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Progress and Open Problems 2018

Arrival date: Saturday, September 8.
Departure date: Wednesday afternoon, September 12, or Thursday, September 13.

Schedule

TBA

This conference will be immediately followed by our Second annual meeting held at the Simons Foundation in New York City.

Alex Waldron: Lectures

June 4 and June 6, 2018
TITLE: Recent developments in Yang-Mills flow

ABSTRACT: YM flow is the basic evolution equation for a connection on a vector bundle over a Riemannian manifold. It functions both as an important tool in classical gauge theory, and as a model problem for more highly nonlinear parabolic equations (such as Ricci flow or Bryant’s Laplacian flow). In the first of two talks, I will give an introduction to the subject and describe some historical results. In the second I will describe my own results in the four-dimensional case, and briefly discuss some forthcoming work with Goncalo Oliveira characterizing blowup in the exceptional holonomy scenario.

Esther Cabezas-Rivas: Lectures

June 5, 2018
TITLE: Ricci Flow, non-negative curvatures & beyond

ABSTRACT: In order to use the Ricci flow to prove classification results in geometry and control the behaviour of solutions as times goes by, it is crucial to look for properties of the manifold that are preserved under the flow. During the talk we will see that this is typically the case for a large family of non-negative curvature conditions.
In contrast, the condition of almost non-negative curvature operator (e.g. the condition that its smallest eigenvalue is larger than -1) is not preserved under Ricci flow. In this second talk we will present a work (joint with Richard Bamler and Burkhard Wilking) in which we generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds).

June 4, 2018
TITLE: The ABC of Ricci Flow

ABSTRACT: Geometric flows have been used to address successfully key questions in Differential Geometry like isoperimetric inequalities, the Poincaré conjecture, Thurston’s geometrization conjecture, or the differentiable sphere theorem. During this talk we will give an intuitive introduction to Ricci flow, which is sort of a non-linear version of the heat equation for the Riemannian metric. The equation should be understood as a tool to canonically deform a manifold into a manifold with nicer properties, for instance, some kind of constant curvature. We will emphasize the features that convert evolution equations into a powerful tool in geometry.

Celso Viana: Lectures

June 7, 2018
TITLE: The evolution of the Whitney sphere along mean curvature flow

ABSTRACT: In this talk we will be concern with the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in Cn satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.

Anna Fino: Lectures

June 5, 2018
TITLE: Laplacian flow and special metrics

ABSTRACT: We discuss some results on the behaviour of the Laplacian G_2-flow starting from a closed G_2-structure whose induced metric satisfies suitable extra conditions. In particular we consider the cases when the induced metric is warped or the G_2-structure is extremally Ricci pinched.
The talk is based on joint work with Alberto Raffero.

Chung-Jun Tsai: Lectures

June 7, 2018
TITLE: A strong stability condition on minimal submanifolds

ABSTRACT: It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. We identify a strong stability condition on minimal submanifolds that generalizes the above scenario. In particular, if a closed minimal submanifold \Sigma is strongly stable, then:

1. The distance function to \Sigma satisfies a convex property in a neighborhood of \Sigma, which implies that \Sigma is the unique closed minimal submanifold in this neighborhood, up to a dimensional constraint.

2. The mean curvature flow that starts with a closed submanifold in a C^1 neighborhood of \Sigma converges smoothly to \Sigma.

Many examples, including several well-known types of calibrated submanifolds, are shown to satisfy this strong stability condition. This is based on joint work with Mu-Tao Wang.

Slides of lecture

Geometric Flows and Special Holonomy, Imperial College, 4-8 June 2018

Arrival Date: Sunday June 3. Departure Date: Saturday June 9. All but one of our meetings are in Huxley 140. Registration and coffee/tea are in the Huxley 5th floor common room.


Schedule

MON 4 JUN

TUE 5 JUN

WED 6 JUN

THU 7 JUN

FRI 8 JUN

9:00 -9:30

Registration

9:30-10:30

F. Schulze J. Lotay 1 J. Lotay 2 L. Wang R. Bryant

10:30-11:30

Coffee/tea Coffee/tea Coffee/tea Coffee/tea Coffee/tea

11:30-12:30

A. Waldron 1 D. Joyce A. Waldron 2 P. Pandit 2 S. Donaldson 1

12:30-14:30

Lunch Lunch Lunch Lunch Lunch

14:30-15:30

E. Cabezas-Rivas 1 E. Cabezas-Rivas 2 P. Pandit 1 C. Viana S. Donaldson 2
Huxley 340

15:30-16:00

Coffee/tea Coffee/tea Coffee/tea Coffee/tea Coffee/tea

16:00-17:00

T. Wiseman A. Fino Free discussion C.-J. Tsai Meeting ends

19:00

Meeting dinner

Speakers:

Structure of Collapsed Special Holonomy Spaces, Duke University, 9-13 April 2018

Arrival date: Sunday, April 8, 2018. Departure date: Friday, April 13, 2018.

The conference venue is the 21C Hotel in downtown Durham, North Carolina. 


Schedule

MON 9 APR

TUE 10 APR

WED 11 APR

THU 12 APR

FRI 13 APR

8:00 -9:00

Breakfast Breakfast Breakfast Breakfast Breakfast

9:00-10:00

R. Zhang J. Cheeger X. Rong R. Zhang D. Morrison

10:00-10:15

Q+A Q+A Q+A Q+A Q+A

10:15-11:00

Coffee Coffee Coffee Coffee Coffee

11:00-12:00

Y. Zhang R. Mazzeo R. Mazzeo Y. Li Y. Li

12:00-2:00

Lunch/Discussion Lunch/Discussion Lunch/Discussion Lunch/Discussion Lunch/Discussion

2:00-3:00

G. Wei G. Chen L. Foscolo M. Haskins S. Salamon

3:00-3:30

Coffee Coffee Coffee Coffee Coffee

3:30-4:30

R. Zhang Y. Zhang J. Viaclovsky Discussion

7:00-9:00

Conference banquet

Speakers:

 

Sir Simon Donaldson to deliver the Marston Morse Lecture Series

Sir Simon Donaldson, one of our Collaboration’s Principal Investigators, will deliver the Marston Morse Lectures at the Institute for Advanced Study during the first week of April, 2018, under the general title “Exceptional holonomy and related geometric structures.” The detailed schedule of lectures is here.

Robert Bryant delivers the McKnight-Zame Distinguished Lecture

Collaboration Director Dr. Robert Bryant delivered the 2018 McKnight-Zame Distinguished Lecture at the University of Miami on March 1, 2018. The lecture, entitled “From Mechanics, to Algebra, to Geometry: The Notion of Holonomy,” introduced a general mathematical audience to the main theme of this Collaboration. Further details about the lecture series and Dr. Bryant’s lecture can be found here.