Tag - Mathematical physics
In 1991 a new type of transfer to the Moon was operationally demonstrated by the Japanese spacecraft, Hiten, using ballistic capture. It was designed by this speaker and James Miller. This is capture about the Moon which is automatic so that no rocket engines are required. It was accomplished due to the existence of regions in phase space called weak stability boundaries, where ballistic capture occurs. These are complex fractal regions of unstable chaotic motion. Until recently it was thought that such a transfer to Mars was not feasible. Recent work by Francesco Topputo and this speaker has shown that a new type of ballistic capture transfer exists to Mars, with interesting implications. It will be described how to design these transfers given realistic constraints and why is so challenging.
Feynman categories are a new universal categorical framework for generalizing operads, modular operads and twisted modular operads. The latter two appear prominently in Gromov-Witten theory and in string field theory respectively. Feynman categories can also handle new structures which come from different versions of moduli spaces with different markings or decorations, e.g., open/closed versions or those appearing in homological mirror symmetry. For any such Feynman category there is an associated Feynman category of universal operations. These give rise to Gerstenhaber's famous bracket, the pre-Lie structure of string topology, as well as to the Lie bracket underlying the three geometries of Kontsevich built from symplectic vector spaces. As time permits, we will also briefly discuss bar, co-bar and Feynman transforms and how these give rise to master equations, such as the Maurer-Cartan equation or the BV master equation.
These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that allows to prove the existence of phase transitions. Finally, we will discuss certain probabilistic representations and their consequences.
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