Hadamard spaces have traditionally played important roles in geometry and geometric group theory. More recently, they have additionally turned out to be a suitable framework for convex analysis, optimization and non-linear probability theory. The attractiveness of these emerging subject fields stems, inter alia, from the fact that some of the new results have already found their applications both in mathematics and outside. Most remarkably, a gradient flow theorem in Hadamard spaces was used to attack a conjecture of Donaldson in Kähler geometry. Other areas of applications include metric geometry and minimization of submodular functions on modular lattices. There have been also applications into computational phylogenetics and image processing.

We survey recent developments in Hadamard space analysis and optimization with the intention to advertise various open problems in the area. We also point out several fallacies in the existing proofs.

阿达玛空间传统上在几何和几何群论中发挥着重要作用。最近，它们也被证明是凸分析、优化和非线性概率论的合适框架。这些新兴学科领域的吸引力尤其源于这样一个事实：一些新成果已经在数学和外部找到了应用。最引人注目的是，哈达玛空间中的梯度流定理被用来攻击卡勒几何中唐纳森的猜想。其他应用领域包括度量几何和模格子上子模函数的最小化。也已应用于计算系统发育学和图像处理。

我们调查了 Hadamard 空间分析和优化的最新进展，旨在宣传该领域的各种开放问题。我们还指出了现有证明中的几个谬误。