2. GEOMETRIC TOPOLOGY MAIN GOAL:
TO PROVE TOPOLOGICAL RESULTS ABOUT SMOOTH MANIFOLDS BY ENDOWING THEM WITH ADDITIONAL GEOMETRIC STRUCTURES

3. GEOMETRIC TOPOLOGY OF LOW DIMENSIONAL MANIFOLDS
SYMPLECTIC FOUR DIMENSIONAL MANIFOLDS
CONTACT THREE DIMENSIONAL MANIFOLDS

4. Property P

10. CONTACT THREE DIMENSIONAL MANIFOLDS

11. Frobenius Theorem

13. Contact forms

14. Contact structure

22. Legendrian curve
A curve in a contact 3-manifold is called Legendrian if it is everywhere tangent to the contact planes.

23. Overtwisted Disk

24. Tight versus overtwisted

26. Tight versus overtwisted

27. Darboux’s Theorem

28. Contact Topology

29. Global structure

30. Global structure

32. Classification of overtwisted contact structures
Martinet+Lutz+Eliashberg
Overtwisted contact structures are classified:
There is, up to isotopy, a unique overtwisted contact structure in every homotopy class of oriented plane fields.

33. Classification of tight contact structures?

34. Convex surfaces

38. 2002 Giroux’s ICM talk in Beijing

40. Open books

41. Complement of the Hopf link in the 3-sphere fibers over the circle

42. Abstract open books

43. Mapping torus M

46. Stabilization of an open book

47. Stabilization of an open book

48. Open books and contact structures

49. Etnyre’s Lemma

53. Our recent contribution