The field of geometric evolution equations has seen tremendous progress in the past twenty years. Analytic, geometric, and numerical techniques are used in the setting of differential geometry to solve pure and applied problems in diverse fields which include global geometry, mathematical physics, algebraic geometry, material science, image processing and optimization. A plethora of important geometric heat flows are of current interest, including Ricci and Kaehler-Ricci flow, mean and inverse mean curvature flows, porous medium equation, Yamabe flow, and the harmonic map heat flow. These flows are characterized by the deformation of geometric objects such as metrics, mappings, and submanifolds by geometric quantities such as curvature and consist of partial differential equations of parabolic type. Via geometric evolution equations, the powerful methods of nonlinear and numerical analysis can be applied to mathematical problems that can be approached geometrically.
During this past decade the theory of formation of singularities was developed for Ricci flow and mean curvature flow, which has had a large impact on other geometric flows. Hamilton has developed a comprehensive program to approach Thurston’s Geometrization Conjecture (which subsumes the Poincare Conjecture) by Ricci flow methods. Recently there have been spectacular breakthroughs on Hamilton’s program by Perelman, which are in the process of being verified as a complete proof. The ideas of Hamilton’s program and Perelman’s recent progress have profoundly impacted the entire field of geometric flows and should have more wide ranging implications for Riemannian geometry, geometric analysis and applied subjects where flow methods are used. Mean curvature and inverse mean curvature flows have been used to solve long-standing problems in general relativity (such as the Riemannian Penrose conjecture). Most recently, Bray discovered a deep relation between the Penrose inequality in General Relativity and the Yamabe invariant of 3-manifolds, given by the inverse mean curvature flow. The recent theoretical progress on geometric flows, especially on understanding weak solutions and singularities, together with the recent computational progress on geometric flows makes this an opportune time to hold a workshop which will bring together mathematicians working on the theoretical and numerical aspects of geometric flows. In particular, there are a number of basic open theoretical problems that should benefit greatly from numerical calculations. On the theoretical side, we propose to have speakers discuss the latest theoretical advances in their areas and describe which open problems will benefit from numerical calculations. On the numerical side, we similarly propose that the speakers discuss the latest numerical advances in their areas and both describe the impact their results will have on the theoretical study of flows and which types of problems are accessible by their methods. The interaction between geometric analysts and numerical analysts should prove very fruitful in developing both new theoretical conjectures and numerical techniques to support them.
Geometric flows appear in many real world applications. For example, surface tension along moving interfaces in fluids and materials is proportional to mean curvature: mean curvature flow and affine mean curvature flow are useful for morphological image processing. Numerical computation of moving interfaces and geometric flows is quite challenging due to dynamic deformation of geometry, nonlinearity and possible development of singularities, especially topological changes. Recently great success has been made in computational methods for moving interfaces such as the level set method. In this workshop, numerical methods, computations and applications of geometric flows will be presented. However, rigorous convergence proofs and error estimates are needed for these numerical algorithms. Mathematical theories for geometric flows may provide useful tools and insights for these proofs. Also in many applications, numerical computations have to be continued after topological changes. So mathematical understanding of the formation of singularities and continuation past singularities can be used to verify current numerical methods or to construct more appropriate numerical schemes.
This five-day workshop at IPAM will bring together experts in geometry with those in numerical analysis for each to explain open problems in their area that might benefit from tools and insights from the other area. Topics to be covered during the workshop include:
This workshop will provide a wonderful opportunity for geometric and numerical analysts to begin collaborations on open problems such as understanding singularities and global existence and convergence, which are amenable to both theoretical and numerical investigations.
Ben Chow (University of California at San Diego)
Panagiota Daskalopoulos (Columbia University)
Richard Hamilton (Columbia University)
Gerhard Huisken (MPI Albert Einstein Institute)
Hong-Kai Zhao (University of California at Irvine)