The unreasonable effectiveness of mathematics that Eugene Wigner wrote of in 1960 [E. Wigner 1960 Comm. Pure and Appl. Math. 13:1] has faced growing challenges as mathematicians have turned high performance computers towards understanding the world around us. Weather, financial markets, the human brain, each are made up of simple parts, but their interesting behaviour arises because of the way those parts interact. Highly complex behaviour then emerges. We even call these complex'' systems, suggesting that complexity in their description is unavoidable, and ever higher demands are placed on computers to model them. Leave your computers at the door! This Fellowship will seek simple explanations for complex behaviour in the fundamentals of interaction itself. A famous example of complex behaviour arising from simple laws is chaos, which explains why the weather is unpredictable despite deterministic underlying equations. Another example is a canard explosion, a rapid cascade of dynamics that occurs too abruptly to observe (the term canard'' alludes to an old French joke, portraying Nature as a trickster selling a mathematician un canard a moitie, or half a duck). A problem arises because these phenomena, highly important to everyday life, often depend on infinitesimally small quantities that are difficult or impossible to calculate. This Fellowship aims to find universal laws linking these small quantities to the nature of interaction. Small quantities can, counterintuitively, rise to dominate a system when an interaction occurs between systems with differing internal time scales, like a fast switch in a slowly changing electric current. When the mismatch between interacting timescales is extreme, sudden jumps can occur between different dynamical regimes. We can approximate the jump by a sharp discontinuity (a true jump in the mathematical sense), and studying the shape or configuration of the system near the discontinuity. This approach has explained why slipping objects suffer episodes of sticking, and how impacts lead to chattering. More surprisingly, it has recently led to a new description of canards, and a new non-deterministic form of chaos. These previously unknown phenomena are so violent that, if seen in experiments, one is unlikely to have conceived of them having such simple mathematical origins. So the theory based on discontinuities predicts qualitatively new phenomena, but it cannot quantify them exactly. The objective of the Fellowship is to bring together the geometrical theory unhindered by small quantities, and the small quantity theory capable of exact calculations. This will give a rigorous explanation for the newly discovered discontinuity-induced phenomena, and provide much simpler geometry based methods to calculate emergent phenomena in complex systems. The simpler these are to calculate, the more effective they are to study real world problems in engineering and medicine. As examples we will study models of neuron signalling in the brain, and electronic switches in power systems. The missing ingredient is how to tie discontinuity geometry to small quantities which are already hard to calculate. For this we will borrow methods for calculating small quantities from wave physics. We study the dynamics in `complex time' and seek stationary solutions. This is an abstract but naturally geometric approach, and has not been applied to discontinuity-induced phenomena before. Wigner said of the unreasonable effectiveness of mathematics, we should be grateful for it and hope that it will remain valid in future research. There is no doubt that systems with many parts can produce complicated dynamics, but the effectiveness of mathematics comes from finding behaviours that are simple and robust, and the Fellowship aims to study these in the mathematics describing interactions, as a crucial step towards the ultimate goal of understanding the full intricacy that complex systems can produce.