# 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: | March 31, 2023 |

Time: | 2:00pm |

Location: | BLOC 624 |

Speaker: | Weichen Gu, University of New Hampshire |

Title: | On the zeta-function of some non-commutative semigroups |

Abstract: | In this talk we introduce a framework on the zeta-functions of some non-commutative semigroups, including the Thompson semigroup and braid semigroups. A generalization $\kappa(n)$ of the Möbius function related to the Thompson group is given, and we will use $\kappa(n)$ to extend the zeta-function of the hompson semigroup to the complex half plane with real part greater than 1/2, and prove that the real pole closest to 1 is a simple pole. |

Date: | April 12, 2023 |

Time: | 2:00pm |

Location: | BLOC 302 |

Speaker: | Jiayin Pan, University of California, Santa Cruz |

Title: | Nonnegative Ricci curvature, fundamental groups, and asymptotic geometry |

Abstract: | We survey the recent developments on fundamental groups of open manifolds with nonnegative Ricci curvature. By studying the equivariant asymptotic geometry, we obtain structure results of fundamental groups such as finite generation and virtual nilpotency/abelianness. |

Date: | April 14, 2023 |

Time: | 1:00pm |

Location: | ZOOM |

Speaker: | Haluk Sengun, University of Sheffield |

Title: | Local theta correspondence via C*-algebras of groups |

Abstract: | Theta correspondence is a major theme in the theory of automorphic forms and in representation theory. The local version of the correspondence sets up a bijection between certain subsets of admissible duals of suitable pairs of reductive groups. There are two special cases in which the correspondence is known to enjoy extra features, the ‘equal rank’ case where temperedness is preserved and the ‘stable range’ case where unitarity is preserved. In joint work with Bram Mesland (Leiden), we show that in these special cases, the local theta correspondence is actually given by a Morita equivalence of certain C*-algebras. There are interesting applications and some global questions that follow this result. Time permitting, I will discuss some of these. |

Date: | April 19, 2023 |

Time: | 2:00pm |

Location: | BLOC 302 |

Speaker: | Arturo Jaime, University of Hawaii |

Title: | Finite complexity rank for C*-algebras |

Abstract: | Motivated by two earlier ideas of decomposability in coarse geometry (Guentner, Tessera, and Yu) and dynamics (Guentner, Willett, and Yu) and nuclear dimension of C*-algebras (Winter and Zacharias), Willett and Yu introduced the notion of complexity rank for C*-algebras: essentially the number of times a C*-algebra can be cut into well-interacting parts until you reach something finite-dimensional. In this talk we will discuss results relating a weaker notion of complexity rank to nuclear dimension one and real rank zero. We then discuss torsion vanishing in K_1 for unital C*-algebras having finite complexity rank at most one. Finally we fit these results into the context of the broader work of Willett and Yu on the UCT. This is based on joint work with Rufus Willett. |

Date: | April 25, 2023 |

Time: | 4:00pm |

Location: | BLOC 302 |

Speaker: | Yasuyuki Kawahigashi, University of Tokyo |

Title: | (Colloquium) Quantum symmetries in operator algebras and mathematical physics |

Abstract: | A notion of symmetry is fundamental in mathematics and physics. A new type of "quantum" symmetry has emerged in operator algebras, quantum groups, quantum invariants in low dimensional topology, integrable systems, vertex operator algebras, quantum field theory and condensed matter physics since 1980's. I will present development in these areas from an operator algebraic viewpoint. Emphasis is given to the Jones theory of subfactors, chiral conformal field theory and two-dimensional topological order. No knowledge on these topics is assumed as a prerequisite. |

Date: | April 27, 2023 |

Time: | 4:00pm |

Location: | BLOC 302 |

Speaker: | Koichi Oyakawa, Vanderbilt University |

Title: | Bi-exactness of relatively hyperbolic groups |

Abstract: | Bi-exactness is an analytic property of groups defined by Ozawa and of fundamental importance to the study of operator algebras. In this talk, I will show that finitely generated relatively hyperbolic groups are bi-exact if and only if all peripheral subgroups are bi-exact. This is a generalization of Ozawa's result which claims that finitely generated relatively hyperbolic groups are bi-exact if all peripheral subgroups are amenable. |

Date: | May 1, 2023 |

Time: | 2:00pm |

Location: | BLOC 302 |

Speaker: | Jintao Deng, University of Waterloo |

Title: | The $K$-theory of Roe algebras and the coarse Baum-Connes conjecture |

Abstract: | The coarse Baum-Connes conjecture claims that a certain assembly is an isomorphism. It has important applications in the study of the existence of a metric with positive scalar curvature and the Novikov conjecture on the homotopy invariance of the higher signature on a manifold. In this talk, I will talk about the Roe algebras which encode the large-scale geometry of a metric space. The higher index of an elliptic operator is an element of the K-theory of this algebra. The coarse Baum-Connes conjecture provides an algorithm to compute its $K$-theory. I will talk about our recent result that the coarse Baum-Connes conjecture holds for the relative expanders constructed by Arzhantseva and Tessera which is not coarsely embeddable into Hilbert space. I will also talk about a recent result on the equivariant coarse Baum-Connes conjecture. |