Hyperkähler mirror symmetry and Langlands duality

A flow of ideas from string theory has inspired spectacular progress in modern mathematical research. This intersectional phenomenon arises from the pursuit of symmetries exhibited by fundamental theories in physics and mathematics. One of these symmetries is S-duality, whose complex behaviour has remained inaccessible to our current mathematical techniques.A theoretical breakthrough came in 2006. Kapustin and Witten conjectured that S-duality gives rise to an equivalence between two types of boundary conditions - branes of type ""BBB"" and ""BAA"", conditions that come from a hyperkähler geometric structure. Their conjecture sets out a clear route to a new mathematical understanding of S-duality: construct the proposed equivalence and shine light on the underlying geometry. The action addresses this challenge using modern tools from algebraic geometry. New methods are developed within three vast fields of research: mirror symmetry, the geometric Langlands program, and wall-crossing. This broad range of ideas will be harnessed to seek powerful and novel insights into branes and their dualities. Specifically, the research objectives are:(1) Construct a categorical equivalence between BBB and BAA-branes in special cases (i.e. prove hyperkähler enhancements of homological mirror symmetry). (2) Develop a twistor theoretic approach to the geometric Langlands program, designed to encode the hyperkähler geometry of BBB and BAA-branes.(3) Apply wall-crossing to construct a class of branes with a particularly rich geometry and study their behaviour under S-duality. The implementation brings together my PhD research, wall-crossing techniques developed by the supervisor Dr. Feyzbakhsh, and spectacular recent progress on the geometric Langlands program by Gaitsgory et al. In doing so, new frontiers are sought in rapidly developing fields of research, from which myself and the academic community will benefit in the years to come.""