Flow, flow, flow your boat

A highly biased survey of flow techniques in the search for canonical geometries

 

 

Text Box: The following outline is only tentative and will evolve in response to the interests of the participants.

 

 

The flavor of flows: a sampler

*      Dimension n=1: the curve shortening flow

*      Dimension n=2: the Yamabe flow

*      Codimension 1: the mean curvature flow

*      Dimension n≥3: the Ricci flow

Ricci curvature and what it means for humanity

*      Ricci curvature and volume

*      Ricci curvature as a differential operator

*      Ricci flow as a gradient flow

*      Heuristics

*      A sampler of special solutions

Self-similar solutions and what they tell us

*      Solitons

*      Examples from mean curvature flow

*      Examples from Ricci flow

*      Examples from Kähler–Ricci flow

*      Classifying solitons

Singularities and why they’re special

*      Preserved and attractive cones

*      Blowing up singularities

*      Singularity types

*      Injectivity radius estimates

*      Singularity models

*      The case in dimension n=3

*      The case in higher dimensions

*      The Huisken–Sinestrari classification for mean curvature flow

The perspicacity of Perelman

*      Monotonicity in geometric evolution

*      Entropy

*      Reduced geometry

*      Examples

*      Localization

Where do we go from here?

*      A discussion of some open problems