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Luke Williams

About Me:

I am originally from Wolverhampton. As an undergraduate I studied Mathematical & Theoretical Physics at Aberystwyth University (MMath), graduating in 2013. I then joined the MASDOC DTC, receiving an MSc in Mathematics & Statistics from Warwick in 2014, where I am now a 1st year PhD student.


My main interest is in the breaking or preservation of symmetries in atomistic systems at positive temperature, specifically the breaking of rotational symmetry. The main methods used come from the idea of M. Heydenrich et al to apply a rigidity estimate, a tool from continuum theories developed by S. Müller et al, to an appropriate interpolant generated from configurations. This yields an order parameter in the form of a distance from a constant rotation which allows us to make statements about how ordered a configuration of points is. The goal is to find sufficient conditions that allow this parameter to be controlled, and how many kinds of defects can be included in configurations before this notion breaks down. My current work is based on generalising new results in this field to obtain results that allow for finite, uniform concentrations of defects for large systems, which has been a problem in recent work due to the scaling behaviour of constants in a generalised rigidity estimate. Both of these concepts were introduced by S. Aumann.