The classification of 1-dim. and 2-dim. manifolds is easy. Perelman may have recently used Differential Geometry to prove conjectures of Poincare and Thurston.
This would be a fantastic advance in the study of 3-manifolds. The theory of smooth 4-dim. manifolds uses symplectic and algebraic geometry.
In dimensions larger than 4 surgery theory is used to reduce classification questions to questions in algebraic topology and quadratic forms.
To volunteer to give a talk, or for any other questions regarding this schedule, contact Sara Miller