# Noncommutative Geometry Seminar

## Spring 2023

Date: | January 25, 2023 |

Time: | 2:00pm |

Location: | BLOC 302 |

Speaker: | Shiqi Liu, Penn State |

Title: | Analysis of the hypoelliptic Laplacian |

Abstract: | Invented by Jean-Michel Bismut, the hypoelliptic Laplacian is the centerpiece of a new type of index theory. It leads to a remarkable trace formula and reveals completely new insights into geometry and representation theory. However, the subject relies on analysis that is made difficult by the non-ellipticity of the hypoelliptic Laplacian operator. Recently, with techniques from noncommutative geometry, we have shown that the hypoelliptic Laplacian is actually elliptic under a new calculus. This will significantly reduce the complexity of the analysis. This is joint work with N. Higson, E. MacDonald, F. Sukochev, and D. Zanin. |

Date: | February 2, 2023 |

Time: | 09:30am |

Location: | ZOOM |

Speaker: | Christina Sormani, CUNYGC/Lehman College |

Title: | Currents on Metric Spaces and Intrinsic Flat Convergence |

Abstract: | First I will provide a brief introduction to Ambrosio-Kirchheim’s Theory of Currents on Metric Spaces. Then I will review joint work with Wenger defining integral current spaces and intrinsic flat convergence. This will provide sufficient background needed to follow the talk of Antoine Song. |

Date: | February 2, 2023 |

Time: | 10:45am |

Location: | ZOOM |

Speaker: | Antoine Song, Caltech |

Title: | Spherical Plateau problem and applications |

Abstract: | I will discuss an area minimization problem in certain quotients of the Hilbert sphere by countable groups. An early version of that setting appears in Besson-Courtois-Gallot’s work on the entropy inequality. As an application of this minimization problem, we obtain some stability results. For instance, consider a closed surface of genus at least $2$ endowed with a Riemannian metric $g$, and let $(S,g)$ be its universal cover. After normalizing $g$ so that the volume entropy of $(S,g)$ is $1$, it is well-known that the first eigenvalue $\lambda$ is at most $\frac14$, and equality holds if $g$ is a hyperbolic metric. The hyperbolic plane is in fact stable: if $\lambda$ is close to the upper bound $\frac14$, then $(S,g)$ is close to the hyperbolic plane in a Benjamini-Schramm topology. |

Date: | February 15, 2023 |

Time: | 2:00pm |

Location: | BLOC 302 |

Speaker: | Bo Zhu, TAMU |

Title: | Metric structure of Riemannian manifolds involving their curvature |

Abstract: | In this talk, we will first introduce some basic concepts(size of Riemannian manifolds) and discuss their relationships with other metric structure quantities. Then, we will introduce some conjectures in this field, which are introduced by Gromov and Yau. Finally, we will present our progress on the estimate of the Uryson 1-width for Riemannian manifolds with uniformly positive mean curvature. |

Date: | February 16, 2023 |

Time: | 09:30am |

Location: | ZOOM |

Speaker: | Xin Zhou, Cornell |

Title: | Recent Developments in Constant Mean Curvature Hypersurfaces I |

Abstract: | We will survey some recent existence theory of closed constant mean curvature hypersurfaces using the min-max method. We hope to discuss some old and new open problems on this topic as well. |

Date: | February 16, 2023 |

Time: | 10:45am |

Location: | ZOOM |

Speaker: | Liam Mazurowski, Cornell |

Title: | Recent Developments in Constant Mean Curvature Hypersurfaces II |

Abstract: | Continuing from the previous talk, we will first discuss two min-max theorems for constructing prescribed mean curvature hypersurfaces in non-compact spaces. The first concerns the existence of prescribed mean curvature hypersurfaces in Euclidean space, and the second concerns the existence of constant mean curvature hypersurfaces in asymptotically flat manifolds. Following this, we will introduce the half-volume spectrum of a manifold M. This is analogous to the usual volume spectrum, except that we restrict to p-sweepouts whose slices are each required to enclose half the volume of M. We use the Allen-Cahn min-max theory to find hypersurfaces associated to the half-volume spectrum. Each hypersurface consists of a constant mean curvature component enclosing half the volume of M plus a (possibly empty) collection of minimal components. |

Date: | February 17, 2023 |

Time: | 1:00pm |

Location: | ZOOM |

Speaker: | Jesus Sanchez Jr, Washington University in St Louis |

Title: | Equivariant Local Index Theory for Lie Groupoids |

Abstract: | In recent work, Higson and Yi developed a new perspective on Getzler’s symbol calculus, reinterpreting the latter in terms of a convolution algebra of sections of a multiplicative vector bundle over the tangent groupoid of a spin manifold. In joint work with S. Liu, Y. Loizides, and A.R.H.S. Sadegh we generalize the construction in two directions; to the equivariant setting, and to the adiabatic groupoid of any Lie groupoid. We discuss applications including an equivariant longitudinal local index theorem for Lie groupoids with a closed space of units. |

Date: | February 22, 2023 |

Time: | 4:00pm |

Location: | BLOC 628 |

Speaker: | Ruobing Zhang, Princeton University |

Title: | (Joint with Topology Seminar) Metric geometry of Einstein 4-manifolds with special holonomy |

Abstract: | Studying the geometry of Einstein spaces and their generalizations is a major theme in current areas of differential geometry. This talk will introduce major developments in the metric geometry and global analysis of Einstein 4-manifolds. Recently geometric studies have culminated in a complete understanding of Einstein 4-manifolds with special holonomy.
Among the tremendous progress in this area, an essential chapter is to understand how Einstein moduli spaces can be compactified and how Einstein metrics can degenerate, especially when volume collapse occurs. Understanding collapsing geometry is much more challenging than studying the non-collapsing case so that little progress was made during the past three decades. I will particularly summarize recent breakthrough in this direction on the accurate characterization of singularity formation and the complete classification of degeneration limits on all scales. |

Date: | March 1, 2023 |

Time: | 2:00pm |

Location: | BLOC 302 |

Speaker: | Bo Zhu, TAMU |

Title: | Metric structure of Riemannian manifolds involving their curvature II |

Date: | March 2, 2023 |

Time: | 09:30am |

Location: | ZOOM |

Speaker: | Simon Brendle, Columbia |

Title: | Scalar curvature rigidity of polytopes |

Abstract: | We will discuss a scalar curvature rigidity theorem for convex polytopes. The proof uses the Fredholm theory for Dirac operators on manifolds with boundary, as well as an estimate due to Fefferman and Phong. |

Date: | March 2, 2023 |

Time: | 10:45am |

Location: | ZOOM |

Speaker: | Florian Johne, Columbia |

Title: | Intermediate curvature and a generalization of Geroch's conjecture |

Abstract: | In this talk we explain a non-existence result for metrics of positive m-intermediate curvature (a notion of curvature reducing to positive Ricci curvature for m=1, and positive scalar curvature for m=n−1) on closed orientable manifolds with topology Nn=Mn−m×Tm for n≤7. Our proof uses a slicing constructed by minimization of weighted areas, the associated stability inequality, and estimates on the gradients of the weights and the second fundamental form of the slices. This is joint work with Simon Brendle and Sven Hirsch. |

Date: | March 3, 2023 |

Time: | 1:00pm |

Location: | ZOOM |

Speaker: | Jinmin Wang , Texas A&M University |

Title: | Index pairing and quantitative control of scalar curvature |

Abstract: | In this talk, I will speak about an index theoretical approach to a series of Gromov’s questions related to positive scalar curvature. Our approach is to study the index pairing of the Dirac operator and vector bundles. The concrete construction of index pairing reveals a quantitative control of positive scalar curvature, which results from a new uncertainty principle for Fourier transforms. This is based on joint work with Zhizhang Xie and Guoliang Yu. |

Date: | April 26, 2023 |

Time: | 2:00pm |

Location: | BLOC 302 |

Speaker: | Yasuyuki Kawahigashi, University of Tokyo |

Title: | TBA |

Date: | April 27, 2023 |

Time: | 4:00pm |

Location: | BLOC 302 |

Speaker: | Koichi Oyakawa, Vanderbilt University |

Title: | TBA |