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

 6/4/2018 and 6/6/2018: Recent developments in YangMills flow
 6/4/2018 and 6/6/2018: Recent developments in YangMills flow
June 4 and June 6, 2018
TITLE: Recent developments in YangMills 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 fourdimensional case, and briefly discuss some forthcoming work with Goncalo Oliveira characterizing blowup in the exceptional holonomy scenario.
Esther CabezasRivas: Lectures

 6/5/2018: Ricci Flow, nonnegative curvatures & beyond
 6/4/2018: The ABC of Ricci Flow
 6/5/2018: Ricci Flow, nonnegative curvatures & beyond
June 5, 2018
TITLE: Ricci Flow, nonnegative 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 nonnegative curvature conditions.
In contrast, the condition of almost nonnegative 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 nonnegative 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 noncollapsed manifolds with almost nonnegative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a shorttime existence result for the Ricci flow on open manifolds with almost nonnegative 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 nonlinear 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

 6/5/2018: Laplacian flow and special metrics
 6/5/2018: Laplacian flow and special metrics
June 5, 2018
TITLE: Laplacian flow and special metrics
ABSTRACT: We discuss some results on the behaviour of the Laplacian G_2flow starting from a closed G_2structure whose induced metric satisfies suitable extra conditions. In particular we consider the cases when the induced metric is warped or the G_2structure is extremally Ricci pinched.
The talk is based on joint work with Alberto Raffero.
ChungJun 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 is strongly stable, then:
1. The distance function to satisfies a convex property in a neighborhood of , which implies that 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 converges smoothly to .
Many examples, including several wellknown types of calibrated submanifolds, are shown to satisfy this strong stability condition. This is based on joint work with MuTao Wang.
Geometric Flows and Special Holonomy, Imperial College, 48 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  
9:3010:30 
F. Schulze  J. Lotay 1  J. Lotay 2  L. Wang  R. Bryant 
10:3011:30 
Coffee/tea  Coffee/tea  Coffee/tea  Coffee/tea  Coffee/tea 
11:3012:30 
A. Waldron 1  D. Joyce  A. Waldron 2  P. Pandit 2  S. Donaldson 1 
12:3014:30 
Lunch  Lunch  Lunch  Lunch  Lunch 
14:3015:30 
E. CabezasRivas 1  E. CabezasRivas 2  P. Pandit 1  C. Viana  S. Donaldson 2 Huxley 340 
15:3016:00 
Coffee/tea  Coffee/tea  Coffee/tea  Coffee/tea  Coffee/tea 
16:0017:00 
T. Wiseman  A. Fino  Free discussion  C.J. Tsai  Meeting ends 
19:00 
Meeting dinner 
Speakers:
 Robert Bryant (Duke University), On solitons for the closed Laplacian flow
 Esther CabezasRivas (GoetheUniversität Frankfurt), The ABC of Ricci Flow
 Esther CabezasRivas (GoetheUniversität Frankfurt), Ricci Flow, nonnegative curvatures & beyond
 Sir Simon Donaldson (Imperial College London and SCGP), G_{2} manifolds with boundary
 Sir Simon Donaldson (Imperial College London and SCGP), Adiabatic limits, multivalued harmonic functions and the NashMoserZehnder theory
 Anna Fino (University of Turin), Laplacian flow and special metrics
 Dominic Joyce (Oxford University), On Mirror Symmetry, Fukaya categories, and Bridgeland stability, with a view towards Lagrangian Mean Curvature Flow
 Jason Lotay (University College London), The G_{2} Laplacian flow: introduction and overview
 Jason Lotay (University College London), The G_{2} Laplacian flow: progress and outlook
 Pranav Pandit (University of Vienna), Categorical Kähler Geometry
 Pranav Pandit (University of Vienna), Gradient flows, iterated logarithms, and semistability
 Felix Schulze (University College London), Singularity formation in Lagrangian mean curvature flow
 ChungJun Tsai (National Taiwan University), A strong stability condition on minimal submanifolds
 Celso Viana (University College London), The evolution of the Whitney sphere along mean curvature flow
 Alex Waldron (SCGP), Recent developments in YangMills flow
 Lu Wang (University of Wisconsin, Madison), Properties of selfsimilar solutions of mean curvature flow
 Toby Wiseman (Imperial College London), Some applications of Ricci flow in physics
Structure of Collapsed Special Holonomy Spaces, Duke University, 913 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 
9:0010:00 
R. Zhang  J. Cheeger  X. Rong  R. Zhang  D. Morrison 
10:0010:15 
Q+A  Q+A  Q+A  Q+A  Q+A 
10:1511:00 
Coffee  Coffee  Coffee  Coffee  Coffee 
11:0012:00 
Y. Zhang  R. Mazzeo  R. Mazzeo  Y. Li  Y. Li 
12:002:00 
Lunch/Discussion  Lunch/Discussion  Lunch/Discussion  Lunch/Discussion  Lunch/Discussion 
2:003:00 
G. Wei  G. Chen  L. Foscolo  M. Haskins  S. Salamon 
3:003:30 
Coffee  Coffee  Coffee  Coffee  Coffee 
3:304:30 
R. Zhang  Y. Zhang  J. Viaclovsky  Discussion  
7:009:00 
Conference banquet 
Speakers:
 Jeff Cheeger (Courant), Curvature and injectivity radius estimates for Einstein 4manifolds
 Gao Chen (IAS), Classification of gravitational instantons
 Lorenzo Foscolo (HeriotWatt), Examples of collapsing 4dimensional hyperkähler metrics
 Mark Haskins (Bath), Uncollapsing highly collapsed G_{2} holonomy metrics
 Yang Li (Imperial College London), Nonlinear perspective of collapsing CY metrics on K3 fibred 3folds
 Yang Li (Imperial College London), A gluing construction of collapsing CY metrics on K3 fibred 3folds
 Rafe Mazzeo (Stanford), The largescale structure of the Hitchin moduli space
 Rafe Mazzeo (Stanford), Analysis of elliptic operators on complete spaces with asymptotically regular collapsing geometry
 Dave Morrison (UC Santa Barbara), Collapsing metrics in string theory and Mtheory
 Xiaochun Rong (Rutgers), Collapsed manifolds with Ricci local bounded covering geometry
 Simon Salamon (King’s College London), Automorphisms of Fano contact manifolds
 Jeff Viaclovsky (UC Irvine), Gravitational collapsing of K3 surfaces I
 Guofang Wei (UC Santa Barbara), Manifolds with integral curvature bounds
 Ruobing Zhang (Stony Brook), Introduction to Ricci curvature and the convergence theory
 Ruobing Zhang (Stony Brook), Quantitative nilpotent structure and regularity theorems of collapsed Einstein manifolds
 Ruobing Zhang (Stony Brook), Gravitational collapsing of K3 surfaces II
 Yuguang Zhang (Imperial College London), Collapsing of CalabiYau manifolds and special lagrangian submanifolds
 Yuguang Zhang (Imperial College London), Collapsing of HyperKahler manifolds
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 McKnightZame Distinguished Lecture
Collaboration Director Dr. Robert Bryant delivered the 2018 McKnightZame 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.