Organizers: TszKiu 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 lowdimensional 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 DiracType 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 3Manifolds with Positive Scalar Curvature
[Li19] Li The Dihedral Rigidity Conjecture for nPrisms
[WXY] Wang, Xie, Yu On Gromov’s Dihedral Extremality and Rigidity Conjectures
Date 
Speaker 
Title and abstract 

Jan 25th 

Organizational Meeting 
Feb 1st 
Yipeng Wang 
The C^0 Limit Theorem for Scalar Curvature Show/hide AbstractsThe 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 CheegerColdingNaber 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 