, let us exclude them.
Edit : If it is not possible to give a complete listing, please mention some main areas along with canonical references. To give a better idea of what kind of things I am looking for here are two examples Advanced CFT and Differetntial Topology and QFT though suggestions do not have to be in these directions.
 A: 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.
A: 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.
A: 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.
