Traditional risk measures in finance, predominantly based on the second moment of return distributions or tail risk heuristics (VaR/CVaR), fail to account for the intrinsic geometric structure of market dynamics. This paper introduces a rigorous mathematical framework utilizing Topological Data Analysis (TDA) to quantify risk as the structural instability of the reconstructed phase space. By applying Takens' Delay Embedding Theorem to cryptocurrency log-returns, we generate a point cloud representation of the underlying attractor. We analyze the evolution of the filtration of Vietoris-Rips complexes to compute persistent homology groups $H_k$. We define a "Topological Persistence Norm" to characterize market regimes and propose a leverage calibration heuristic based on the persistence of 1-dimensional cycles. This approach provides a coordinate-free, stability-invariant metric for risk assessment that is robust to high-frequency noise.
We introduce semitopologies, a generalisation of point-set topology that removes the restriction that intersections of open sets need necessarily be open. The intuition is that points are participants in some distributed system, and an open set is a collection of participants that can collaborate to update their local state by taking a distributed collaborative action; we call this an actionable coalition. What constitutes an actionable coalition depends on what actions we want to model. Intuitive examples include 'a group of people that is collectively strong enough to lift a rock', where the state update is very simply 'holding rock low' to 'holding rock high' and this update is common to all participants in the actionable coalition. Or, consider 'two people wishing to barter a can of juice for a bar of chocolate', in which case the coalition is any such pair and the state updates differ between participants to flip them between 'has/has no juice' and 'has/has no chocolate'. A characteristic of these systems is that state updates are local to the coalition, voluntary, may vary between participants, and are not assumed subject to permission or synchronisation by a central authority. Peer-to-peer computer networks, including filesharing and blockchain systems, provide motivating examples from computing. This monograph presents a comprehensive view of semitopologies which includes point-set semitopology, algebra, and logic inspired by these considerations. This is interesting in and of itself and it provides a conceptual framework within which to understand a useful class of distributed systems.
Abstract We introduce semitopology, a generalization of point-set topology that removes the restriction that intersections of open sets need necessarily be open. The intuition is that points represent participants in a decentralized system, and open sets represent collections of participants that collectively have the authority to collaborate to update their local state; we call this an actionable coalition. Examples of actionable coalition include: majority stakes in proof-of-stake blockchains; communicating peers in peer-to-peer networks; and even pedestrians working together to not bump into one another in the street. Where actionable coalitions exist, they have in common that collaborations are local (updating the states of the participants in the coalition, but not immediately those of the whole system); collaborations are voluntary (up to and including breaking rules); participants may be heterogeneous in their computing power or in their goals (not all pedestrians want to go to the same place); participants can choose with whom to collaborate; and they are not assumed subject to permission or synchronization by a central authority. We develop a topology-flavoured mathematics that goes some way to explaining how and why these complex decentralized systems can exhibit order, and gives us new ways to understand existing practical implementations. Semitopology is also interesting in and of itself, having a rich and interesting theory that quickly deviates from standard accounts on topological spaces. It soon becomes clear that the most interesting semitopologies are rather ill-behaved from the usual viewpoint, as they are never Hausdorff. A notion of ‘transitive open sets’ (topens) becomes central to the story, as topens define subsets of participants who should decide the same value in a distributed system that tries to achieve consensus, and points are called ‘regular’ when they have a topen neighbourhood. The theory is then further developed by introducing intertwined points, closures, closed sets and two interesting characterizations of regularity.