We classify automorphisms of finite order on compact connected simple Lie groups by which the induced Cartan embedding is an austere embedding. Furthermore, we consider the problem whether an austere Cartan embedding is weakly reflective or not.

Let $X$ be a compact Riemann surface of genus $g\geq 2$. In this paper we study how two memorable stratifications of the moduli space of Higgs bundles of rank $4$ over $X$, Shatz stratification and Bialynicki-Birula stratification, relate to each other and provide dimensions for the Harder-Narasimhan type of a semistable rank 4 Higgs bundle over $X$. In particular, we prove that the two stratifications do not coincide. Thus, we extend to rank 4 the work of Hausel [7], who proved that both stratifications coincide in rank 2, and of Gothen and Zúñiga-Rojas [6], who proved that such a thing no longer occurred in rank 3.

In this paper we consider the following Cauchy problem for the linear wave equation with time-dependent propagation speed:\begin{align} \label{Abstract.Cauchy} \tag{$\star$}\begin{cases}u_{tt}-\lambda^2(t)\omega^2(t)\Delta u=0, & (t,x)\in[0,\infty)\times \mathbb{R}^n, \\u(0,x)=u_0(x), ~~ u_t(0,x)=u_1(x), & x\in\mathbb{R}^n,\end{cases}\end{align}where $\lambda=\lambda(t)$ is an increasing shape function and $\omega=\omega(t)$ is a bounded oscillating function which has very fast oscillations.

The goal is to prove $L^p-L^q$ estimates on the conjugate line for Sobolev solutions of the Cauchy problem \eqref{Abstract.Cauchy} in the case that $\omega=\omega(t)$ has very fast oscillations. Basically, we apply the WKB analysis for the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. Then, the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter. This research continues the research of the paper [17].

In this paper, we derive evolution equations for the first eigenvalue of geometric operators $-\Delta+cR^a$ under the Ricci flow and the normalized Ricci flow, respectively. As applications, we obtain some monotonicity results.

We examine isolated line singularities, transversal $A_1$-type non-isolated hypersurface singularities studied by D. Siersma, in the context of algebraic analysis. By using Poincaré-Birkhoff-Witt algebra, we explicitly compute a Gröbner basis of the annihilator $\text{Ann}_{D_X[s]}f^s$ in a non-commutative ring associated with these singularities. We compute local cohomology solutions of the associated holonomic $D$-modules by utilizing the Gröbner basis of $\text{Ann}_{D_X[s]}f^s$ and determine in particular the monodromy structure of the local cohomology solutions along a singular stratum of hypersurfaces. As a byproduct, we obtain micro-local $b$-functions of isolated line singularities in an explicit manner.

Multivariate Bessel processes describe Calogero-Moser-Sutherland particle models. They depend on a root system and a multiplicity $k$. Recently, several stochastic limit theorems for $k\to\infty$ were derived where the limits depend on the solutions of associated ODEs in these freezing regimes. In this paper we study these ODEs which are singular on the boundaries of their domains. We prove that for arbitrary initial conditions on the boundary, the ODEs have unique solutions in their domains for $t \gt 0$.

We determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. In the even dimensional case non-isolated singularities occur as ‘zigzag trains’.