Viaclovsky, Jeff

University of Wisconsin Madison, Madison, WI, United States

In more technical terms, the main component of research of this award is the study of critical points of curvature functionals on Riemannian manifolds. An important problem is to understand compactness of moduli spaces and existence of critical metrics, such as anti-self-dual and extremal Kaehler metrics. With certain geometric noncollapsing assumptions, the appropriate moduli spaces can be compactified by adding metrics with orbifold-like singularities. A long-term goal is to extend the compactness theorem to include the possibility of collapsing, and to find other applications to the differential topology of four-manifolds. In dimension four, critical points of the Weyl energy are known as Bach-flat metrics, which contains the class of anti-self-dual metrics. Such metrics have very interesting properties, and can be studied using twistor theory. In higher dimensions, the PI will investigate quadratic curvature functionals, and their variational properties, such as stability and rigidity of critical points. This has applications to volume comparison theorems and gluing results. The PI is also interested in non-compact examples of critical metrics, such as asymptotically locally Euclidean critical metrics, and obtaining optimal decay rates for such spaces. This has applications to the understanding of the structure of moduli spaces, and to the removal of singularities.

An important motivation for the research of the PI is to understand the relationship between the geometry and the topology of a space. The latter, topology, is the study of properties of a space which are invariant under continuous stretching or bendings of a space, while the former, geometry, involves understanding distances and is more rigid. For example, the surface of our planet is a sphere, and one measures distances on it by computing arclengths of great circles (the Earth is actually an oblate spheroid, but it is very close to being perfectly spherical). One can imagine deforming the Earth by pushing in or pulling on small or large regions to warp the geometry. Such a deformation is less appealing that the familiar round Earth, and there are many ways to make this notion very precise in terms of minimizing some sort of total energy measurement. This is directly related to physical principles which say that the state of a physical system will tend towards a final configuration which minimizes the total energy. This idea can be generalized to higher-dimensional objects called manifolds, which are generalized versions of the surface of the our planet. For example, the space that we live in is three-dimensional, and if one includes time, we are in a four-dimensional universe. In order to understand these types of higher-dimensional objects, one attempts to find the best way to measure distances on them which use the least amount of energy, and maximize the symmetries of the space. The projects described above are to define appropriate energies on such spaces, and to seek out the important optimal geometries which minimize the total energy.

An important motivation for the research of the PI is to understand the relationship between the geometry and the topology of a space. The latter, topology, is the study of properties of a space which are invariant under continuous stretching or bendings of a space, while the former, geometry, involves understanding distances and is more rigid. For example, the surface of our planet is a sphere, and one measures distances on it by computing arclengths of great circles (the Earth is actually an oblate spheroid, but it is very close to being perfectly spherical). One can imagine deforming the Earth by pushing in or pulling on small or large regions to warp the geometry. Such a deformation is less appealing that the familiar round Earth, and there are many ways to make this notion very precise in terms of minimizing some sort of total energy measurement. This is directly related to physical principles which say that the state of a physical system will tend towards a final configuration which minimizes the total energy. This idea can be generalized to higher-dimensional objects called manifolds, which are generalized versions of the surface of our planet. For example, the space that we live in is three-dimensional, and if one includes time, we are in a four-dimensional universe. In order to understand these types of higher-dimensional objects, one attempts to find the best way to measure distances on them which use the least amount of energy, and maximize the symmetries of the space. The projects in this proposal are to define appropriate energies on such spaces, and to seek out the important optimal geometries which minimize the total energy. In more technical terms, the research of the PI is, broadly speaking, to use solutions of partial differential equations which are geometric in origin to study properties of differentiable manifolds. The main areas of concentration of the PI's research are the existence of critical metrics generalizing the Einstein condition, the study of quadratic curvature functionals on Riemannian manifolds, the study of critical ALE metrics and orbifolds, and properties of moduli spaces of critical metrics. In joint work with Matt Gursky, the PI has proved existence of critical metrics on various four-manifolds, and is currently engaged in further study the properties of the moduli space of such solutions. This is related to orbifold compactness theorems previously studied by Tian-Viaclovsky. The PI has previously demonstrated non-solvability of the Yamabe problem on certain compact orbifolds, which showed that the orbifold Yamabe problem is more subtle than in the case of smooth manifolds. The PI has further explored this phenomenon, and of connections with the notion of mass of ALE spaces. There has been a considerable amount of research on the existence of anti-self-dual metrics on compact manifolds; they have been shown to exist in abundance. Progress has also been made in understanding global properties of the moduli spaces in certain cases; especially for orbifold-cone anti-self-dual metrics. Finally, the PI is committed to integrating research and education and cultivating intellectual development on many levels. The PI has been active in outreach and organization of conferences in the mathematics community.

- Agency
- National Science Foundation (NSF)
- Institute
- Division of Mathematical Sciences (DMS)
- Type
- Standard Grant (Standard)
- Application #
- 1105187
- Program Officer
- Joanna Kania-Bartoszynska

- Project Start
- Project End
- Budget Start
- 2011-08-15
- Budget End
- 2014-07-31
- Support Year
- Fiscal Year
- 2011
- Total Cost
- $198,488
- Indirect Cost

- Name
- University of Wisconsin Madison
- Department
- Type
- DUNS #

- City
- Madison
- State
- WI
- Country
- United States
- Zip Code
- 53715