Complex Analysis, Harmonic Analysis, & Complex Geometry Seminar

Abstract: Optimal degeneration is a 'test configuration' of a projective variety X which degenerates the variety X to a variety Y K-semistable in an extended sense. It is characterized as a maximizer of a certain quantity for test configurations. The conjectural existence of optimal degeneration is important as it completes the proof of the YTD conjecture and more importantly it clarifies the right treatment of K-unstable varieties in moduli theory. 

There are various frameworks of optimal degeneration depending on the choice of the quantity (normalized Donaldson-Futaki invariant, H-entropy, mu-entropy, ...), which corresponds to choosing a certain extended framework of K-stability for 'spacetime' (relative K-stability, modified D-stability, muK-stability, ...). 

I would like to sketch the blueprint of optimal degeneration problem and report where we are now in the long way to go. 

 The notion of first integral came from the classical mechanics.  The constants of motion, namely, the total enegy and the angular momentum in the Newtonian mechanics are typical examples of the first integral.   I will survey briefly how these notions have been  generalized along the lines of exterior differential systems, including the Frobenius theorem on integrability.   Then we shall discuss the partial integrability and the notion of weak first integral,  and as  applications,  the affine  control  problems with prescribed obstacles and the stability of population dynamics of Kolmogorov type.