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Symplectic geometry, the field of research of this thesis, originated from a branch of physics called classical mechanics. In classical mechanics, we study the motion of objects, like the motion of the planets around the sun. The objects that we study in symplectic geometry are called symplectic manifolds and can be used to describe the equations of motion of a classical mechanical system.

The first part of this thesis studies so-called symplectic capacities. A symplectic capacity is a function that takes a symplectic manifold as input and gives a positive number as output. Intuitively, a symplectic capacity measures the size of symplectic manifolds.

One of the main results of this thesis shows that one can use symplectic capacities to distinguish between symplectic manifolds. In other words, given two different symplectic manifolds, then there exists a symplectic capacity which does not give the same number for these two symplectic manifolds. This answers a question of Helmut Hofer (IAS Princeton) et al.

In the second part of this thesis, we look at which symplectic manifolds can fit inside other symplectic manifolds. Another main result of this thesis shows that some subsets of symplectic manifolds can fit inside boxes that are arbitrarily small. Such subsets can be thought of as having symplectic size zero. In the proof, we construct a way of folding these subsets in half repeatedly until they fit in the desired box.