Talks

I will first consider the geometry of the complex hyperbolic plane and immersed surfaces therein, in particular the cases of Lagrangian and complex surfaces. The complex surfaces are all minimal, but there are many others as well. As it is a symmetric space, the more general case of harmonic maps from a Riemann surface into the complex hyperbolic plane naturally generates holomorphic data of a Higgs bundle. We impose a compactness condition to relate our study of minimal surfaces to Higgs bundles. Let S be a closed Riemann surface of genus at least 2. Consider then harmonic immersions of the universal cover of S into the complex hyperbolic plane which are equivariant with respect to some representation of the fundamental group of S into the group P U(2, 1) of holomorphic isometries of the complex hyperbolic plane. In this case, the nonlinear Hodge correspondence applies and thus there is a (poly-)stable Higgs bundle over S. In the standard case of the group GL(n, C), this consists of a...

The eigenstate thermalization hypothesis (ETH) provides a sufficient condition for thermalization and equilibration from any initial states in isolated systems with a large number of degrees of freedom. One part of this statement is that for large systems, the diagonal matrix elements of typical observables in the Hamiltonian eigenstate basis, that is, the expectation values of the energy eigenstates, depend smoothly on the corresponding energy values. This statement is highly related to the fact that the variance of the energy eigenstate expectation values is small. Indeed, it is conjectured that this variance shows power-law decay with respect to the Hilbert space dimension.

On the other hand, in general relativity, it is expected that heavy objects undergo gravitational collapse. Additionally, it is known that gravitational systems with black holes exhibit maximally chaotic properties. By using the holographic principle, or specifically the AdS/CFT correspondence, these facts imp...