|Friday, April 11|
Robert Bryant, Duke University
Rolling surfaces and exceptional geometry
The mechanical system of one rigid surface rolling over another without twisting or slipping is a staple of non-holonomic mechanics and has been studied from a number of different points of view. The differential equations that describe this motion turn out to be a special case of a system of PDE studied by Élie Cartan in 1910. Remarkably, Cartan showed that such systems can have a symmetry group with dimension as large as 14 (and that, in this case, the symmetry group is isomorphic to the exceptional group $G_2$). For example, it turns out that a sphere of radius~$1$ rolling over a sphere of radius~$3$ belongs to this highly symmetric case.
In recent years, interest in these systems have come from a number of different sources, and there have been some surprising developments. P. Nurowski has shown that there is a close connection of Cartan's work with split-conformal geometry in dimension~$5$, and T. Willse has shown that there is a connection with pseudo-Riemannian metrics of special holonomy in dimension~$7$. Quite recently, Nurowski and An have used this connection to discover a remarkable convex surface in $3$-space whose differential constraints that describe its rolling over the flat plane have $G_2$-symmetry, which raises the question of how many such pairs of surfaces might exist.
In this talk, I will describe the history of this problem, the geometry that goes into its study, and the above recent developments in this area, including some recent results of my own that provide progress in classifying the pairs of surfaces whose rolling constraints have exceptional symmetry.
|Saturday, April 12|
Robert Haslhofer, New York University
André Neves, Imperial College
Song Sun, SCGP and Stony Brook University
Matthew Gursky, Notre Dame University
The main application is an existence result, using two well-known Einstein manifolds
as building blocks: the Fubini-Study metric on $CP^2$, and the product metric
on $S^2 x S^2$. Using these metrics in various gluing configurations,
critical metrics are found on connected sums.
The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on $CP^2$, and the product metric on $S^2 x S^2$. Using these metrics in various gluing configurations, critical metrics are found on connected sums.
Alice Chang, Princeton University
|Sunday, April 13|
Kenji Fukaya, SCGP and Stony Brook University
Herman Gluck, University of Pennsylvania
Mihalis Dafermos, Princeton University