Georgia Institute of Technology College of Sciences

Fruitful interactions between arithmetic geometry and dynamical systems have emerged in recent years. In this talk I will illustrate how insights from complex dynamics can be employed to study problems from arithmetic geometry. And conversely how arithmetic geometry can be used in the study of dynamical systems. The motivating questions are inspired by a recurring phenomenon in arithmetic geometry known as `unlikely intersections' and conjectures of Pink and Zilber therein. More specifically, I will discuss work toward understanding the distribution of preperiodic points in subvarieties of families of rational maps.

The flow of compressible fluids is governed by the Euler equations, and understanding the dynamics for large times is an outstanding open problem whose full resolution is unlikely to happen in our lifetimes. The main source of difficulty is that any global-in-time theory must incorporate singularities in the PDEs, a fact we have known even in one spatial dimension since Riemann’s 1860 work. In this 1D setting, mathematicians have successfully spent the past 160 years painting a nearly-full picture of fluid dynamics that incorporates singularities.

There is a monumental gap in our understanding of compressible fluids in the physical 3D setting compared to the 1D case. This is due in large to the (provable) inaccessibility of the technical PDE tools used in 1D when quantifying the dynamics in 3D. Nevertheless, Christodoulou’s 2007 celebrated breakthrough on shock singularities for the Euler equation has sparked a dramatic wave of results and ideas in multiple space dimensions that have the potential to make the first meaningful dent in the global-in-time theory of compressible fluids. Roughly, shocks are a form of singularity where the fluid solution remains regular but certain first derivatives blow up.

In this talk I will discuss the recent culmination of the wave of results initiated by Christodoulou: my work on the maximal classical development (MCD) for compressible fluids, joint with J. Speck. Roughly speaking, the MCD describes the largest region of spacetime where the Euler equations admit a classical solution. For an open set of smooth data, my work reveals the intimate relationship between shock singularity formation and the full structure of the MCD. This fully solves the 162 year old open problem of extending Riemann’s historic 1D result to 3D without symmetry assumptions. In addition to the mathematical contribution, the geo-analytic information of the MCD is precisely the correct “initial data” needed to physically describe the fluid “past” the initial shock singularity in a weak sense. I will also briefly discuss the countless open problems in the field, all of which can be viewed as “building blocks” which will shine the first lights onto the outstanding global-in-time open problem of fluids.

Structural graph theory has traditionally focused on graph classes that are closed under both vertex- and edge-deletion (such as, for each surface Σ, the class of all graphs which embed in Σ). A more recent trend, however, is to require only closure under vertex-deletion. This is typically the right approach for graphs with geometric, rather than topological, representations. More generally, it is usually the right approach for graphs that are dense, rather than sparse. I will discuss this paradigm, taking a closer look at classes with a forbidden vertex-minor.