In geometry the classification of complex manifolds via special/canonical metrics is of central importance, and can be seen as the search of a higher dimensional analogue to the uniformization theorem for Riemann surfaces. This question still remains open in general, and has inspired a beautiful theory connecting existence of special metrics (e.g. Kähler-Einstein metrics, constant scalar curvature metrics) with stability in algebraic geometry, following work of Calabi, Yau, Donaldson, Tian and many others. In this talk I will survey some of my work related to obstructions and existence of Calabi's canonical metrics on (possibly non-projective) Kähler manifolds. I will also present recent and ongoing work on effective algebraic/numerical criteria for existence of solutions to related geometric PDE in complex geometry, mirroring concepts of "wall-and-chamber decompositions" from Bridgeland stability and opening up to using computers for studying stability in the future. Along the way I will highlight open questions and potential points of contact with colleagues at the department.
After the lecture, the Department will host a small reception in Vandrehallen, building 1530.