一句话：用无点拓扑与几何理论构造实数指数和对数，建立其代数规则并发展提升与粘合技术。
论文在无点拓扑（point-free topology）框架下构造实数指数与对数，即定义 exp:(0,∞)×R→(0,∞) 与 log:(1,∞)×(0,∞)→R 并建立熟悉的代数法则。其方法将空间点视为几何理论的模型而非集合元素，因此构造是构造性的，并可作用于非 Set 的 topos 中的点。作者发展了无点拓扑中的提升（lifting）与粘合（gluing）新技术，揭示 Q 的性质如何决定实数指数性质。该工作由通过 topos 理论发展 adelic geometry 的研究计划驱动，尤其希望构造 Q 的 places 的分类 topos，以几何视角理解 R 与 Q_p 的关系。评估方式为构造性数学证明与代数规则验证。
We define a point-free construction of real exponentiation and logarithms, i.e.\ we construct the maps $\exp\colon (0, \infty)\times \mathbb{R} \rightarrow \!(0,\infty),\, (x, \zeta) \mapsto x^\zeta$ and $\log\colon (1,\infty)\times (0, \infty) \rightarrow\mathbb{R},\, (b, y) \mapsto \log_b(y)$, and we develop familiar algebraic rules for them. The point-free approach is constructive, and defines the points of a space as models of a geometric theory, rather than as elements of a set - in particular, this allows geometric constructions to be applied to points living in toposes other than Set. Our geometric development includes new lifting and gluing techniques in point-free topology, which highlight how properties of $\mathbb{Q}$ determine properties of real exponentiation. This work is motivated by our broader research programme of developing a version of adelic geometry via topos theory. In particular, we wish to construct the classifying topos of places of $\mathbb{Q}$, which will provide a geometric perspective into the subtle relationship between $\mathbb{R}$ and $\mathbb{Q}_p$, a question of longstanding number-theoretic interest.