PhD defence: Non-local Differential Relations

PLEASE NOTE: If a candidate gives a layman's talk, the livestream will start fifteen minutes earlier.

One of the basic problems in Mathematics is understanding differential constraints and their solutions. Such constraints commonly take the following form: For some fixed manifolds M and N (i.e. 'smooth' spaces such as Euclidean spaces), we want to find all smooth maps f:M->N such that for every point x in the domain, some statement about the value and derivatives of f at x must be true.

This statement can be an equation in the simplest case, and the study of these differential equations has become a cornerstone of our modern world. More broadly, the constraint may also involve inequalities that provide some sort of regularity, such as asking for a non-vanishing derivative. Especially for the latter situation, the study of Gromov's h-principles has been a fruitful avenue.

This thesis explores a wider class of differential constraints that are non-local in that they may depend on the values and derivatives at different points in the domain. Our aim is in particular to lay the groundwork for h-principles of non-local relations. A central construction towards this goal is a 'weighted blow-up' of multijet space, for which we introduce a general framework that should also be useful outside of this context.