Characteristic classes, which are abstract topological invariants associated
with vector bundles, have become an important notion in modern physics with
surprising real-world consequences. As a representative example, the incredible
properties of topological insulators, which are insulators in their bulk but
conductors on their surface, can be completely characterized by a specific
characteristic class associated with their electronic band structure, the first
Chern class. Given their importance to next generation computing and the
computational challenge of calculating them using first-principles approaches,
there is a need to develop machine learning approaches to predict the
characteristic classes associated with a material system. To aid in this
program we introduce the {\emph{Haldane bundle dataset}}, which consists of
synthetically generated complex line bundles on the $2$-torus. We envision this
dataset, which is not as challenging as noisy and sparsely measured real-world
datasets but (as we show) still difficult for off-the-shelf architectures, to
be a testing ground for architectures that incorporate the rich topological and
geometric priors underlying characteristic classes.

用于预测与材料系统相关的特征类的机器学习方法的开发，引入了 Haldane 捆绑数据集，该数据集由在 2 维环面上合成生成的复线捆绑组成，旨在成为结合丰富的拓扑和几何先验的架构的测试基地。

Haldane 集束：学习预测环绕圆环上线束的陈数的数据集

Haldane Bundles: A Dataset for Learning to Predict the Chern Number of  Line Bundles on the Torus

We survey both old and new developments in the theory of algorithms in real
algebraic geometry -- starting from effective quantifier elimination in the
first order theory of reals due to Tarski and Seidenberg, to more recent
algorithms for computing topological invariants of semi-algebraic sets. We
emphasize throughout the complexity aspects of these algorithms and also
discuss the computational hardness of the underlying problems. We also describe
some recent results linking the computational hardness of decision problems in
the first order theory of the reals, with that of computing certain topological
invariants of semi-algebraic sets. Even though we mostly concentrate on exact
algorithms, we also discuss some numerical approaches involving semi-definite
programming that have gained popularity in recent times.

本文调查了实代数几何算法理论的新、旧发展，主要涵盖关于真实数的一阶理论中的量词消除和计算半代数集拓扑不变量等问题，重点分析了这些算法的复杂性和背后问题的计算难度，并对一些近年来越来越受欢迎的半定规划数值方法进行了讨论。