Symplectic geometry is the geometry of a closed skew-symmetric form. It turns out to be very different from the with which we are familiar. One impor… - Dusa McDuff

Over the past 15 years symplectic geometry has developed its own identity, and can now stand alongside traditional Riemannian geometry as a rich and meaningful part of mathematics. The basic definitions are very natural from a mathematical point of view: one studies the geometry of a skew-symmetric bilinear form ω rather than a symmetric one. However, this seemingly innocent change of symmetry has radical effects. For example, one dimensional measurements vanish since ω(v, v) = −ω(v, v) by skew-symmetry. ...
The theory has two faces. There are two kinds of geometric subobjects in a symplectic manifolds, hypersurfaces and Lagrangian submanifolds that appear in dynamical constructions, and even-dimensional symplectic submanifolds that are closely related to Riemannian and complex geometry. As we shall see, the analog of a geodesic in a symplectic manifold is a two-dimensional surface called a .

... Gelfand amazed me by talking of mathematics as if it were poetry. He tried to explain to me what von Neumann had been trying to do and what the ideas were behind his work. That was a revelation for me — that one could talk about mathematics that way. It is not just some abstract and beautiful construction but is driven by the attempt to understand certain basic phenomena that one tries to capture in some idea or theory. If you can’t quite express it one way, you try another. If that doesn’t quite work, you try to get further by some completely different approach. There is a whole undercurrent of ideas and questions.

On two different occasions recently, (male) mathematicians asked me in all innocence: But you surely never suffered any discrimination?