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Thesis by D. Joksimovic MSc

PhD Defense: Locally uniform existence of leafwise fixed points for C0-small Hamiltonian flows...

The content of this thesis belongs to a field of mathematics called symplectic geometry. Symplectic geometry studies spaces equipped with a symplectic form. Such a form measures the area of each two-dimensional subspace. Symplectic geometry originated as a mathematical way to formalize classical mechanics, where phase space carries a canonical symplectic form. The time-evolution of a mechanical system is governed by Hamilton's equation, and as such, it preserves symplectic form.

This thesis studies two types of objects in symplectic geometry. The first main result is an existence result for leafwise fixed points. Leafwise fixed points are a natural generalization of fixed points of Hamiltonian motions, i.e. the points which return to their initial state after some prescribed period of time. On the other hand a leafwise fixed point is a point whose trajectory changes only up to a shift in time after a time-dependent perturbation of the Hamiltonian system (i.e. such as an earthquake).

The other two results of this thesis study generating systems of symplectic capacities. A symplectic capacity is a map which assigns a positive number to each symplectic space and which satisfies the assumptions of monotonicity (which means that to a bigger space we assign a bigger number) and conformality (which means that if we enlarge the space k times, its capacity gets larger |k| times). An important question in symplectic geometry is whether symplectic capacities uniquely determine a given symplectic space. This would mean that in some sense we would be able to understand symplectic spaces by just looking at some set of numbers assigned to them, namely their capacities. Since this set can be huge, it is, therefore, natural to look for minimal sets of symplectic capacities which in some sense contain the information about all other symplectic capacities. These we call generating systems of symplectic capacities.

Morally, the other two main results say that generating systems are as hard to deal with as the set of all capacities. More precisely, the main conclusion is that in general, every generating system of symplectic capacities has (almost) the same cardinality as the set of all symplectic capacities. In other words, every such system contains the same number of capacities as the set of all capacities.

Start date and time
End date and time
Location
University Hall, Domplein 29
PhD candidate
D. Joksimovic MSc
Dissertation
Locally uniform existence of leafwise fixed points for C0-small Hamiltonian flows & generating systems of symplectic capacities
PhD supervisor(s)
prof. dr. M.N. Crainic
Co-supervisor(s)
dr. F.J. Ziltener
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