This is a readable review of recent work on non-Hermitian bound state problems with complex potentials. A particular example is the generalization of the harmonic oscillator with the potentials: $$V\left(x\right)=\frac{{\omega }^2}{2}{\left(x-\frac{2i\beta }{\omega }\right)}^2-\frac{\omega }{2}.$$ Other examples include complex generalizations of the Morse potential, the spiked radial harmonic potential, the Kratzer-Coulomb potential, the Rosen Morse oscillator and others. Instead of demanding Hermiticity $H=H^{*}$ the condition required is $H=PTHPT$ where $P$ changes the parity and $T$ transforms $i$ to $-i$.

The author constructs the gauged Skyrme model by introducing the skyrmion bundle as follows: instead of considering maps $U:M\to {\text{SU}}_{N_F}$ he thinks of the meson fields as of global sections in a bundle $B(M,{\text{SU}}_{N_F},G)=P(M,G){\times }_G{\text{SU}}_{N_F}$. For calculations within the skyrmion bundle the author introduces by means of the so-called equivariant cohomology an analogue of the topological charge and the Wess-Zumino term. The final result of this paper is the following Theorem. For the skyrmion bundle with $N_F\le 6$, one has $$H^{*}\left(EG{\times }_G{\text{SU}}_{N_F}\right)\cong H^{*}{\left({\text{SU}}_{N_F}\right)}^G\cong \text{S}\left({\underline{G}}^{*}\right)\otimes H^{*}\left({\text{SU}}_{N_F}\right)\cong H^{*}\left(BG\right)\otimes H^{*}\left({\text{SU}}_{N_F}\right),$$ where $EG(BG,G)$ is the universal bundle...

Summary: We give an introduction to the Skyrme model from a mathematical point of view. Hereby, we show that it is difficult to solve the field equation even by means of the classical ansatz, the so-called hedgehog ansatz. Our main result is an extended existence proof for solutions of the field equation in the hedgehog ansatz.

Summary of the three lectures. These notes are available electronically at http://www.ma.huji.ac.il/~drorbn/Talks/Srni-9901/notes.html.

Let $M$ be a $C^{\infty }$-manifold with a Riemannian conformal structure $C$. Given a regular curve $\gamma$ on $M$, the authors define a linear operator on the space of (differentiable) vector fields along $\gamma$, only depending on $C$, called the Fermi-Walker connection along $\gamma$. Then, the authors introduce the concept of Fermi-Walker parallel vector field along $\gamma$, proving that such vector fields set up a linear space isomorphic to the tangent space at a point of $\gamma$. This allows to consider the Fermi-Walker horizontal lift of...

Summary: Geodesics and curvature of semidirect product groups with right invariant metrics are determined. In the special case of an isometric semidirect product, the curvature is shown to be the sum of the curvature of the two groups. A series of examples, like the magnetic extension of a group, are then considered.

Summary: All algebraic objects in this note will be considered over a fixed field $k$ of characteristic zero. If not stated otherwise, all operads live in the category of differential graded vector spaces over $k$. For standard terminology concerning operads, algebras over operads, etc., see either the original paper by J. P. May [“The geometry of iterated loop spaces”, Lect. Notes Math. 271 (1972; Zbl 0244.55009)], or an overview [J.-L. Loday, “La renaissance des opérads”, Sémin. Bourbaki 1994/95,...

Let $F({ℝ}^n,k)$ denote the configuration space of pairwise-disjoint $k$-tuples of points in ${ℝ}^n$. In this short note the author describes a cellular structure for $F({ℝ}^n,k)$ when $n\ge 3$. From results in [F. R. Cohen, T. J. Lada and J. P. May, The homology of iterated loop spaces, Lect. Notes Math. 533 (1976; Zbl 0334.55009)], the integral (co)homology of $F({ℝ}^n,k)$ is well-understood. This allows an identification of the location of the cells of $F({ℝ}^n,k)$ in a minimal cell decomposition. Somewhat more detail is provided by the main result here,...

In the joint paper of the author with K. P. Tod [J. Reine Angew. Math. 491, 183-198 (1997; Zbl 0876.53029)] they showed all local solutions of the Einstein-Weyl equations in three dimensions, where the background metric is homogeneous with unimodular isometry group. In particular, they proved that there are no solutions with Nil or Sol as background metric. In this note, these two special cases are presented.

Summary: It is proven that the Poisson algebra of a locally conformal symplectic manifold is integrable by making use of a convenient setting in global analysis. It is also observed that, contrary to the symplectic case, a unified approach to the compact and non-compact case is possible.

Summary: Here, a relation between integrals and variational multipliers of a system of second-order ordinary differential equations is studied. A simple necessary and sufficient local condition for the existence of a multiplier is given.