Gauge theory with infinite group actions

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Tropical geometry connects discrete dynamical systems with automata. We used it to induce an analytic comparison theorem between mutually different rational dynamics or PDEs. We applied the method to theory of automata groups, particularly to Burnside groups and obtained a new example of infinitely quasi-recursive rational dynamical systems.We constructed moduli theory of holomorphic curves over infinite dimensional almost Kaehler manifolds with high symmetry. As an application, we constructed Hamiltonian deformation of groups acting on trees, and verified that for some class of Hamiltonians, they must be always uniformly infinite in ours sense.