Ryosuke Takahashi will speak in the geometry seminar on the 10th October. The talk will take place in the Salle de Profs (9th floor, NO) at 10.30am. Ryosuke’s title is *“A new parabolic flow approach to the Kähler-Einstein problem”* and his abstract is below. **Please note the time, which is earlier than usual!** (This is to give people the possibility to attend Cédric Villani’s lecture which will take place at KULeuven, before the award of his honorary doctorate.)

We introduce the “Mabuchi flow”, a new parabolic flow of Monge-Ampère type which is designed to deform a given Kähler metric to the Kähler-Einstein one. We study the Mabuchi flow from the view point of Geometric Invariant Theory. We provide three results about the Mabuchi flow:

- Long-time existence and smooth convergence on the canonically polarized manifolds
- GIT stability relation between the self-similar solutions of the Mabuchi flow and extremal Kähler metrics
- Relation between the Mabuchi flow and optimal destabilizer on toric Fano manifolds
In the canonically polarized case, the large time behavior of the Mabuchi flow is similar to that of the Kähler-Ricci flow. On the other hand, the Mabuchi flow on anti-canonically polarized manifolds has a different aspect. We observe that it encodes the optimal destabilizing information about Kähler/algebro-geometric structure in a suitable sense when the manifold is toric. In addition, the Mabuchi flow has many properties similar to the Calabi flow. For instance, we can compare the GIT-stabilities for their solitons. This talk is based on a joint work with T. Collins (Harvard Univ) and T. Hisamoto (Nagoya Univ).