2021-05-13 Person of the Week

I'm Eiji INOUE, a new member of iTHEMS as a special postdoctoral researcher in mathematics. My current interest is Kahler geometry of algebraic variety. Algebraic variety is (locally) the solution set of polynomials, say x^2 + y^2 -1 = 0. While its origins trace back to ancient Greek, it still fascinates many mathematicians: you can find many Fields medalists, including all Japanese medalists, are awarded for their monumental works on algebraic variety.
Calabi-Yau variety is a special class of algebraic varieties attracting attention in string theory. A Calabi-Yau variety admits a Kahler-Einstein metric, which can be thought of as a canonical 'shape' of the variety. Though a general variety in other classes does not necessarily admit such canonical metrics, it is gradually believed by not a few specialists that any variety has a unique degeneration to another variety admitting a canonical metric in some sense. My recent study gives a mathematical formulation of this problem.
This framework has a special aspect: it naturally possesses a new parameter λ which plays a role analogous to the inverse temperature. When λ is sufficiently low, canonical metrics, which you may see as 'equilibrium states' of the variety, are unique if it exists. On the other hand, when λ is sufficiently high, canonical metrics are not unique and the absolutely stable states may break the symmetry of the variety. It is reminiscent of phase transition. I am looking forward to discussing this phenomenon with researchers in other areas.