Symplectic aspects of some modern topics in singularity theory

My primary research interest is in symplectic geometry; I am especially drawn to problems in Floer theory with interdisciplinary applications. The present research proposal has three separate lines of enquiry, all connected by symplectic aspects of singularity theory as a common theme. Singularities are ubiquitous in mathematics and physics, often appearing as limiting configurations of smooth objects. For example, imagine a bowling ball on a trampoline, and then that the ball's weight slowly increased. Assuming the mat never tore, the endpoint of this process would be a singularity -- the mat would be infinitely sharply curved at the bottom. Such a situation is closely related to certain singularities in general relativity, which we know as black holes. Since singularities are unavoidable, it is necessary to develop tools in a variety of contexts in order to best understand a theory as a whole. In the above situation, for example, it would not be possible to claim that we understand general relativity without understanding black holes. We focus on singularities in the context of symplectic and algebraic geometry, where there are classically two `obvious' approaches to their study. Firstly, one could work topologically, looking for something smooth which is `close to' the singular space. The symplectic geometry of which smooth spaces are nearby (if any) and how singularities form as these spaces degenerate can then be used to understand the singularity itself. The second method comes from algebraic geometry, where one simply cuts out the singular point and glues back in something smooth. One can then investigate the singularity in question by understanding the different ways in which this can be done. In conjunction, these two operations are extremely useful; however, it is not clear whether they provide complete, the same, complementary or overlapping information. In Project (A), we will study the symplectic geometry of smoothings of certain threefold singularities -- called compound du Val -- and compare them with the algebraic geometry of the same singularity. Roughly speaking, the goal is to show that, in these cases, the symplectic and algebro-geometric approaches do, in fact, contain the same information. In Project (B), we will study cusp singularities, which are an important class of surface singularities arising in pairs, and appearing in the boundary of moduli spaces of surfaces of general type. Roughly speaking, our goal is to show that the number of distinct ways which one can symplectically smooth a cusp is predicted by the algebraic geometry of its dual. Heuristically, this can be thought of as showing that the algebraic and symplectic approaches to studying cusp singularities contain complementary information. Project (C) is of a distinctly different flavour to the previous two projects, and is in the field of quantum singularity theory. This seeks to understand singularities and the symmetries of their defining equations directly, without any smoothing or resolving; however, the premise of this project is still in comparing the symplectic and algebraic approaches. Roughly speaking, we will aim to show that the structure of a certain analytically defined invariant of a class of curve singularities is tightly connected with the algebraic geometry of the same curve singularities. This is a non-trivial prediction, since, as we saw above, different approaches to studying the same singularity may or may not be expected to be related.