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.