Stability conditions on triangulated categories were introduced by Bridgeland, based on ideas from string theory. Conjecturally, they control existence of solutions to the deformed Hermitian Yang-Mills equation and the special Lagrangian equation (on the A-side and B-side of mirror symmetry, respectively). I will focus on the symplectic side and sketch a program which replaces special Lagrangians by "spectral networks", certain graphs enhanced with algebraic data. Based on joint work in progress with Katzarkov, Konstevich, Pandit, and Simpson.

In the talk I will discuss the  following result and related analytic and geometric questions:   On the boundary of any convex body in the Euclidean space there exists at least one infinite geodesic.