Lagrangian mean curvature flow is the name given to the phenomenon that, in a Calabi-Yau manifold, the class of Lagrangian submanifolds is preserved under mean curvature flow. An influential conjecture of Thomas and Yau, refined since by Joyce, proposes to utilise the Lagrangian mean curvature flow to prove that certain Lagrangian submanifolds may be expressed as a connect sum of volume minimising 'special Lagrangians'.

 

This talk is an exposition of recent joint work with Wei-Bo Su and Chung-Jun Tsai, in which we exhibit a Lagrangian mean curvature flow which exists for infinite time and converges to an immersed special Lagrangian. This demonstrates one mechanism by which the above decomposition into special Lagrangians may occur, and is also the first example of an infinite -time singularity of Lagrangian mean curvature flow. The work is a parabolic analogue of work of Dominic Joyce and Yng-Ing Lee on desingularisation of special Lagrangians with conical singularities, and is inspired by the work of Simon Brendle and Nikolaos Kapouleas on ancient solutions of the Ricci flow.

Self-dual Yang-Mills famously have a description in terms of twistors; one of the outstanding questions is how to promote it to full (non-self-dual) Yang-Mills and learn about its dynamics. In this talk, I will present some progress in this direction in the "most symmetric" Yang-Mills theory; namely N=4 super Yang-Mills in four dimensions. I will express the full Yang-Mills theory as a deformation of self-dual Yang-Mills. By treating the deformation perturbatively and using the formalism of twistors, I will write down the loop-integrands of correlation functions of determinant operators in the planar limit at any order in the 't Hooft coupling. Interestingly, the final expression is given by a partition function of a "dual" matrix model. This in turn manifests a ten-dimensional structure that combines spacetime and R-charge symmetries of SYM.