CFUC · Calabi flows with unbounded curvature
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2016-05-26 → 2018-05-25
- EU contribution
- €183,455
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
Lines connect the coordinator with its partners.
Results in brief
Calabi flows with unbounded curvature
The Calabi flow was introduced by Calabi in the 1950s, aiming to find constant scalar curvature Kähler (cscK) metrics. An important class of cscK metrics is the Ricci flat Kähler metrics. The Calabi conjecture was about the existence of such metrics and its resolution was part of Yau's Fields medal work. Nowadays, the existence of cscK metrics has been intensively studied and its development reveals the link between differential geometry and algebraic geometry, which is now known as the Yau-Tian-Donaldson conjecture. Geometric flows have been successful in finding canonical metrics. The theory of Ricci flow developed by Hamilton and Perelman has successfully solved the conjecture of Poincare and Thurston, one of the seven $1 million Clay Mathematics Institute Millennium Prizes. This proposal concerns singularity analysis of the Calabi flow, when the curvature is unbounded. The cscK metrics are the stationary solutions of the Calabi flow and the Kähler metrics with cone singularities generally do not have bounded curvature. Consequently, cscK metrics with cone singularities serve as a natural singularity model of the Calabi flow with unbounded curvature. During the fellowship, the researcher pioneered the development and established a theory of cscK metrics with cone singularities. The primary impact of the project is knowledge exchange between the researcher and his collaborators. The broad development of fundamental mathematics such as the research in this proposal is giving new ways to understand our universe and is leading to applications that affect our everyday life. For example, geometric flows, as studied in this proposal, are being used for image processing.
Data: CORDIS, © European Union
Project objective
In the 1950s, Calabi proposed a program in Kahler geometry and then introduced the Calabi flow, aiming to find the constant scalar curvature Kahler (cscK) metrics. When the first Chern class is zero, the cscK metric reduces to Ricci flat Kahler metric. The problem to find such metrics is called Calabi conjecture. Its resolution was Yau's Fields medal work. Generally, it is known as the Yau-Tian-Donaldson conjecture. Geometric flow provides an effective way to find canonical metrics. E.g., the theory by Hamilton and Perelman of Ricci flow has achieved great success to solve the conjecture of Poincare and Thurston, one of the seven $1 million Clay Mathematics Institute Millennium Prizes. X.X. Chen conjectured the Calab flow has long time existence. This proposal concerns singularity analysis of the Calabi flow, when the curvature gets unbounded. Warwick leads a major new project funded by an EPSRC grant 'Singularities of Geometric PDEs', together with Imperial and Cambridge, making it a natural host for this proposal. The supervisor Topping is the Principal investigator of this project. He is a leading expert on geometric flows and nonlinear PDEs. He has considerable experience in supervising research: 14 postdocs and 8 PhD students. Currently, he is working on Ricci flows with unbounded curvature and presented an invited 45-minute lecture on this topic at Seoul ICM in 2014.Zheng completed his PhD at the Chinese Academic of Sciences under the supervision of W.Y. Ding and X.X. Chen. From his advisors, Zheng gained intimate understanding of Kahler geometry. He worked as a postdoc at the Institut Fourier in France and then Leibniz Universitaet in Germany. Up to May 2015, his research experience has entirely been outside UK. He is ambitious to establish himself as an independent researcher at a prestigious UK institution. He has published 8 papers in high reputation international journals. This project will help him to integrate himself into the UK research system.
Original text from CORDIS.
Participants
- UNIVERSITY OF WARWICK · COVENTRYCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union