Monday, June 27th, 2022
Lorenzo Mazzieri
Time: 9:00 AM - 10:00 AM
Location: SCGP 102
Title: Geometric inequalities and potential theory
Speaker: Lorenzo Mazzieri
Abstract: Geometric inequalities and potential theoryAbstract: We describe through some selected examples an approach based on potential theory toward the proof of relevant geometric inequalities, holding in classical and curved frameworks. Time permitting, we also discuss some applications of interest in general relativity, including the positive mass theorem and the Riemannian Penrose inequality.
Title: Geometric inequalities and potential theory
Speaker: Lorenzo Mazzieri
Abstract: Geometric inequalities and potential theoryAbstract: We describe through some selected examples an approach based on potential theory toward the proof of relevant geometric inequalities, holding in classical and curved frameworks. Time permitting, we also discuss some applications of interest in general relativity, including the positive mass theorem and the Riemannian Penrose inequality.
Christina Sormani
Time: 10:00 AM - 11:00 AM
Location: SCGP 102
Title: Compactness and Stability Conjectures
Speaker: Christina Sormani
Abstract: Christina SormaniTitle: Compactness and Stability ConjecturesAbstract: Gromov has conjectured that a sequence of compact three dimensional Riemannian manifolds with nonnegative scalar curvature converges in the intrinsic flat sense to a limit space with generalized nonnegative scalar curvature. He has also conjectured the stability of the scalar torus rigidity theorem: that a sequence of three tori with scalar >-1/j has a subsequence which converges in the intrinsic flat sense to a flat torus. We will survey examples and results in this direction including joint work with Allen and Perales introducing a notion we call VADB convergence that has been applied by Cabrera Pacheco, Perales, and Ketterer to prove the stability of the scalar torus rigidity theorem. We will suggest a number of open problems applying VADB convergence to test both these conjectures and others.
Title: Compactness and Stability Conjectures
Speaker: Christina Sormani
Abstract: Christina SormaniTitle: Compactness and Stability ConjecturesAbstract: Gromov has conjectured that a sequence of compact three dimensional Riemannian manifolds with nonnegative scalar curvature converges in the intrinsic flat sense to a limit space with generalized nonnegative scalar curvature. He has also conjectured the stability of the scalar torus rigidity theorem: that a sequence of three tori with scalar >-1/j has a subsequence which converges in the intrinsic flat sense to a flat torus. We will survey examples and results in this direction including joint work with Allen and Perales introducing a notion we call VADB convergence that has been applied by Cabrera Pacheco, Perales, and Ketterer to prove the stability of the scalar torus rigidity theorem. We will suggest a number of open problems applying VADB convergence to test both these conjectures and others.
Daniel Stern
Time: 11:30 AM - 12:30 PM
Location: SCGP 102
Title: Harmonic maps and extremal Schrodinger operators on higher-dimensional manifolds
Speaker: Daniel Stern
Abstract: I'll discuss recent progress on the existence theory for harmonic maps from arbitrary Riemannian manifolds of dimension >2 to a large class of target manifolds, including 3-manifolds of positive Ricci curvature and manifolds of dimension > 3 with positive isotropic curvature. As a special case, every Riemannian manifold (M^n,g) of dimension n>2 admits a family of nontrivial stationary harmonic maps u_k to the standard spheres S^k for k>2, smooth away from a singular set of dimension at most n-7 for k sufficiently large. I'll explain how these maps give rise to solutions of an optimization problem for Schrodinger operators related to work of Grigor'yan-Netrusov-Yau and Grigor'yan-Nadirashvili-Sire, generalizing to higher dimensions the maximization of Laplace eigenvalues on surfaces of fixed conformal type. (Based on joint work with Mikhail Karpukhin.)
Title: Harmonic maps and extremal Schrodinger operators on higher-dimensional manifolds
Speaker: Daniel Stern
Abstract: I'll discuss recent progress on the existence theory for harmonic maps from arbitrary Riemannian manifolds of dimension >2 to a large class of target manifolds, including 3-manifolds of positive Ricci curvature and manifolds of dimension > 3 with positive isotropic curvature. As a special case, every Riemannian manifold (M^n,g) of dimension n>2 admits a family of nontrivial stationary harmonic maps u_k to the standard spheres S^k for k>2, smooth away from a singular set of dimension at most n-7 for k sufficiently large. I'll explain how these maps give rise to solutions of an optimization problem for Schrodinger operators related to work of Grigor'yan-Netrusov-Yau and Grigor'yan-Nadirashvili-Sire, generalizing to higher dimensions the maximization of Laplace eigenvalues on surfaces of fixed conformal type. (Based on joint work with Mikhail Karpukhin.)
Jinmin Wang
Time: 2:00 PM - 3:00 PM
Location: SCGP 102
Title: Dihedral rigidity conjecture and Stoker's problem
Speaker: Jinmin Wang
Abstract: The Stoker Conjecture states that the dihedral angles of a convex Euclidean polyhedron completely determine the angles of each face. In this talk, I will present my recent work joint with Zhizhang Xie and Guoliang Yu that answers positively to the Stoker Conjecture in all dimensions. Our work proves a more general dihedral rigidity theorem, which concerns the comparison of scalar curvature, mean curvature, and dihedral angles for convex polyhedrons, or more general, manifolds with polytope boundary. We use index theory on manifolds with polytope boundary and the Dirac operator methods.
Title: Dihedral rigidity conjecture and Stoker's problem
Speaker: Jinmin Wang
Abstract: The Stoker Conjecture states that the dihedral angles of a convex Euclidean polyhedron completely determine the angles of each face. In this talk, I will present my recent work joint with Zhizhang Xie and Guoliang Yu that answers positively to the Stoker Conjecture in all dimensions. Our work proves a more general dihedral rigidity theorem, which concerns the comparison of scalar curvature, mean curvature, and dihedral angles for convex polyhedrons, or more general, manifolds with polytope boundary. We use index theory on manifolds with polytope boundary and the Dirac operator methods.
Zhizhang Xie
Time: 3:00 PM - 4:00 PM
Location: SCGP 102
Title: On Gromov’s dihedral extremality and rigidity conjectures
Speaker: Zhizhang Xie
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for manifolds with corners (and more generally manifolds with polyhedral boundary). They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. In this talk, I will explain some recent work on positive solutions to these conjectures. The talk is based on my joint papers with Jinmin Wang and Guoliang Yu.
Title: On Gromov’s dihedral extremality and rigidity conjectures
Speaker: Zhizhang Xie
Abstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature, mean curvature and dihedral angle for manifolds with corners (and more generally manifolds with polyhedral boundary). They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. In this talk, I will explain some recent work on positive solutions to these conjectures. The talk is based on my joint papers with Jinmin Wang and Guoliang Yu.
Tuesday, June 28th, 2022
Richard Bamler
Time: 9:00 AM - 10:00 AM
Location: SCGP 102
Title: Towards a theory of Ricci flow in dimension 4 (and higher)
Speaker: Richard Bamler
Abstract: The Ricci flow (with surgery) has proven to be a powerful tool in the study of 3-dimensional topology — its most prominent application being the verification of the Poincaré and Geometrization Conjectures by Perelman about 20 years ago. Since then further research has led to a satisfactory understanding of the flow and surgery process in dimension 3. In dimensions 4 and higher, on the other hand, Ricci flows have been understood relatively poorly and a surgery construction seemed distant. Recently, however, there has been some progress in the form of a new compactness and partial regularity theory for higher dimensional Ricci flows. This theory relies on a new geometric perspective on Ricci flows and provides a better understanding of the singularity formation and long-time behavior of the flow. In dimension 4, in particular, it may eventually open up the possibility of a surgery construction or a construction of a "flow through singularities". The goal of this talk will be to describe this new compactness and partial regularity theory and the new geometric intuition that lies behind it. Next, I will focus on 4-dimensional flows. I will present applications towards the study of singularities of such flows and discuss several conjectures that provide a possible picture of a surgery construction in dimension 4. Lastly, I will discuss potential topological applications.
Title: Towards a theory of Ricci flow in dimension 4 (and higher)
Speaker: Richard Bamler
Abstract: The Ricci flow (with surgery) has proven to be a powerful tool in the study of 3-dimensional topology — its most prominent application being the verification of the Poincaré and Geometrization Conjectures by Perelman about 20 years ago. Since then further research has led to a satisfactory understanding of the flow and surgery process in dimension 3. In dimensions 4 and higher, on the other hand, Ricci flows have been understood relatively poorly and a surgery construction seemed distant. Recently, however, there has been some progress in the form of a new compactness and partial regularity theory for higher dimensional Ricci flows. This theory relies on a new geometric perspective on Ricci flows and provides a better understanding of the singularity formation and long-time behavior of the flow. In dimension 4, in particular, it may eventually open up the possibility of a surgery construction or a construction of a "flow through singularities". The goal of this talk will be to describe this new compactness and partial regularity theory and the new geometric intuition that lies behind it. Next, I will focus on 4-dimensional flows. I will present applications towards the study of singularities of such flows and discuss several conjectures that provide a possible picture of a surgery construction in dimension 4. Lastly, I will discuss potential topological applications.
Christos Mantoulidis (no recording)
Time: 10:00 AM - 11:00 AM
Location: SCGP 102
Title: Decomposing PSC 4-manifolds
Speaker: Christos Mantoulidis
Abstract: We will discuss a decomposition result involving 0- and 1-surgeries for closed, oriented, topologically PSC 4-manifolds.
Title: Decomposing PSC 4-manifolds
Speaker: Christos Mantoulidis
Abstract: We will discuss a decomposition result involving 0- and 1-surgeries for closed, oriented, topologically PSC 4-manifolds.
Claude LeBrun
Time: 11:30 AM - 12:30 PM
Location: SCGP 102
Title: Kodaira Dimension and the Yamabe Problem, Revisited
Speaker: Claude LeBrun
Abstract: Dimension four provides a surprisingly idiosyncratic setting for the interplay between scalar curvature and differential topology. This peculiarity becomes especially pronounced when discussing the Yamabe invariant (or “sigma constant”) of a smooth compact manifold; and Seiberg-Witten theory makes this especially apparent for those 4-manifolds that arise as compact complex surfaces. For compact complex surfaces of Kaehler type, I showed in the late 1990s that the sign of the Yamabe invariant is always determined by the Kodaira dimension, and moreover calculated the Yamabe invariant exactly in all cases where it is non-positive. In this talk, I will describe recent joint work with Michael Albanese that generalizes these results to all complex surfaces of non-Kaehler type. However, the complex surfaces of class VII actually violate many of the expected patterns, and navigating around this hazard represents a key aspect of our story.
Title: Kodaira Dimension and the Yamabe Problem, Revisited
Speaker: Claude LeBrun
Abstract: Dimension four provides a surprisingly idiosyncratic setting for the interplay between scalar curvature and differential topology. This peculiarity becomes especially pronounced when discussing the Yamabe invariant (or “sigma constant”) of a smooth compact manifold; and Seiberg-Witten theory makes this especially apparent for those 4-manifolds that arise as compact complex surfaces. For compact complex surfaces of Kaehler type, I showed in the late 1990s that the sign of the Yamabe invariant is always determined by the Kodaira dimension, and moreover calculated the Yamabe invariant exactly in all cases where it is non-positive. In this talk, I will describe recent joint work with Michael Albanese that generalizes these results to all complex surfaces of non-Kaehler type. However, the complex surfaces of class VII actually violate many of the expected patterns, and navigating around this hazard represents a key aspect of our story.
YITP Event: Special seminar: Nat Tantivasadakarn (Harvard)
Time: 11:30 AM - 12:30 PM
Location:
Title: Pauli Stabilizer models based on abelian anyons theories
Abstract: stabilizers models are an appealing class of models due to its simplicity, transparent use for quantum error correction, and the surprising connection to topological phases of matter. However, so far the only known examples of topological Pauli stabilizer codes in two spatial dimensions in the literature are equivalent to a number of copies of the toric code. I will demonstrate that in fact, a Pauli stabilizer model can be constructed for every abelian anyon theory that admits a gapped boundary (the so-called twisted quantum double models), giving a vast simplification of previous models based on such theories, which are not Pauli. My primary example will be the construction of a code based on the double semion anyon theory using four-dimensional qudits.
This talk is based on arXiv:2112.11394
Note: the seminar will be held in person at the YITP Common Room, Math 6-125. But we have set up a Zoom link for remote participants:
Join Zoom Meeting
https://stonybrook.zoom.us/j/94206502475?pwd=S0ZDa0NuOTBCQjIza0V5VWkxOXovQT09
Meeting ID: 942 0650 2475
Passcode: 993742
Title: Pauli Stabilizer models based on abelian anyons theories
Abstract: stabilizers models are an appealing class of models due to its simplicity, transparent use for quantum error correction, and the surprising connection to topological phases of matter. However, so far the only known examples of topological Pauli stabilizer codes in two spatial dimensions in the literature are equivalent to a number of copies of the toric code. I will demonstrate that in fact, a Pauli stabilizer model can be constructed for every abelian anyon theory that admits a gapped boundary (the so-called twisted quantum double models), giving a vast simplification of previous models based on such theories, which are not Pauli. My primary example will be the construction of a code based on the double semion anyon theory using four-dimensional qudits.
This talk is based on arXiv:2112.11394
Note: the seminar will be held in person at the YITP Common Room, Math 6-125. But we have set up a Zoom link for remote participants:
Join Zoom Meeting
https://stonybrook.zoom.us/j/94206502475?pwd=S0ZDa0NuOTBCQjIza0V5VWkxOXovQT09
Meeting ID: 942 0650 2475
Passcode: 993742
David Maxwell
Time: 2:00 PM - 3:00 PM
Location: SCGP 102
Title: Sobolov-Class Asymptotically Hyperbolic Manifolds and the Yamabe Problem
Speaker: David Maxwell
Abstract: We consider asymptotically hyperbolic manifolds whose metrics have Sobolev-class regularity. Building on prior work by Allen, Isenberg, Lee, and Allen-Stavrov in the Hölder category, we introduce two new function spacesfor metrics potentially having a large amount of interior differentiability measured in Sobolev scales, but whose regularity implies only a Hölder continuous conformal structure. We establish Fredholm theorems for elliptic operators arising from metrics in these families.To demonstrate utility of our methods, we solve the Yamabe problem in this category. As a special limiting case, we show that the asymptotically hyperbolic Yamabe problem is solvable so long as the metric admits a $W^{1,p}$ conformal compactification, with $p$ greater than the dimension of the manifold.(Joint with Paul T. Allen, Lewis & Clark and John M. Lee, University of Washington)
Title: Sobolov-Class Asymptotically Hyperbolic Manifolds and the Yamabe Problem
Speaker: David Maxwell
Abstract: We consider asymptotically hyperbolic manifolds whose metrics have Sobolev-class regularity. Building on prior work by Allen, Isenberg, Lee, and Allen-Stavrov in the Hölder category, we introduce two new function spacesfor metrics potentially having a large amount of interior differentiability measured in Sobolev scales, but whose regularity implies only a Hölder continuous conformal structure. We establish Fredholm theorems for elliptic operators arising from metrics in these families.To demonstrate utility of our methods, we solve the Yamabe problem in this category. As a special limiting case, we show that the asymptotically hyperbolic Yamabe problem is solvable so long as the metric admits a $W^{1,p}$ conformal compactification, with $p$ greater than the dimension of the manifold.(Joint with Paul T. Allen, Lewis & Clark and John M. Lee, University of Washington)
Wednesday, June 29th, 2022
Joachim Lohkamp
Time: 10:00 AM - 11:00 AM
Location: SCGP 102
Title: Removal of Singularities
Speaker: Joachim Lohkamp
Abstract: Minimal hypersurfaces are important to study general scalar curvatureconstraints. The occurence of singularitiesis an infamous problem one may solve from an inductive dimensionalreduction scheme with built-in regularization. We will discuss thisparticular process and indicated some ways how it can be applied.
Title: Removal of Singularities
Speaker: Joachim Lohkamp
Abstract: Minimal hypersurfaces are important to study general scalar curvatureconstraints. The occurence of singularitiesis an infamous problem one may solve from an inductive dimensionalreduction scheme with built-in regularization. We will discuss thisparticular process and indicated some ways how it can be applied.
Dan Lee
Time: 11:30 AM - 12:30 PM
Location: SCGP 102
Title: The positive mass theorem with boundaries, complete ends, and scalar curvature shields
Speaker: Dan Lee
Abstract: In joint work with M. Lesourd and R. Unger, we give a non-spinor proof of the spacetime positive mass theorem with weakly outer trapped boundary. It turns out that this implies a version of the Riemannian positive mass theorem that does not require nonnegative scalar curvature everywhere, so long as the negative scalar curvature is “shielded” from the asymptotically flat end by a region with sufficiently positive scalar curvature. By an argument of Lesourd, Unger, and Yau, this latter fact also implies the Riemannian positive mass theorem for arbitrary complete ends.
Title: The positive mass theorem with boundaries, complete ends, and scalar curvature shields
Speaker: Dan Lee
Abstract: In joint work with M. Lesourd and R. Unger, we give a non-spinor proof of the spacetime positive mass theorem with weakly outer trapped boundary. It turns out that this implies a version of the Riemannian positive mass theorem that does not require nonnegative scalar curvature everywhere, so long as the negative scalar curvature is “shielded” from the asymptotically flat end by a region with sufficiently positive scalar curvature. By an argument of Lesourd, Unger, and Yau, this latter fact also implies the Riemannian positive mass theorem for arbitrary complete ends.
Talk by Valentina Prilepina
Time: 1:30 PM - 2:30 PM
Location:
Title: Thermal Stress Tensor Correlators through the Lens ofAdS/CFT: the OPE and Holography
Speaker: Valentina Prilepina in collaboration withRobin Karlsson, Andrei Parnachev, and Samuel Valach
Abstract: In this talk, I will work in the context of strongly coupled conformal field theories(CFTs) with a large central charge. In such theories, an important set of operators are the stress tensor and its composites, referred to as the multi-stress tensors. Here I will examine the OPE expansion of two-point functions of the stress tensor, restricting attention to the contributions due to single and double-stress tensor exchange in the OPE. Throughout, I will study CFTs in four space time dimensions. I will show that comparing these results to the holographic finite temperature two-point functions empowers us to read off CFT data beyond the leading order in the large central charge expansion. Specifically, this procedure allows us to extract corrections to the OPE coefficients that completely fix the near-lightcone behavior of the correlators. In addition, it yields the anomalous dimensions of the double-stress tensor operators. I will begin by describing our overall goal and methods. I will then proceed to the bulk side, discussing how to compute the stress tensor two-point function in a near-boundary expansion.Next, I will cross over to the CFT side and elucidate the OPE expansion of the stress tensor thermal two-point functions. Thereafter, I will compare these CFT results to their bulk counterparts and expound how to read off the CFT data.
Title: Thermal Stress Tensor Correlators through the Lens ofAdS/CFT: the OPE and Holography
Speaker: Valentina Prilepina in collaboration withRobin Karlsson, Andrei Parnachev, and Samuel Valach
Abstract: In this talk, I will work in the context of strongly coupled conformal field theories(CFTs) with a large central charge. In such theories, an important set of operators are the stress tensor and its composites, referred to as the multi-stress tensors. Here I will examine the OPE expansion of two-point functions of the stress tensor, restricting attention to the contributions due to single and double-stress tensor exchange in the OPE. Throughout, I will study CFTs in four space time dimensions. I will show that comparing these results to the holographic finite temperature two-point functions empowers us to read off CFT data beyond the leading order in the large central charge expansion. Specifically, this procedure allows us to extract corrections to the OPE coefficients that completely fix the near-lightcone behavior of the correlators. In addition, it yields the anomalous dimensions of the double-stress tensor operators. I will begin by describing our overall goal and methods. I will then proceed to the bulk side, discussing how to compute the stress tensor two-point function in a near-boundary expansion.Next, I will cross over to the CFT side and elucidate the OPE expansion of the stress tensor thermal two-point functions. Thereafter, I will compare these CFT results to their bulk counterparts and expound how to read off the CFT data.
Pengzi Miao
Time: 2:00 PM - 3:00 PM
Location: SCGP 102
Title: Mass and capacitary potential of asymptotically flat 3-manifolds
Speaker: Pengzi Miao
Abstract: I will discuss some relation between the mass and positive harmonic functions decaying to zero on asymptotically flat 3-manifolds.
Title: Mass and capacitary potential of asymptotically flat 3-manifolds
Speaker: Pengzi Miao
Abstract: I will discuss some relation between the mass and positive harmonic functions decaying to zero on asymptotically flat 3-manifolds.
David Wiygul
Time: 3:00 PM - 4:00 PM
Location: SCGP 102
Title: Asymptotic estimates for the Bartnik mass of small metric balls
Speaker: David Wiygul
Abstract: For all the various definitions of quasilocal mass in general relativity it is natural to investigate their behavior on regions shrinking to a point. In particular one can seek a Taylor expansion of the mass of geodesic balls of small radius and fixed center. I will present some recent estimates for the Bartnik mass in this regime, including one that is obtained by estimating the ADM mass of asymptotically flat static vacuum extensions with "small" Bartnik boundary data to second order in the data.
Title: Asymptotic estimates for the Bartnik mass of small metric balls
Speaker: David Wiygul
Abstract: For all the various definitions of quasilocal mass in general relativity it is natural to investigate their behavior on regions shrinking to a point. In particular one can seek a Taylor expansion of the mass of geodesic balls of small radius and fixed center. I will present some recent estimates for the Bartnik mass in this regime, including one that is obtained by estimating the ADM mass of asymptotically flat static vacuum extensions with "small" Bartnik boundary data to second order in the data.
Thursday, June 30th, 2022
Bernhard Hanke
Time: 9:00 AM - 10:00 AM
Location: SCGP 102
Title: Lipschitz rigidity for scalar curvature
Speaker: Bernhard Hanke
Abstract: Let M be a closed connected smooth spin manifold of even dimension n, let g be a Riemannian metric of regularity W^{1,p}, p > n, on M whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by n(n-1), and let f : (M,g) \to S^n be a 1-Lipschitz continuous map of non-zero degree to the standard round n-sphere.
Then f is a metric isometry.
This generalizes a result of Llarull (1998) and answers a question of Gromov (2019) in his ``four lectures’’. Our proof, which is a joint work with Simone Cecchini and Thomas Schick, combines spectral properties of Dirac operators for metrics with low regularity and twisted with Lipschitz bundles with the theory of quasiregular maps due to Reshetnyak.
Title: Lipschitz rigidity for scalar curvature
Speaker: Bernhard Hanke
Abstract: Let M be a closed connected smooth spin manifold of even dimension n, let g be a Riemannian metric of regularity W^{1,p}, p > n, on M whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by n(n-1), and let f : (M,g) \to S^n be a 1-Lipschitz continuous map of non-zero degree to the standard round n-sphere.
Then f is a metric isometry.
This generalizes a result of Llarull (1998) and answers a question of Gromov (2019) in his ``four lectures’’. Our proof, which is a joint work with Simone Cecchini and Thomas Schick, combines spectral properties of Dirac operators for metrics with low regularity and twisted with Lipschitz bundles with the theory of quasiregular maps due to Reshetnyak.
Guoliang Yu
Time: 10:00 AM - 11:00 AM
Location: SCGP 102
Title: Higher index theory and scalar curvature
Speaker: Guoliang Yu
Abstract: I will give an introduction to higher index theory of Dirac operators and its applications to scalar curvature.In particular, I will discuss my recent joint work with Shmuel Weinberger and Zhizhang Xie on higher index theory at infinityand its applications to Gromov's compactness conjecture on scalar curvature.
Title: Higher index theory and scalar curvature
Speaker: Guoliang Yu
Abstract: I will give an introduction to higher index theory of Dirac operators and its applications to scalar curvature.In particular, I will discuss my recent joint work with Shmuel Weinberger and Zhizhang Xie on higher index theory at infinityand its applications to Gromov's compactness conjecture on scalar curvature.
Demetre Kazaras
Time: 11:30 AM - 12:30 PM
Location: SCGP 102
Title: Comparison geometry and spacetime harmonic functions
Speaker: Demetre Kazaras
Abstract: Comparison theorems are the basis for our geometric understanding of Riemannian manifolds satisfying a given curvature condition. A remarkable example is the Gromov-Lawson toric band inequality, which bounds the distance between the two sides of a Riemannian torus-cross-interval with positive scalar curvature by a sharp constant inversely proportional to the scalar curvature's minimum. We will give a new qualitative version of this and similar band-type inequalities in dimension 3 using the notion of spacetime harmonic functions, which recently played the lead role in our recent proof of the positive mass theorem. Other applications include new versions of Bonnet-Meyer's diameter estimate for positive Ricci curvature manifolds and Llarull's theorem which do not require a completeness assumption. This is joint work with Sven Hirsch, Marcus Khuri, and Yiyue Zhang.
Title: Comparison geometry and spacetime harmonic functions
Speaker: Demetre Kazaras
Abstract: Comparison theorems are the basis for our geometric understanding of Riemannian manifolds satisfying a given curvature condition. A remarkable example is the Gromov-Lawson toric band inequality, which bounds the distance between the two sides of a Riemannian torus-cross-interval with positive scalar curvature by a sharp constant inversely proportional to the scalar curvature's minimum. We will give a new qualitative version of this and similar band-type inequalities in dimension 3 using the notion of spacetime harmonic functions, which recently played the lead role in our recent proof of the positive mass theorem. Other applications include new versions of Bonnet-Meyer's diameter estimate for positive Ricci curvature manifolds and Llarull's theorem which do not require a completeness assumption. This is joint work with Sven Hirsch, Marcus Khuri, and Yiyue Zhang.
Rudolf Zeidler
Time: 2:00 PM - 3:00 PM
Location: SCGP 102
Title: Distance estimates under lower scalar curvature bounds in the spin setting and beyond
Speaker: Rudolf Zeidler
Abstract: We will discuss several situations in which lower scalar curvature bounds can be used to infer certain distance estimates using the Dirac operator on spin manifolds. This includes Gromov’s question on the distance between the boundary components of „bands“, rigidity properties of warped products, and related questions in the context of Witten’s spinor proof of the positive mass theorem. Finally, we will contrast this with a recently obtained result in the non-spin setting concerning the non-existence of complete psc metrics on certain manifolds which admit a compact incompressible hypersurface without psc. Based on joint works with S. Cecchini and D. Räde.
Title: Distance estimates under lower scalar curvature bounds in the spin setting and beyond
Speaker: Rudolf Zeidler
Abstract: We will discuss several situations in which lower scalar curvature bounds can be used to infer certain distance estimates using the Dirac operator on spin manifolds. This includes Gromov’s question on the distance between the boundary components of „bands“, rigidity properties of warped products, and related questions in the context of Witten’s spinor proof of the positive mass theorem. Finally, we will contrast this with a recently obtained result in the non-spin setting concerning the non-existence of complete psc metrics on certain manifolds which admit a compact incompressible hypersurface without psc. Based on joint works with S. Cecchini and D. Räde.
Daniel Räde
Time: 3:00 PM - 4:00 PM
Location: SCGP 102
Title: Scalar and mean curvature comparison for Riemannian bands
Speaker: Daniel Räde
Abstract: We use $\mu$-bubbles to compare Riemannian bands in scalarcurvature, mean curvature and width to warped products over scalar flatmanifolds with $\log$-concave warping functions. Furthermore we explainhow these comparison results can be used to prove two conjectures due toGromov resp. Rosenberg and Stolz in dimensions $\leq 7$. Thistalk is, in part, based on joint work with S. Cecchini and R. Zeidler.
Title: Scalar and mean curvature comparison for Riemannian bands
Speaker: Daniel Räde
Abstract: We use $\mu$-bubbles to compare Riemannian bands in scalarcurvature, mean curvature and width to warped products over scalar flatmanifolds with $\log$-concave warping functions. Furthermore we explainhow these comparison results can be used to prove two conjectures due toGromov resp. Rosenberg and Stolz in dimensions $\leq 7$. Thistalk is, in part, based on joint work with S. Cecchini and R. Zeidler.
Friday, July 1st, 2022
Carla Cederbaum
Time: 9:00 AM - 10:00 AM
Location: Zoom
Title: Coordinates are messy
Speaker: Carla Cederbaum
Abstract: Asymptotically Euclidean initial data sets $(M,g,K)$ are characterized by the existence of asymptotic coordinates in which the Riemannian metric $g$ and second fundamental form $K$ decay to the Euclidean metric $\delta$ and to $0$ suitably fast, respectively. Provided their matter densities satisfy suitable integrability conditions, they have well-defined (ADM-)energy, (ADM-)linear momentum, and (ADM-)mass as was shown by Bartnik in 1986. To study their (ADM-)angular momentum and (BORT-)center of mass, one usually assumes the existence of so-called Regge—Teitelboim coordinates. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge—Teitelboim coordinates and explain other “non-features” of the Regge—Teitelboim coordinate conditions. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences of these findings for the definition of the center of mass, relying on joint work with Nerz and with Sakovich.
Title: Coordinates are messy
Speaker: Carla Cederbaum
Abstract: Asymptotically Euclidean initial data sets $(M,g,K)$ are characterized by the existence of asymptotic coordinates in which the Riemannian metric $g$ and second fundamental form $K$ decay to the Euclidean metric $\delta$ and to $0$ suitably fast, respectively. Provided their matter densities satisfy suitable integrability conditions, they have well-defined (ADM-)energy, (ADM-)linear momentum, and (ADM-)mass as was shown by Bartnik in 1986. To study their (ADM-)angular momentum and (BORT-)center of mass, one usually assumes the existence of so-called Regge—Teitelboim coordinates. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge—Teitelboim coordinates and explain other “non-features” of the Regge—Teitelboim coordinate conditions. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences of these findings for the definition of the center of mass, relying on joint work with Nerz and with Sakovich.
Zhongshan An
Time: 10:30 AM - 11:30 AM
Location: SCGP 102
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Speaker: Zhongshan An
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.
Title: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data
Speaker: Zhongshan An
Abstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation, as well as constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.