UBC Math Department Rising Stars Colloquium: Tristan Collins

Mirror symmetry describes a surprising duality between symplectic geometry and complex geometry, coming from a duality between certain models of string theory. Of particular interest in these theories is the question: "what are the observable particles?". These particles are described by certain nonlinear partial differential equations; the deformed Hermitian-Yang-Mills equation, and the special Lagrangian equation. These equations do not always have solutions, corresponding to the fact that not all particles are stable-- some particles decay into small constituent parts. It has long been expected that these decays are predicted by purely algebraic structures. I will describe how such an algebraic structure appears in the study of the deformed Hermitian-Yang-Mills equation through the study of certain convex functions on infinite dimensional manifolds. Time permitting, I will also explain how these results, coupled with mirror symmetry, provide insight on the special Lagrangian equation. This is joint work with S.-T. Yau.