Vorträge in der Woche 17.04.2023 bis 23.04.2023

Mittwoch, 19.04.2023: A realisation of some Schubert varieties as quiver Grassmannians

Giulia Iezzi (RWTH Aachen)

Quiver Grassmannians are projective varieties parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. For instance, this method was used to study linear degenerations of flag varieties, obtaining characterizations of flatness, irreducibility and normality via rank tuples. We give a construction for smooth quiver Grassmannians of a specific wild quiver, realising a class of Schubert varieties inside flag varieties. This allows for a definition of linear degenerations of Schubert varieties.

Donnerstag, 20.04.2023: Corresponding CMC initial data sets and their KIDs

Anna Sancassani

Donnerstag, 20.04.2023: Dynamics of dissipative systems over a harmonic oscillator Hilbert space

Paul Gondolf (Tübingen)

The dynamics of open quantum systems over a Hilbert space of the Harmonic oscillator has been a useful tool for modelling for example Laser and Maser systems in the past. With the increasing interest in quantum computation, similar dissipative systems have become important models for the dynamics of continuous variables (CV) and in particular the Schrödinger CAT codes. However, due to the infinite-dimensional nature of the underlying Hilbert space, these systems are difficult to handle and there is only limited knowledge about their mathematical properties, including the well-posedness of the dynamics. In our work, we propose a mathematical framework to address these issues. We specifically provide necessary and sufficient conditions that are easy to verify for an operator comprising of annihilation and creation operators to qualify as a generator. These conditions not only establish the well-posedness, existence, and uniqueness of the dynamics in the form of a trace and positivity preserving C0-semigroup but also guarantee a property that we call Sobolev preservation. This property enables us to conduct perturbation theory and appears to be a promising tool also in various other contexts.

Donnerstag, 20.04.2023: Charge Operators for the Dirac Quantum Field

Pablo Costa Rico (TU München)

We will discuss the existence of localised charge operators in the Dirac quantum field, that is, whether for each measurable Borel set it is possible to associate a densely defined self-adjoint operator in the fermionic Fock space, which gives us information about the charge in that system. We will establish the natural definition of this operator and study its properties for finite and infinite dimensional systems. While in finite dimension, this operator satisfies the expected properties, in infinite dimension it presents problems that make this definition discouraging and probably ill-defined, since vectors such as the vacuum do not lie in its domain for cube-shaped regions.