# Geometric flows in mathematics and physics (11w5010)

Arriving in Banff, Alberta Sunday, April 17 and departing Friday April 22, 2011

## Organizers

Mauro Carfora (University of Pavia)

Zindine Djadli (Institut Fourier, Université Grenoble 1)

Gerhard Huisken (Universitaet Tuebingen)

Lei Ni (University of California, San Diego)

Eric Woolgar (University of Alberta)

## Objectives

One primary objective of this meeting is to generate well-defined mathematical problems which are motivated by physics and which can be addressed using geometric flow equations. A second primary objective is to enable researchers to report progress on such problems as have already been identified. Secondary objectives are to provide a forum for mathematicians to discuss recent progress in geometric flows, to expose physicists to this progress, and, conversely, to familiarize mathematicians with potential physical applications of their work.

To enable us to achieve these objectives, we will leave lots of time during the workshop for informal discussions. Morning sessions will have a small number of hour-long lectures aimed at giving both physicists and mathematicians a broad perspective of the current state of affairs. Afternoon sessions will contain talks on recent research results, which we hope to limit to 30 minutes to provide adequate free time for discussion. Not all of the participants will speak.

Past workshops in the area have focused on the role of geometric flow equations in quantum field theory (QFT). Geometric flows arise as approximations to renormalization group (RG) flows in QFT (as well as possibly arising as models for string dynamics). This has focused attention on ancient solutions of the flow, which are now well-understood by mathematicians in certain cases, such as the Ricci flow in 3 dimensions. An open problem that has come out of this is the question of the relationship between Perelmanï¿½s entropy functionals and the C-function of RG flow. This may give insight into the possibility that RG flow (for the 2-dimensional nonlinear sigma model) is a gradient flow. This discussion has also motivated interest in the problem of mean curvature flow of a submanifold inside a target manifold undergoing Ricci flow. Finally, this perspective has led to the realization that truncations of RG flow that go beyond the first order Ricci flow term lead to new geometric flows with a rich structure of self-similar and stationary solutions.

For the proposed workshop, we plan to explore other, more recent developments and related applications of Ricci flow. We will, for example, consider the Ricci flow on noncompact complete manifolds and manifolds-with-boundary. This is of interest in mathematics as a possible tool for extending the classification of closed 3-manifolds to the case of complete 3-manifolds. In physics, it has the potential to address a number of outstanding conjectures concerning static metrics in general relativity. An example is the static minimization conjecture of Robert Bartnik, which is important in the theory of quasi-local mass in general relativity. Another possible example is the construction of black hole metrics that arise in the Randall-Sundrum scenario (i.e., a noncompact manifold with a static metric, a boundary called a ï¿½braneï¿½, and an asymptotic end). These are both examples of Ricci flow as a possible relaxation method for finding Ricci-flat metrics. This motivates the study of numerical methods for Ricci flow, an as yet wide open field to which we will also dedicate some time during the workshop. It also leads to the study of Ricci flow coupled to other, more complicated flows. Two of these are the flow of stationary metrics and the harmonic-map-heat-flow-like evolutions of sections of certain bundles. Evolution equations of this nature will also comprise part of our workshop programme.

A second area to be explored during the workshop will be progress in complex geometric flows, including the Kï¿½hler-Ricci flow and Lagrangian mean curvature flow. This continues to be mathematically a very active field, providing good source of interaction with physics because of the applications to string theory and to supersymmetric RG flow. Numerical Kï¿½hler-Ricci flow has the possibility of yielding Calabi-Yau metrics and, more importantly, spectra of the corresponding Laplacians. This would be of considerable use in string theory and is related to the study of various Monge-Ampï¿½re type equations and minimal surface equations, a subject of active research.

A third area of research to be explored during the workshop will be recent progress in the characterization of Ricci curvature on general metric spaces and the interplay between Ricci flow and optimal transport theory. This is a very active field lying at the boundary between Mathematics and Physics with many applications to both fields. On the mathematical side, optimal transport processes on Riemannian manifolds and metric spaces have a natural feedback with Ricci flow which one would like to explore in depth. On the physical side the corresponding diffusion processes may significantly advance our understanding of renormalization group flows for QFT.

Finally, we have suggested dates for the worskshop that are two weeks to either side of the ICIAM 2011 congress in Vancouver. Colleagues at the University of Alberta are planning an ICIAM satellite meeting in Edmonton on numerical methods for geometric problems. The satellite workshop will focus on numerical methods. It is expected that there will be synergy between this and the numerical aspect of our BIRS workshop, with some people participating in both workshops. To take maximum advantage of this, our hope is to hold the BIRS workshop two weeks before or after the ICIAM, with the satellite meeting occurring in the week that will lie between these two events.

To enable us to achieve these objectives, we will leave lots of time during the workshop for informal discussions. Morning sessions will have a small number of hour-long lectures aimed at giving both physicists and mathematicians a broad perspective of the current state of affairs. Afternoon sessions will contain talks on recent research results, which we hope to limit to 30 minutes to provide adequate free time for discussion. Not all of the participants will speak.

Past workshops in the area have focused on the role of geometric flow equations in quantum field theory (QFT). Geometric flows arise as approximations to renormalization group (RG) flows in QFT (as well as possibly arising as models for string dynamics). This has focused attention on ancient solutions of the flow, which are now well-understood by mathematicians in certain cases, such as the Ricci flow in 3 dimensions. An open problem that has come out of this is the question of the relationship between Perelmanï¿½s entropy functionals and the C-function of RG flow. This may give insight into the possibility that RG flow (for the 2-dimensional nonlinear sigma model) is a gradient flow. This discussion has also motivated interest in the problem of mean curvature flow of a submanifold inside a target manifold undergoing Ricci flow. Finally, this perspective has led to the realization that truncations of RG flow that go beyond the first order Ricci flow term lead to new geometric flows with a rich structure of self-similar and stationary solutions.

For the proposed workshop, we plan to explore other, more recent developments and related applications of Ricci flow. We will, for example, consider the Ricci flow on noncompact complete manifolds and manifolds-with-boundary. This is of interest in mathematics as a possible tool for extending the classification of closed 3-manifolds to the case of complete 3-manifolds. In physics, it has the potential to address a number of outstanding conjectures concerning static metrics in general relativity. An example is the static minimization conjecture of Robert Bartnik, which is important in the theory of quasi-local mass in general relativity. Another possible example is the construction of black hole metrics that arise in the Randall-Sundrum scenario (i.e., a noncompact manifold with a static metric, a boundary called a ï¿½braneï¿½, and an asymptotic end). These are both examples of Ricci flow as a possible relaxation method for finding Ricci-flat metrics. This motivates the study of numerical methods for Ricci flow, an as yet wide open field to which we will also dedicate some time during the workshop. It also leads to the study of Ricci flow coupled to other, more complicated flows. Two of these are the flow of stationary metrics and the harmonic-map-heat-flow-like evolutions of sections of certain bundles. Evolution equations of this nature will also comprise part of our workshop programme.

A second area to be explored during the workshop will be progress in complex geometric flows, including the Kï¿½hler-Ricci flow and Lagrangian mean curvature flow. This continues to be mathematically a very active field, providing good source of interaction with physics because of the applications to string theory and to supersymmetric RG flow. Numerical Kï¿½hler-Ricci flow has the possibility of yielding Calabi-Yau metrics and, more importantly, spectra of the corresponding Laplacians. This would be of considerable use in string theory and is related to the study of various Monge-Ampï¿½re type equations and minimal surface equations, a subject of active research.

A third area of research to be explored during the workshop will be recent progress in the characterization of Ricci curvature on general metric spaces and the interplay between Ricci flow and optimal transport theory. This is a very active field lying at the boundary between Mathematics and Physics with many applications to both fields. On the mathematical side, optimal transport processes on Riemannian manifolds and metric spaces have a natural feedback with Ricci flow which one would like to explore in depth. On the physical side the corresponding diffusion processes may significantly advance our understanding of renormalization group flows for QFT.

Finally, we have suggested dates for the worskshop that are two weeks to either side of the ICIAM 2011 congress in Vancouver. Colleagues at the University of Alberta are planning an ICIAM satellite meeting in Edmonton on numerical methods for geometric problems. The satellite workshop will focus on numerical methods. It is expected that there will be synergy between this and the numerical aspect of our BIRS workshop, with some people participating in both workshops. To take maximum advantage of this, our hope is to hold the BIRS workshop two weeks before or after the ICIAM, with the satellite meeting occurring in the week that will lie between these two events.