Monday

Scalar Curvature: Structure and Regularity – Part 1/2

  • Speaker: Robin Neumayer
  • Time: 9:00-10:00
  • Abstract: A broad theme in geometric analysis aims to understand the geometric structure of Riemannian manifolds that satisfy constraints on curvature. Over the past 25 years, a rather complete picture has been given for the structure and a priori regularity of Riemannian manifolds with lower bounds on Ricci curvature, as well as their limit spaces. On the other hand, when one assumes only a lower bound on scalar curvature, the trace of the Ricci curvature, the situation is much less well understood. The scalar curvature asymptotically governs the volumes of balls of small radii, and arises in various settings in general relativity and differential geometry. This mini-course will focus on the structure and regularity of Riemannian manifolds with scalar curvature lower bounds.

 

Stable minimal hypersurfaces in 4-manifolds (Part 1/2)

  • Speaker: Chao Li
  • Time: 10:10-11:10
  • Abstract: Stable minimal surfaces in 3-manifolds have been extensively used to probe the topology of the ambient space. Important results include rigidity theorems in 3-manifolds with nonnegative Ricci or scalar curvature. In this minicourse, I will discuss how one may extend these results to stable minimal hypersurfaces in 4-manifolds. I will start with a few examples illustrating various non-rigidity phenomena and prove a rigidity theorem under the combination of sectional and scalar curvature conditions.

Hodge theory of almost complex 4-manifolds – Part 1/2

  • Speaker: Joana Cirici
  • Time: 11:20-12:20
  • Abstract: In this mini-course we will review the Hodge theory of compact complex surfaces from a new viewpoint: we will prove general properties for almost complex 4-manifolds, and observe how these collapse to well-known and strong properties the case of complex surfaces. Along the way, we will define Hodge numbers for arbitrary almost complex manifolds and will also discuss obstructions to integrability using Hodge and homotopy theory.

 

Scalar Curvature: Structure and Regularity – Part 2/2

  • Speaker: Robin Neumayer
  • Time: 2:00-3:00
  • Abstract: A broad theme in geometric analysis aims to understand the geometric structure of Riemannian manifolds that satisfy constraints on curvature. Over the past 25 years, a rather complete picture has been given for the structure and a priori regularity of Riemannian manifolds with lower bounds on Ricci curvature, as well as their limit spaces. On the other hand, when one assumes only a lower bound on scalar curvature, the trace of the Ricci curvature, the situation is much less well understood. The scalar curvature asymptotically governs the volumes of balls of small radii, and arises in various settings in general relativity and differential geometry. This mini-course will focus on the structure and regularity of Riemannian manifolds with scalar curvature lower bounds.

Stable minimal hypersurfaces in 4-manifolds (Part 2/2)

  • Speaker: Chao Li
  • Time: 3:10-4:10
  • Abstract: Stable minimal surfaces in 3-manifolds have been extensively used to probe the topology of the ambient space. Important results include rigidity theorems in 3-manifolds with nonnegative Ricci or scalar curvature. In this minicourse, I will discuss how one may extend these results to stable minimal hypersurfaces in 4-manifolds. I will start with a few examples illustrating various non-rigidity phenomena and prove a rigidity theorem under the combination of sectional and scalar curvature conditions.

Hodge theory of almost complex 4-manifolds – Part 2/2

  • Speaker: Joana Cirici
  • Time: 4:20-5:20
  • Abstract: In this mini-course we will review the Hodge theory of compact complex surfaces from a new viewpoint: we will prove general properties for almost complex 4-manifolds, and observe how these collapse to well-known and strong properties the case of complex surfaces. Along the way, we will define Hodge numbers for arbitrary almost complex manifolds and will also discuss obstructions to integrability using Hodge and homotopy theory.