Open Problems list
Solution 1:
Open Problem Garden http://garden.irmacs.sfu.ca/ is a good place to start.
You can also check out https://mathoverflow.net/questions/48299/more-open-problems for a comprehensive list.
Solution 2:
I think in general it is too hard to reach public open problems, but it is good to get a flavor.
To comply the spirit of MathSE making answers self-contained, I pasted the content from the wiki article here: http://en.wikipedia.org/wiki/Shing-Tung_Yau#Open_problems
Yau has compiled an influential set of open problems in geometry.
- Harmonic functions with controlled growth
One of Yau’s problems is about bounded harmonic functions, and harmonic functions on noncompact manifolds of polynomial growth. After proving non-existence of bounded harmonic functions on manifolds with positive curvatures, he proposed the Dirichlet problem at infinity for bounded harmonic functions on negatively curved manifolds, and then proceeded to harmonic functions of polynomial growth. Dennis Sullivan tells a story about Yau's geometric intuition, and how it led him to reject an analytical proof of Sullivan's. Michael Anderson independently found the same result about bounded harmonic function on simply connected negatively curved manifolds using a geometric convexity construction.
- Rank rigidity of nonpositively curved manifolds
Again motivated by Mostow's strong rigidity theorem, Yau called for a notion of rank for general manifolds extending the one for locally symmetric spaces, and asked for rigidity properties for higher rank metrics. Advances in this direction have been made by Ballmann, Brin and Eberlein in their work on non-positive curved manifolds, Gromov's and Eberlein's metric rigidity theorems for higher rank locally symmetric spaces and the classification of closed higher rank manifolds of non-positive curvature by Ballmann and Burns-Spatzier. This leaves rank 1 manifolds of non-positive curvature as the focus of research. They behave more like manifolds of negative curvature, but remain poorly understood in many regards.
- Kähler–Einstein metrics and stability of manifolds
It is known that if a complex manifold has a Kähler–Einstein metric, then its tangent bundle is stable. Yau realized early in 1980s that the existence of special metrics on Kähler manifolds is equivalent to the stability of the manifolds. Various people including Simon Donaldson have made progress to understand such a relation.
- Mirror symmetry
He has collaborated with string theorists including Strominger, Vafa and Witten, and as post-doctorals from theoretical physics with B. Greene, E. Zaslow and A. Klemm . The Strominger–Yau–Zaslow program is to construct explicitly mirror manifolds. David Gieseker wrote of the seminal role of the Calabi conjecture in relating string theory with algebraic geometry, in particular for the developments of the SYZ program, mirror conjecture and Yau–Zaslow conjecture.
Solution 3:
How about the list of problems each of which has a $1 million prize for their resolution? I believe it is called "The Millennium List". [see the preface to Stewart& Tall's "Algebraic Number Theory and Fermat's Last Theorem" which was one of the problems on the list, I believe--not sure]