Topology Unveiled: A New Horizon for Economic and Financial Modeling

Sinceits introduction in the 19th century to address geometric problems, topology as a methodology has undergone a series of evolutions, encompassing branches of geometric topology, point-set topology (analytic topology), algebraic topology, and differential topology, gradually permeating into various interdisciplinary applied fields. Starting from disciplines with typical geometric characteristics such as geography, physics, biology, and computer science, topology has found its way to economic fields in the 20th century. Given that the introduction of topology to economics is relatively new and presents features of being fragmented and non-systematic, this review aimed to provide scholars with a systematic evolution map to refine the characteristics of topology as a methodology applied in economics and finance, thereby aiding future potential interdisciplinary developments in these fields. By collecting abundant literature indexed in SCOPUS/WoS and other famous databases, with a qualitative analysis to classify and summarize it, we found that topological methods were introduced to modern economics when dealing with dynamic optimization, functional analysis, and convex programming problems, including famous applications such as uncovering equilibrium with fixed-point theorems in Walrasian economics. Topology can help uncover and refine the topological properties of these function space transformations, thus finding unchangeable features. Meanwhile, in contemporary economics, topology is being used for high-dimension reduction, complex network construction, and structural data mining, combined with techniques of machine learning, and applied to high-dimensional time series and structure analysis in financial markets. The most famous practical applications include the use of topological data analysis (TDA) and topological machine learning (TML) for different applied problems.