Abstracts

Shuli Chen: Positive scalar curvature metrics and aspherical summands 

It has been a classical question which manifolds admit Riemannian metrics with positive scalar curvature. A closed manifold is called aspherical if it has contractible universal cover. It has been conjectured since the 80s that all closed aspherical manifolds do not admit metric with positive scalar curvature. By works of Gromov—Lawson, Chodosh—Li, Gromov, and Chodosh—Li—Liokumovich, for n = 3, 4, 5, if a closed n-manifold M admits a map of nonzero degree to a closed aspherical n-manifold, then M does not admit a metric with positive scalar curvature. We prove for n = 3,4,5 that the connected sum of a closed aspherical n-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. In particular, a special case of our result answers a question of Gromov. More generally, for n=3,4,5, we give a partial classification result of complete n-manifolds with positive scalar curvature and prove that no complete n-manifold with a non-zero degree map to a closed aspherical n-manifold can have positive scalar curvature. This result confirms the validity of Gromov's non-compact domination conjecture for closed aspherical manifolds of dimensions 3, 4, and 5. This is joint work with Jianchun Chu and Jintian Zhu.

Jintao Deng: The K-theory of relative group C*-algebras 

Abstract: For the Dirac operator on a manifold with boundary, we can define its relative higher index which lies in the K-theory of the relative group $C^*$-algebra of fundamental groups. The relative Baum-Connes conjecture claims that a certain relative assembly map is an isomorphism. It provides an algorithm of the computation of the K-theory of relative group C*-algebras. In my talk, I will present several cases when the relative assembly maps are isomorphic (or injective). The strong relative Novikov conjecture states that the relative assembly map is injective. I will also talk about the applications of the strong Novikov conjecture in geometry and topology, especially about the relative higher signatures of manifolds with boundary. This is based on a joint work with G. Tian, Z. Xie and G. Yu.

Conghan Dong: Stability for the Positive Mass Theorem and the Penrose inequality 

The Positive Mass Theorem in dimension 3 states that if $(M^3, g)$ is asymptotically flat and has nonnegative scalar curvature, then its ADM mass $m(g)$ satisfies $m(g) \geq 0$, and equality holds only when $(M, g)$ is the flat Euclidean 3-space $\mathbb{R}^3$. The Penrose inequality in dimension 3 states that if $(M^3, g)$ is asymptotically flat with an outermost minimal boundary, and has nonnegative scalar curvature, then its ADM mass $m(g)$ satisfies $m(g) \geq \sqrt{\frac{\mathrm{Area} (\partial M)} {16 \pi} }$, and equality holds only when $(M, g)$ is isometric to the Schwarzschild 3-manifold. In this talk, we show that the Euclidean 3-space and the Schwarzschild 3-manifold are stable for the Positive Mass Theorem and the Penrose inequality, respectively, in the Gromov-Hausdorff topology modulo negligible spikes. This talk is based on joint work with Antoine Song.

Elliott Gesteau: Bootstrapping spectral gaps of hyperbolic spin surfaces 

In this talk, I will describe new bounds on Laplace and Dirac spectral gaps of compact orientable hyperbolic surfaces and orbifolds equipped with a spin structure. The central technique underlying these bounds is known as the conformal bootstrap, which originated in the study of conformal field theories in theoretical physics. This method produces a variety of nearly optimal upper bounds on Laplace spectral gaps, as well as Dirac spectral gaps conditioned on the former, over various spaces of surfaces, some of which are sensitive to not only topological but also geometric properties like hyperellipticity. This is joint work with Sridip Pal, David Simmons-Duffin and Yixin Xu.

Roberta Iseppi: BV spectral triples: towards a Batalin-Vilkovisky quantization

In this seminar we will argue why noncommutative geometry provides a very interesting mathematical framework to the study of the Batalin-Vilkovisky formalism. After a brief review of the BV construction, where we will recall its original physical motivations and the main steps that compose it, we will then focus on the mathematical setting: noncommutative geometry and the key notion of spectral triple. Beyond their relevance from a purely geometrical perspective, spectral triples also play a relevant role in the mathematical formalization of gauge theories. Knowing the motivations and having described the tools, we will then present our main result: we will see how the BV construction can be inserted in the formalism of noncommutative geometry. A key role will be played by the "BV spectral triple", whose notion we introduce to encode in a finite-dimensional noncommutative manifold the ghost sector of a BV-extended gauge theory. 

Hyun Chul Jang: Instability of minimal volume entropy rigidity for products of rank-one symmetric spaces 

The volume entropy is a geometric invariant of a Riemannian manifold defined by the volume growth of geodesic balls in the universal cover with respect to the pull-back metric. For some special cases, the minimal entropy rigidity results has been established in the literature. In this talk, I will discuss a couple of problems regarding the minimal entropy rigidity of products of rank-one symmetric spaces. Firstly, I will talk about the uniqueness of the spherical plateau solution for such manifolds, which has a natural connection to show the minimal entropy rigidity. Secondly, I will demonstrate the instability of the minimal entropy rigidity with a counterexample, in contrast to the case of locally symmetric spaces.

Demetre Kazaras: Scalar curvature and isolated codimension 2 collapse 

This talk is about the structure of Riemannian 3-manifolds satisfying a scalar curvature lower bound. I will discuss a "drawstring" construction, which modifies a manifold near a given curve, reducing its length and incurring only negligible damage to a scalar curvature lower bound. This extends ideas of Basilio-Sormani and Lee-Naber-Neumayer, and produces unexpected examples with relevance to a few areas, including almost-rigidity problems. This is based on joint work with Kai Xu.

Yi Lai: Riemannian and Kahler flying wing steady Ricci solitons 

Abstract: Steady Ricci solitons are fundamental objects in the study of Ricci flow, as they are self-similar solutions and often arise as singularity models. Classical examples of steady solitons are the most symmetric ones, such as the 2D cigar soliton, the O(n)-invariant Bryant solitons, and Cao’s U(n)-invariant Kahler steady solitons. Recently we constructed a family of flying wing steady solitons in any real dimension n≥3, which confirmed a conjecture by Hamilton in n=3. In the Kahler case, we also constructed a family of Kahler flying wing steady gradient solitons with positive curvature for any complex dimension n≥2, which answers a conjecture by H.-D. Cao in the negative. This is partly collaborated with Pak-Yeung Chan and Ronan Conlon.

Yangyang Li: Existence and regularity of anisotropic minimal hypersurfaces 

Anisotropic area, a generalization of the area functional, arises naturally in models of crystal surfaces. The regularity theory for its critical points, anisotropic minimal hypersurfaces, is significantly more challenging than the area functional case, mainly due to the absence of a monotonicity formula. In this talk, I will discuss how one can overcome this difficulty and obtain a smooth anisotropic minimal surface and optimally regular minimal hypersurfaces for elliptic integrands in closed Riemannian manifolds through min-max construction. This confirms a conjecture by Allard in 1983. If time permitted, I will also discuss how this could be connected to the minimal surface theory. The talk is based on joint work with Guido De Philippis and Antonio De Rosa.

Annachiara Piubello: Mass and Riemannian Polyhedra 

In this talk, we will first give a brief introduction to the ADM mass in general relativity. This concept represents the total mass of an isolated system, modeled by asymptotically flat manifolds, describing gravitating systems where all the matter is contained inside a a compact region. We will then show a formula for this mass as the limit of the total mean curvature plus the total defect of dihedral angle of the boundary of large polyhedra. In the special case of coordinate cubes, we will show an integral formula relating the n-dimensional mass with a geometrical quantity that determines the (n-1)-dimensional mass. This is joint work with Pengzi Miao.

Jesus Sanchez: Refinements in Local Index Theory 

The Atiyah-Singer index theorem is one of the great pillars in global analysis connecting the fields of analysis, topology, and geometry. To date, there are various approaches to the subject of index theory, local index theory having its focus on deriving characteristic forms from the local coefficients of a geometric elliptic operator. For a spin Riemannian manifold, the natural choice of operator is the Dirac operator on spinors. In this talk we will discuss recent refinements on the local geometry of the Dirac operator as well as the local geometry of a spin Riemannian manifold.

Guo Chuan Thiang: Macroscopic index theory from physics 

In 2000, Kitaev introduced a geometric trace formula to describe the unusual quantization found in macroscopic Quantum Hall Effect experiments. I will explain that the mathematical basis lies in pre-NCG work of Helton-Howe-Carey-Pincus on traces of commutators. In modern language, this realizes a local index theory of geometric operators, which trivializes on strictly local operators, but survives in the large-scale, or coarse-geometric limit. Joint work with J. Xia (SUNY Buffalo) and M. Ludewig (Regensburg).

Ryo Toyota: Quantum Gromov-Hausdorff convergence of spectral truncations of discrete groups. 

Spectral truncation is a new scope in noncommutative geometry to approximate the entire noncommutative geometric structure from the partial date of the spectrum. This idea is motivated by the practical issue in quantum physics that we don't have access to all infinitely many possible energy levels of the physical system. On some classes of spectral triples arising from groups, we discuss how the restricted data on the finite range of spectrum (energy level) approximate the entire observables in terms of the quantum analogue of Gromov-Hausdorff convergence.

Zhaoting Wei: Residue currents of coherent sheaves via superconnection 

Residue currents are generalizations of one complex variable residues and can be thought of as currents representing ideals of holomorphic functions. In 2007 Andersson and Wulcan constructed residue current for coherent sheaves. Their construct requires a global resolution by holomorphic vector bundles of the given coherent sheaf, which may not exist on general (non-projective) complex manifolds. I this talk I will introduce the idea of residue current and also show how to get rid of the above restriction with the help of anti-holomorphic flat superconnections.

Rudolf Zeidler: Positive mass theorems and distance estimates for spin initial data sets 

Abstract: We will present variants of the spacetime positive mass theorem in the spin setting: Firstly, its conclusion `$E \geq  |P|$’ holds for a single asymptotically flat (`AF’) end in a spin initial data set satisfying the dominant energy condition, even if the initial data set has other arbitrary complete (but not necessarily AF) ends. Secondly, we can also treat incomplete situations provided that the incompleteness is controlled in terms of the Riemannian distance on the underlying manifold. Moreover, there are versions of these results in the presence of a compact boundary under suitable boundary conditions. On a technical level they are based on a combination of Witten’s spinor proof of the positive mass theorem with recent ideas originating in approaches to Gromov’s `band with estimates’. However, to deal with the general initial data case, a curious new trick is required—we need to formally introduce a second timelike direction to the spinor bundle in addition to the one implicit in the initial data set. Based on joint work with S. Cecchini and M. Lesourd.