My research mainly focuses on the properties and behavior of solutions of (local and nonlocal) partial differential equations arising from different fields like Physics, Biology or Finance and on the mathematical understanding of modern machine learning algorithms, in particular the very successful Artificial Neural Networks (ANNs).
To be more precise, topics I have worked on in the past include
the qualitative behavior of solutions of the obstacle problem and the obstacle problem for the fractional Laplacian, including a partial proof of a long standing conjecture on the classification of global solutions of the obstacle problem,
traveling waves and propagation and non-propagation phenomena in reaction-diffusion equations. I have been able to give explicit a-priori criteria for existence and non-existence of generalized traveling fronts and I have constructed some heteroclinic orbits between traveling wave solutions,
Wasserstein gradient-flows with time-dependent constraints and long-time asymptotics,
the Mathematical understanding of the training algorithms in Artificial Neural Networks. While Artificial Neural Networks have proven very successful in a vast number of areas including image recognition, machine translation, text production, ... and besides the vast amount of research that has been carried out in recent years, from a Mathematical standpoint even very basic questions about their training algorithms are not yet understood,
non-periodic homogenization including ramified domains.
For more details on the precise subjects of my research and my precise results please take a look at my list of publications.