On 2 November 2016, Prof. Rob Stevenson and Dr. Viresh Patel were awarded a TOP Grant by the 'Nederlandse Organisatie voor Wetenschappelijk Onderzoek' (NWO). These Grants were awarded in the TOP Grants Science round 2016, compartment 1 and compartment 2
This project is about optimal adaptive methods for the numerical solution of various partial differential equations. Starting from an initial coarse partition, within a loop these methods gradually refine the mesh, or increase the degree of the polynomial approximation, on those locations where the current error is large. One of the topics that will be studied is their application to time-dependent problems. The traditional solution schemes by time marching have limited possibilities concerning adaptivity, and are inherently sequential, and thus not very suited for a parallel implementation. We will investigate the application of optimal adaptive methods to the problem as a whole in a well-posed simultaneous space-time variational formulation.
One of the main goals of extremal combinatorics is to understand sharp conditions under which a combinatorial structure contains some substructure of interest. Finding such conditions for spanning substructures in graphs is a very active area of research and one of the most fundamental such substructures is the Hamilton cycle, that is a cycle that passes through every vertex of a graph.
There has been great recent success in solving long-standing problems asking whether dense (directed) graphs with certain conditions contain one or more Hamilton cycles. The Szemerédi Regularity Lemma as well as the recent robust expansion technique have been instrumental in this regard. The goal of this project is to attack long-standing open problems about Hamilton cycles in sparse graphs partly by extending the ideas of robust extension. The problems involve the long-standing conjectures of Chvátal on toughness and Hamiltonicity, and Lovász’s conjecture on Hamiltonicity of connected Cayley graphs. In addition we consider the problem of Hamiltonian resilience: given a graph class that is Hamiltonian, how likely is it to remain Hamiltonian under random edge deletion. The final theme is to investigate applications of methods from extremal graph theory to the algorithmic Hamilton cycle problem.