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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.

Toby Wiseman: Lectures

June 4, 2018
TITLE: Some applications of Ricci flow in physics

ABSTRACT: I will review two areas where Ricci flow makes contact with physics. Firstly I will review how Ricci flow arises from the renormalisation group equations of 2d `sigma models’ (I will try to explain what these words mean!). Secondly I will review a more recent link, where Ricci flow may be thought of as an algorithm to numerically find solutions to Einstein’s gravitational equations in exotic settings. In physics in both cases it is interesting to consider how black holes evolve under Ricci flow. Static black holes may be thought of as Riemannian geometries, while stationary black holes cannot, but still may be evolved using a ‘Lorentzian’ signature Ricci flow. I will also discuss the existence of Ricci solitons which are important to understand in the second context.

Lu Wang: Lectures

June 7, 2018
TITLE: Properties of self-similar solutions of mean curvature flow

ABSTRACT: I will survey some known results as well as some open problems about self-similar solutions of the mean curvature flow.

Felix Schulze: Lectures

June 4, 2018
TITLE: Singularity formation in Lagrangian mean curvature flow

ABSTRACT: We will survey results on singularity formation in mean curvature flow, both in codimension one and in higher codimension with a particular focus on Lagrangian mean curvature flow. We will also review different concepts of weak flows through singularities together with geometric applications.

Pranav Pandit: Lectures

June 7, 2018
TITLE: Gradient flows, iterated logarithms, and semistability

ABSTRACT: The formalism of categorical Kähler geometry outlined in the previous lecture leads to the study of certain dynamical systems. A typical example is furnished by the Yang-Mills flow on the space of hermitian metrics on a holomorphic bundle. It turns out that the asymptotic behaviour of these dynamical systems is governed by iterated logarithms. The talk will elaborate on this statement, and explain how it leads to the discovery of a canonical refinement of the Harder-Narasimhan filtration in a variety of contexts. This is a report on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.

Slides of lecture

June 6, 2018
TITLE: Categorical Kähler Geometry

ABSTRACT: After introducing the paradigm of derived geometry, I will outline attempts to formalize and understand the mathematical structures underlying the physical notion of stability for D-branes in string theory using the language of derived noncommutative geometry. These efforts build upon Bridgeland’s notion of stability conditions on triangulated categories, and are inspired by ideas from symplectic geometry, non-Archimedean geometry, dynamical systems, geometric invariant theory, and the Donaldson-Uhlenbeck-Yau correspondence. This talk is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.

Slides of lecture

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