Using the coherent states attached to the complex Jacobi group—the semi-direct product of the Heisenberg-Weyl group with the real symplectic group—we study some of the properties of coherent states based on the manifold which is the product of the $n$-dimensional complex plane with the Siegel upper half plane.

Hofer's metric is a very interesting way of measuring distances between compactly supported Hamiltonian symplectic maps. Unfortunately, it is not known yet how to compute it in general, for example for symplectic maps far away from each other. It is known that Hofer's metric is locally flat, and it can be computed by the so-called oscillation norm of the difference between the Poincare generating functions of symplectic maps close to identity. It is shown here that the same result holds for arbitrary extended generating function types and symplectic maps, as long as the respective generating functions are well defined for the given symplectic maps. This result plays a crucial role is formulating and solving the optimal symplectic approximation problem in Hamiltonian nonlinear dynamics. Applications to beam physics are oulined.

The moduli space of flat $\mathrm{SU}(n)$ connections on Riemann surfaces is of fundamental importance in TQFT. There is an associated representation of the mapping class group on the space of covariantly constant sections of the Verlinde bundle with respect to the AdPW-H connection. J. Andersen showed that this representation is asymptotically faithful. In his proof the Berezin-Toeplitz quantization of compact Kähler manifolds is used. In this contribution the background and some ideas of Andersen's proof is sketched.

We study solutions of Grassmannian sigma-model both in finite-dimensional and infinite-dimensional settings. Mathematically, such solutions are described by harmonic maps from the Riemann sphere $\mathbb C\mathbb P^1$ or, more generally, compact Riemann surfaces to Grassmannians. We describe first how to construct harmonic maps from compact Riemann surfaces to the Grassmann manifold\linebreak $\operatorname{G}_r(\mathbb C^d)$, using the twistor approach. Then we switch to the infinite-dimensional setting and consider harmonic maps from compact Riemann surfaces to the Hilbert-Schmidt Grassmannian $\operatorname{Gr}_{\text{HS}}(H)$ of a complex Hilbert space $H$. Solutions of this infinite-dimensional sigma-model are, conjecturally, related to Yang-Mills fields on $\mathbb R^4$.

In many cases tensor universality implies that multi-partite quantum-state processing is determined by what happens in totally disentangled cases. In independent systems relative time direction for the parts is arbitrary. This hints that time may be linked to entanglement and measurements and that there may be a measurement-only version of quantum mechanics. One-way quantum computation suggests that this may be possible.

We study the symmetry of Maxwell's equations for external moving media together with the additional Minkowski constitutive equations (or Maxwell–Minkowski equations). We established that the system is conformally invariant.

We present a geometric explanation of Kalman filters in terms of a symplectic linear space and a special quadratic form on it. It is an extension of the work of Bougerol with application of a different metric we have introduced earlier. The new results are contained in Theorem 1 and Theorem 4.