Many natural phenomena, such as the motion of objects, the flow of air, or the temperature of objects or spaces, are described by solutions to differential equations. Typically, the solutions gradually approach stationary states (described by constants, time-periodic oscillating functions, etc.) or diverge exponentially, but occasionally there are “singular” behaviors that do not fit into this framework. For example, for u’ = u² (prime is the time derivative), if we take the values of u at time t=0 positive, the solution goes to infinity at t→T for some finite value of T, and the equation becomes “unsolvable” thereafter.

This is a “weird” behavior that cannot occur in typical physical phenomena and is called a “finite-time blow-up” of the solution. Finite-time blow-ups are often identified as mathematical objects of peculiar phenomena, such as thermal runaway associated with heat source ignition. On the other hand, like the above equation, the system itself is often nonsingular, and for a given system, questions “does blow-up occur?” and if so, “when, where, and how?” are nontrivial to answer, and control of such behavior requires a deep understanding of the phenomenon itself. Meanwhile, we are dealing directly with “infinity” for blow-up, which is difficult to capture mathematically and numerically, and this phenomenon itself has been the subject of mathematical research for many years.

One of my research themes is to realize a unified description of “weird behavior” such as blow-ups in a standard way. One concrete idea is to “embed the entire space in a hemisphere or cut paraboloid and represent infinity as its boundary”. This is analogous to the Riemann sphere in Complex Function Theory and compactification in Topology, but here we also pay attention to the scalability of the system, construct an embedding of the entire space in an appropriate (bounded) surface, and express infinity as the boundary: the “horizon”. This idea originates from the way singularities and infinity points are viewed in algebraic geometry. Combining it with dynamical systems theory, which comprehensively describes the possible (qualitative) behavior of all solutions, we obtain the correspondence “Divergent Solutions = Those approaching to sets on the horizon”. Furthermore, the way the solution converges to the horizon makes it possible to accurately describe the blow-up behavior. This is achieved by combining geometric aspects of dynamical systems, algebraic geometry, and asymptotic analysis.

One advantage of this approach is a big compatibility with other theories and techniques, such as numerical analysis; (1): the behavior of the solution is covered, including the presence or absence of blow-ups; (2): blow-up solutions broken by perturbations of initial values are computed numerically with mathematical rigor; and (3): even “complex” blow-ups can be accurately captured through the horizon. These can be achieved by combining rigorous numerics, singularity theory, etc.

Not only blow-ups, but also various “finite-time singularities” are difficult to characterize. However, by changing the way of looking at it as described above, it is gradually becoming clear that it is actually just a straightforward behavior seen from a different viewpoint. This way has been applied to multiscale dynamics in the past, and more recently to tipping points, which can also be “interpreted in a straightforward manner” with knowledge of dynamical systems theory and geometry. The idea of describing these in a comprehensive manner will lead to the view that “weird” = “honest” in many respects.