His dissertation is titled "Enumeration of local and global étale algebras applied to generalized Fermat equations". The research was supervised by Sander Dahmen as part of his NWO-Vidi project "New Diophantine directions".
Casper studied specific classes of Diophantine equations, i.e. polynomial equations that have to be solved in integer unknowns. He obtained far reaching results on generalized Fermat equations, a notoriously difficult class of Diophantine equations. These are equations of the shape x^p+y^q=z^r for given integer exponents p, q, r > 1, to be solved in nonzero integer unknowns x, y, z whose greatest common divisor equals 1. Over the past three decades many equations of this shape have been solved, often driving central developments in number theory and arithmetic geometry, as these are used as underlying techniques to resolve the equations.
However, in the important case that the three exponents p, q, r in the generalized Fermat equation are distinct prime numbers (except when they involve both 2 and 3, the two smallest primes), close to nothing was known about how to resolve such an equation explicitly. In his PhD thesis, Casper developed both theoretical and computational techniques to attack this situation, which is also applicable in a more general setting when we put some nonzero coefficients into the equation. This framework is also fully implemented and available for everybody who likes to study these Diophantine equations. Upon using this framework, one of the remarkable concrete results obtained in the PhD thesis, is that the generalized Fermat equation with exponents p=2, q=5, r=7, has no solutions if y is even.