|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|
Jim Simons, Stony Brook University
Robert Haslhofer, New York University
Mean curvature flow with surgery
We give a new proof for the existence of mean curvature flow with
surgery for 2-convex hypersurfaces. Our proof works in all dimensions,
including mean convex surfaces in $R^3$. We also derive a priori estimates
for a more general class of flows. This is joint work with Bruce
André Neves, Imperial College
Existence of minimal hypersurfaces
I will talk about my recent work with Fernando Marques where we show that positive ricci curvature metrics admit an infinite number of minimal embedded hypersurfaces.
Song Sun, SCGP and Stony Brook University
Kahler-Einstein metrics: Gromov-Hausdorff limits and algebraic geometry
In this talk we will discuss compactification of the moduli space of Kahler-Einstein manifolds using Gromov-Hausdorff limits, in both abstract and explicit terms, and emphasize the relation with algebraic geometry. This is based on joint works with Donaldson, Chen-Donaldson, and Odaka-Spotti.
Matthew Gursky, Notre Dame University
Critical metrics on connected sums of Einstein four-manifolds
I will describe joint work with J. Viaclovsky in which we use a
gluing construction to produce new examples of four-manifolds that are
critical for certain quadratic Riemannian curvature functionals.
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
On positivity of a class of conformal covariant operators
I will report on a recent joint work with Jeffrey Case in which we study the
positivity of a class of non-local conformal covariant operators which are fractional
GJMS operators defined via scattering theory on asymptotic hyperbolic manifolds,
which includes the Dirichlet-Neumann operator as a special case. When the order
of the operator is higher than 2, we will explore the positivity property of this class
of operators via Sobolov trace extension formulas in the setting of metric spaces
|Sunday, April 13|
Kenji Fukaya, SCGP and Stony Brook University
'Hodge Theory' from Floer homology
In this talk I want to report some resent progress
of the project in the title.
In our previous work with Oh, Ohta, Ono,
we established relation between Jacobian ring of super potential
defined by Lagrangian Floer theory on Toric manifolds
and Quantum cohomology.
We also establish certain pairing in Jacobian ring and
Hochshild homology (jointly also with Abouzaid)
and relate it to the Poincare pairing
in Quantum cohomology.
We are now studying, higher residue pairing and primitive
form (of Kyoji Saitoh) in the LG model of super potential
which is Hodge theoretical part of LG model,
and its relation to cyclic homology and also to
$S^1$ equivariant Gromov-Witten theory.
(Many parts of this story are work in progress.)
Herman Gluck, University of Pennsylvania
History of the Geometry Festival
Mihalis Dafermos, Princeton University
On null singularities for the Einstein vacuum equations and the strong cosmic censorship conjecture in general relativity
I will present new results on the emergence of "null singularities" in the interior of generic vacuum black hole spacetimes (without symmetry assumptions), and I will discuss what this means for the status of Penrose's celebrated strong cosmic censorship conjecture in general relativity. This is joint work with Jonathan Luk (MIT).