IAS Working Group 2018

Official IAS Working Group Participants:

Michael Eichmair (Mon-Fri)

Bernhard Hanke (Mon-Fri)

Lan-Hsuan Huang (Mon-Wed)

Sajjad Lakzian (Mon-Fri)

Yashar Memarian (Mon-Fri)

Pengzi Miao (Mon-Fri)

Jacobus Portegies (Mon-Fri)

Raquel Perales (Mon-Fri)

Rick Schoen (Mon-Fri)

Guofang Wei (Mon-Fri)

Schedule:

(all are welcome at talks in Simonyi Hall 101)

(some talks/discussions are closed due to the lack of availability of a large enough room)

Monday

9:00-9:50 Simonyi Hall 101: Christina Sormani: Introduction to Scalar Curvature and Convergence (video)

10:00-11:00 Closed Questioning of Christina Sormani about intrinsic flat convergence and scalar curvature [Sormani-Scalar]

11:00-12:00 Closed Discussion lead by Misha Gromov concerning his Conjectures on Scalar Curvature [Gromov-96] [Gromov-Dirac]

12:00-1:30 Discussion at Lunch

1:30-2:30 Closed Questioning of Rick Schoen about Scalar Curvature, Minimal Surfaces and Mass [Schoen-Yau-PMT]

3:00-4:00 Closed Presentation by Jacobus Portegies Intrinsic Flat Convergence and Filling Volume [PS] [Matveev-Portegies]

4:00-5:00 Discussion at Tea

5:00-6:00 Closed Discussion: reviewing the ideas of the day [Oo] [Llarull] [Goette-Semmelmann]

Tuesday

10:00-11:00 Closed Presentation by Chao Li on Scalar Curvature, Polyhedral Comparison and Singularities (recent video)

11:30-12:00 Closed Presentation by Bernhard Hanke on an Example of a 7D manifold with no PSC metric

12:00-1:30 Discussion at Lunch

1:30-2:30 Simonyi Hall 101: Misha Gromov on Notions of Scalar Curvature (video)

3:00-4:00 Simonyi Hall 101: Colloquium by Rick Schoen on Positive Mass Theorem in all Dimensions (recent video at SCGP)

4:00-5:00 Discussion at Tea

5:00-6:00: Closed Discussion by Pengzi Miao and Chao Li on Shi-Tam vs Li’s Simplicial Rigidity (also [Marques-Neves Sphere Rigidity])

Wednesday

9:30-10:30 Simonyi Hall 101: Lan Hsuan Huang on Stability of the Positive Mass Thm (video)

11:00-12:00 Simonyi Hall 101: Raquel Perales on Limits of Tori with almost Nonnegative Scalar Curvature

12:00-1:00 Discussion at lunch

1:45-2:45 Princeton Fine Hall 1201: Pengzi Miao on Scalar Curvature and Volume [Miao-Tam]

3:00-4:00: Princeton Geometric Analysis Seminar 314: Guofang Wei on Convergence and Eigenvalue Estimates for Integral Curvature

4:30-5:30 Princeton Fine Hall 214: Michael Eichmair and Otis Chodosh Scalar Curvature and Isoperimetry [CESY]

6:00-7:30 IAS White Levy Room: Mathematical Conversation by Gromov

Thursday

9:15-10:15 Closed Discussion lead by Guofang Wei on Tangent Cones of Limit Spaces

10:30-11:30 Closed Presentation by Dan Lee and Jeff Jauregui on Semicontinuity of Mass

11:30-12:00: Continued Closed Dicussions lead by Yashar Memarian

12:00-1:30 Discussion at lunch

1:30-3:00 Simonyi Hall 101: Colloquium by Bernhard Hanke on Index Theory and Flexibility in Positive Scalar Curvature Geometry (video)

3:00-4:00: Closed Discussion lead by Otis Chodosh on a Scalar Splitting Theorem [CEM]

4:00-5:00 Discussion at tea

5:00-6:00: Closed Discussion: reviewing the ideas of the day

Friday

9:30-10:30 Simonyi Hall 101: Brian Allen on Tori of Almost Nonnegative Scalar curvature [AHPPW]

11:00-12:00 Simonyi Hall 101: Open Discussion on Techniques for Intrinsic Flat Convergence and Scalar Curvature [HLS]

12:00-1:30 Discussion at lunch

1:30-2:30 Simonyi Hall 101: Sajjad Lakzian on Smooth Convergence Away from Singularities [LS] [L]

2:30-3:30 Simonyi Hall 101: Open Discussion Formation of Research Teams

3:30-5:00 Discussion at tea

Additional Invited Participants: (funded by various grants including NSF DMS 1309360)

Brian Allen (funded by USMA) [Th-Fri]

Luca Ambrozio (IAS)

Mauricio Bustamante (funded by Augsburg) [Mon-Fri]

Alessandro Carlotto (at IAS) [Mon-Fri]

Otis Chodosh (IAS and Princeton) [Wed-Th]

Fernando Coda Marques (IAS and Princeton) [Mon-Fri]

Brian Frieden (IAS)

Benedikt Hunger (funded by Augsburg) [Mon-Fri]

Jeff Jauregui (Union, funded by NSF) [Th]

Nicos Kapouleas (IAS and Brown)

Demetre Kazaras (funded by NSF)

Anusha Krishnan (U Penn)

Dan Lee (CUNY, funded by NSF)

Chao Li (IAS and Princeton)

Yevgeny Liokumovich (IAS)

Siyuan Lu (Rutgers)

Elena Maeder-Baumdicker (Princeton)

Andrea Malchiodi (IAS)

Robin Neumeyer (IAS)

Andre Neves (IAS and Chicago)

Daniel Rade (funded by Augsburg)

Shengwen Wang (JHU) [Tu]

Ruobing Zhang (Princeton and Stony Brook)

Xin Zhou (IAS and UCSB)

Other members of IAS and faculty at Princeton and Rutgers are welcome to attend events in the lecture hall but contact Christina Sormani if you also wish to attend some closed sessions as there is very little space in the seminar room.

Over the year there will be biweekly meetings with those participants of our workshop who will be at IAS longer [Schoen, Huang and Sormani]. Some of these later meetings might be held at the Courant Institute to include participants located in NYC [Gromov and Lakzian]. Others may join in by skype. The questions are open ended and should lead to a variety of papers by various subgroups of participants in the coming years.

Description of the Topic:

When studying sequences of Riemannian manifolds with nonnegative sectional curvature, one obtains limits under Gromov-Hausdorff convergence to Alexandrov spaces with Alexandrov curvature bounded below (as studied in the work of Gromov and Burago-Gromov-Perelman). For sequences of manifolds with nonnegative Ricci curvature one applies metric measure notions of convergence and obtains metric measure spaces satisfying a variety of notions of generalized nonnegative Ricci curvature (as in the work of Cheeger-Colding, Lott-Villani, Sturm, and Ambrosio-Gigli-Savare). It is not yet clear what notion of convergence is best suited to Riemannian manifolds with nonnegative scalar curvature, although the notion of intrinsic flat convergence introduced by Sormani-Wenger in [SW-JDG] using the work of Ambrosio-Kirchheim seems promising. Even less well understood is an appropriate definition for a generalized notion of nonnegative scalar curvature on a limit space.

Gromov has proposed a variety of conjectures in this direction in [Gromov-12] [Gromov-Dirac] [Gromov-Plateau]. Sormani has refined the statements of some of these conjectures and stated others, including key examples in [Sormani-Scalar]. With the new Schoen-Yau proof of the Positive Mass Theorem in all dimensions, there is a new approach to understanding nonnegative scalar curvature in settings with lower regularity [Schoen-Yau-PMT]. The Positive Mass Theorem states that a complete asymptotically flat manifold with nonnegative scalar curvature has nonnegative ADM mass and the ADM mass is zero iff the manifold is isometric to Euclidean space. Lee-Sormani conjectured that if the ADM mass is small the manifold is close in the intrinsic flat sense to Euclidean space [Lee-Sormani-1]. This has been proven in various special cases by Lee-Sormani, Huang-Lee-Sormani, and Sormani-Stavrov using techniques by Lakzian-Sormani. There are teams currently working on Gromov's conjectured Stability of the Scalar Torus Rigidity Theorem working in the same special cases. However, all progress in this direction has required additional hypotheses. See the Intrinsic Flat Convergence Website for links to all these papers. While these recent results provide insight, they do not address the fundamental question of defining a notion of generalized nonnegative scalar curvature on a limit space.

We propose to bring together a team of experts on a variety of methods of convergence including senior mathematicians and postdocs to work on this problem. We wish to hold this workshop at IAS in the Fall of 2018, because the IAS program on {\em Variational Methods in Geometry} has already convened many experts in scalar curvature. This will be the ideal opportunity to explore the techniques of Marques and Neves, and of Schoen and Huang, all of whom will be at IAS this year. With this funding we can bring in two additional European experts on scalar curvature (Hanke and Eichmair) as well as the experts on a variety of notions of convergence (Portegies, Lakzian, Perales, Memarian, and Wei),

Outcomes:

M Gromov and C Sormani ``IAS Emerging Topics: Scalar Curvature and Convergence'' Report to IAS,

M Gromov ``Scalar Curvature of Manifolds with Boundaries" arxiv preprint

C Sormani and IAS Participants "Conjectures on Convergence and Scalar Curvature" (arxiv)