Student Geometric & Analysis Seminar (Spring 2023): Scalar Curvature and Dihedral Rigidity

Organizers: Tsz-Kiu Aaron Chow, Jingbo Wan, Yipeng Wang

Time: Wednesdays 3:10 pm—5:10 pm

Location:  Room 528 at Columbia Math Department

This seminar dedicates to providing a survey about some known results and open questions related to the structure of Riemannian manifolds with scalar curvature bounded from below. Such manifolds seem could be characterized using metric geometric objects such as polyhedron comparison arguments via dihedral angles between adjacent weakly mean convex faces.  And those phenomena motivate the program proposed by Gromov, to study the delicate effect of scalar curvature on Riemannian manifolds through the shape of more singular metric spaces, namely the Dihedral Rigidity Conjecture. At the same time, we will introduce various analytic techniques such as constructing free boundary minimal hypersurfaces, establishing index theory for manifolds with corners, and solving certain elliptic boundary value problems. Applications in low-dimensional topology and mathematical physics will also be discussed if time permits. 

References: 

[B] Brendle Scalar Curvature Rigidity for Convex Polytopes

[BB11] B ̈ar, Ballmann Boundary Value Problems for Elliptic Differential Operators of First Order

[BB13] B ̈ar, Ballmann Guide to Boundary Value Problems for Dirac-Type Operators 

[G14] Gromov Dirac and Plateau Billiards in Domain with Corners

[G19] Gromov Four Lectures on Scalar Curvature

[G22] Gromov Convex Polytopes, Dihedral Angles, Mean Curvature, and Scalar Curvature 

[GL] Gromov, Lawson Spin and Scalar Curvature

[Li17] Li A Polyhedron Comparison Theorem for 3-Manifolds with Positive Scalar Curvature 

[Li19] Li The Dihedral Rigidity Conjecture for n-Prisms 

[WXY] Wang, Xie, Yu On Gromov’s Dihedral Extremality and Rigidity Conjectures


Title and Abstract (Spring 2023)

Date

Speaker

Title and abstract

Jan 25th


Organizational Meeting

Feb 1st

Yipeng Wang

The C^0 Limit Theorem for Scalar Curvature 

Show/hide Abstracts

The theory of taking limits of Riemannian manifolds is extensively studied given that certain curvature quantities are uniformly bounded from below. For instance, the theory of Alexandrov spaces is developed to study limit spaces with sectional curvature bounded from below, and similar situations apply to spaces with bounded Ricci curvature via Cheeger-Colding-Naber Theory. However, the corresponding results on scalar curvature are still far from being understood. Recently, Gromov proposed to study spaces with positive scalar curvature in terms of polyhedra and use it to prove the compactness theorem. We will discuss a simpler proof proposed via Ricci flow introduced by Bamler. At the same time, we will motivate the corresponding rigidity conjecture and some related questions about scalar curvature. 

 


Feb 8th

Yipeng Wang

TBA

Feb 15th

Tin Yau Tsang (UC Irvine)

TBA

Feb 22nd

Jingbo Wan

TBA

Mar 1st

Jingbo Wan

TBA

Mar 8th

Jingbo Wan

TBA

Mar 15th

No seminar (Spring break)

Mar 22nd


TBA

Mar 29th


TBA

Apr 5th


TBA

Apr 12th


TBA

Apr 19th


TBA

Apr 26th


TBA