Floer homology and low-dimensional topology

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Mathematical knots, visualised as a simple but knotted closed rope, have been studied by scientists since the end of the XIXth century. One of the early motivations was to use these objects as a way to understand properties of atoms and matter. The ties to natural sciences have grown over the years and nowadays, for example, the study of DNA topology is an active, interdisciplinary area of research.From early days of Knot theory, the classification problem for knots has played a central and motivating role. The basic problem seems quite simple: can one tell if two given knots are topologically equivalent or not. Approaches to this problem have led topologists to study various classes of knots and to try to characterize their numerous invariants. Moreover, generations of mathematicians have constructed increasingly complex knot invariants which aim to capture essential properties of knots such as genus, slice genus, unknotting number and so on.One of the many uses of knots, from a mathematical perspective, is that they are intimately related to objects called 3-manifolds, whose classification has been a central problem in Mathematics for more than 60 years. Indeed examples of the relationship between the two (nonts and 3-manifolds) are provided by constructions like Dehn surgery and cyclic branched covers along knots.  Our research is concerned with invariants of knots and such 3-manifolds, the invariants we are interested in being obtained from elliptic partial differential equations on manifolds. In particular, the study of Floer homology theory in its various flavours has helped to prove various conjectures in the area of low-dimensional topology and the applicant's research proposal outlines various problems which may be approached with these powerful invariants.