**GNOSC SEMINAR**

**G****romov, Hanke, Sormani, and Yu's**

**N****ot**** O****nly**** S****calar ****C****urvature**** S****eminar**

**Spring 2022 **

**SEMINAR INFO:**

**Each meeting has a pair of talks: an introductory colloquium and a related seminar on a recent advance in geometry. Not all the talks will concern scalar curvature but perhaps many will. The pair of talks will be followed by a discussion both in person and online. Everyone is welcome to join the session live (****zoom****) or watch the recorded videos of the talks later (****IHES channel****).**

**Zoom starts at 9:00 am NYC Time: **

**click here to register and receive link**** **** **

**IHES Carmin TV Channel Info: **

**https://www.carmin.tv/en/collections/not-only-scalar-curvature-seminar**

**or click a title below for direct access to its video.**

**SCHEDULE: (Spring 2022)**

**January 21: Gromov and Zeidler **

**Colloquium**** (Jan 21 9:00 am EST) **

**Invitation to Scalar Curvature**

Abstract: There are three great domains in geometry, which lie on the boundary of "soft" and "rigid":

(1) low dimensional, especially 4-dimensional topology/geometry;

(2) symplectic topology/geometry;

(3) scalar curvature bounded from below.

I will try to elucidate in my lecture common features of these three and explain the specificity of the problems arising with the scalar curvature.

**Seminar**** (Jan 21 10:15am EST) **

**Scalar and mean curvature comparison via the Dirac operator**

Abstract: I will explain a spinorial approach towards a comparison and rigidity principle involving scalar and mean curvature for certain warped products over intervals. This is motivated by recent scalar curvature comparison questions of Gromov, in particular distance estimates under lower scalar curvature bounds on Riemannian bands $M \times [-1,1]$ and Cecchini's long neck principle. I will also exhibit applications of these techniques in the context of the positive mass theorem with arbitrary ends. This talk is based on joint work with Simone Cecchini.

**February 4: Sormani and Li**

**Colloquium (Feb 4 9:00 am EST) **

**Sequences of Manifolds with Lower Bounds on their Scalar Curvature**

Abstract: If one has a weakly converging sequence of manifolds with a uniform lower bound on their scalar curvature, what properties of scalar curvature persist on the limit space? What additional hypotheses might be added to provide stronger controls on the limit space? What hypotheses are required to be guaranteed that a sequence has a converging subsequence? What notions of convergence might we consider? I will present examples and conjectures.

**Seminar (Feb 4 10:15am EST) **

**Scalar Curvature and the Dihedral Rigidity Conjecture**** **

Abstract: In 2013, Gromov proposed a geometric comparison theorem for metrics with nonnegative scalar curvature, formulated in terms of the dihedral rigidity phenomenon for Riemannian polyhedrons. In this talk, I will discuss recent progress towards this conjecture, and its connection to other rigidity/almost rigidity questions in scalar curvature.

**February 18: Weinberger and Yu**

**Colloquium (Feb 18 9:00 am EST)**

**Some introductory remarks on the Novikov conjecture**

Abstract: I will explain a few simple ideas about the Novikov conjecture and related problems.

**Seminar (Feb 18 10:15am EST)**

**The Novikov conjecture and scalar curvature**

Abstract: I will discuss some connections between the Novikov conjecture and scalar curvature.

**March 4: Hanke and Ebert ****(9am ****NYC, 3****pm ****Paris, 10:00 pm China)**

**Colloquium (Mar 4 9:00 am ****NYC 3:00 pm Paris, 10:00 pm China)**

**Surgery, bordism and scalar curvature**

Abstract: One of the most influential results in scalar curvature geometry, due to Gromov-Lawson and Schoen-Yau, is the construction of metrics with positive scalar curvature by surgery. Combined with powerful tools from geometric topology, this has strong implications for the classification of such metrics. We will give an overview of the method and point out some recent developments.

**Seminar (Mar 4 10:15am ****NYC Time, 4:15pm Paris, 11:15 pm China)**

**Rigidity theorems for the diffeomorphism action on spaces of positive scalar curvature**

Abstract: The diffeomorphism group, Diff(M), of a closed manifold acts on the space, R+(M), of positive scalar curvature metrics. For a basepoint, g, we obtain an orbit map

σg : Diff(M) → R+(M)

which induces a map on homotopy groups

(σg)∗ : π∗(Diff(M)) → π∗( R+(M)).

The rigidity theorems from the title assert that suitable versions of the map (σg)∗ factors through certain bordism groups. A special case of our main result asserts that (σg)∗ has finite image if M is simply connected, stably parallelizable, and of dimension at least 6. The results of this talk are from joint work of the speaker with Oscar Randal–Williams.

**March 18: Sesum and Burkhardt-Guim (9am NYC, 2pm Paris, 9pm China)**

**Colloquium (Mar 18 9:00am ****NYC, 2:00pm Paris, 9:00pm China)**

Abstract: We will introduce the Ricci flow and discuss singularity formation in the flow. Ancient solutions occur as singularity models in the flow and we will mention some instances in which they can be well understood.

**Seminar (Mar 18 10:15am ****NYC, 3:15 pm Paris, 10:15 pm China)**

**Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach**

Abstract: We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.

**April 1: Xie and Wang ****(9am ****NYC****, ****3****pm ****Paris, ****9pm China)**

**Colloquium (Apr 1 9:00 am ****NYC****, 3:00 ****pm Paris****) **

**Comparisons of scalar curvature, mean curvature and dihedral angle, and their applications**

Abstract: In this talk, I will review Gromov’s dihedral extremality and rigidity conjectures regarding comparisons of scalar curvature, mean curvature and dihedral angle for compact manifolds with corners. These conjectures have profound implications in geometry and mathematical physics such as the positive mass theorem. I will explain the recent work on positive solutions to these conjectures, and some related applications (such as a positive solution to the Stoker conjecture). The talk is based on my joint works with Jinmin Wang and Guoliang Yu.

**Seminar (Apr 1 10:15am ****NYC****, ****4****:15 pm ****Paris****)**

**Gromov's dihedral rigidity conjecture and index theory on manifolds with corners**

Abstract: In this talk, I will explain the key ideas of our recent work on positive solutions to Gromov's dihedral extremality and rigidity conjectures. One of the main ingredients is a new index theory on manifolds with corners (more generally, manifolds with polytope singularities), which is of independent interest on its own. Our approach is based on the analysis of differential operators arising from conical metrics. The comparison of dihedral angles enters into the study of these differential operators in an essential way. This is based on my joint works with Zhizhang Xie and Guoliang Yu.

**April 8: Zhang and Su ****(9am NYC, 3pm Paris, 9pm China)**

**Colloquium (Apr 8 9:00 am ****NYC****, 3:00 pm ****Paris****) **

**Deformations of Dirac operators**

Abstract: Deformations of Dirac operators have played important roles in various aspects in geometry and topology. In this expository talk I will discuss some of these applications.

**Seminar (Apr 8 10:15am ****NYC****, ****4:15**** pm ****Paris****) **

**Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds**

Abstract：Let $(M,g^{TM})$ be a noncompact complete Riemannian manifold of dimension $n$, and $F\subseteq TM$ be an integrable subbundle of $TM$. Let $g^F=g^{TM}|_{F}$ be the restricted metric on $F$ and $k^F$ be the associated leafwise scalar curvature. Let $f:M\to S^n(1)$ be a smooth area decreasing map along $F$, which is locally constant near infinity and of non-zero degree. We show that if $k^F> {\rm rk}(F)({\rm rk}(F)-1)$ on the support of ${\rm d}f$, and either $TM$ or $F$ is spin, then $\inf (k^F)<0$. As a consequence, we prove Gromov's sharp foliated $\otimes_\varepsilon$-twisting conjecture. Using the same method, we also extend two famous non-existence results due to Gromov and Lawson about $\Lambda^2$-enlargeable metrics (and/or manifolds) to the foliated case. This is a joint work with Xiangsheng Wang and Weiping Zhang.

**April 22: Shi and Zhu ****(9am NYC, 3pm Paris, 9pm China)**

**Colloquium (Apr 22 9:00am NYC, ****3:00 pm ****Paris) **

**Quasi-local mass and geometry of scalar curvature**

Abstract: Quasi-local mass is a basic notion in General Relativity. Geometrically, it can be regarded as a geometric quantity of a boundary of a 3-dimensional compact Riemannian manifold. Usually, it is in terms of area and mean curvature of the boundary. It is interesting to see that some of quasi-local masses, like Brown-York mass, have deep relation with Gromov’s fill-in problem of metrics with scalar curvature bounded below. In this talk, I will discuss these relations. This talk is based on some of my recent joint works with J.Chen, P.Liu, W.L. Wang , G.D.Wei and J. Zhu etc.

**Seminar (Apr 22 10:15am NYC, 4:15**** pm ****Paris) **

**Incompressible hypersurface, positive scalar curvature and positive mass theorem**

Abstract: In this talk, I will introduce a positive mass theorem for asymptotically flat manifolds with fibers (like ALF and ALG manifolds) under an additional but necessary incompressible condition. I will also make a discussion on its connection with surgery theory as well as quasi-local mass and present some new results in these fields. This talk is based on my recent work joint with J. Chen, P. Liu and Y. Shi.

**April 2****9****: Bär and Cecchini (9am NYC, 3pm Paris, 9pm China)**

**Colloquium (Apr 2****9 ****9:00 am NYC, 3:00 pm Paris) **

**Boundary value problems for Dirac operators**

Abstract: This introduction to boundary value problems for Dirac operators will

not focus on analytic technicalities but rather provide a working

knowledge to anyone who wants to apply the theory, i.e. in the study of

positive scalar curvature. We will systematically study "elliptic

boundary conditions" and discuss the following topics:

* typical examples of such boundary conditions

* regularity of the solutions up to the boundary

* Fredholm property and index computation

* geometric applications

**Seminar (Apr 2****9 ****10:15am NYC, 4:15 pm Paris)**

**Distance estimates in the spin setting and the positive mass theorem**

Abstract:The positive mass theorem states that a complete asymptotically Euclidean manifold of nonnegative scalar curvature has nonnegative ADM mass. It relates quantities that are defined using geometric information localized in the Euclidean ends (the ADM mass) with global geometric information on the ambient manifold (the nonnegativity of the scalar curvature). It is natural to ask whether the positive mass theorem can be ``localized’’, that is, whether the nonnegativity of the ADM mass of a single asymptotically Euclidean end can be deduced by the nonnegativity of the scalar curvature in a suitable neighborhood of E.

I will present the following localized version of the positive mass theorem in the spin setting. Let E be an asymptotically Euclidean end in a connected Riemannian spin manifold (M,g). If E has negative ADM-mass, then there exists a constant R > 0, depending only on the geometry of E, such that M must either become incomplete or have a point of negative scalar curvature in the R-neighborhood around E in M. This gives a quantitative answer, for spin manifolds, to Schoen and Yau's question on the positive mass theorem with arbitrary ends. Similar results have recently been obtained by Lesourd, Unger, and Yau without the spin condition in dimensions <8 assuming Schwarzschild asymptotics on the end E. I will also present explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end E. The bounds obtained are reminiscent of Gromov's metric inequalities with scalar curvature. This is joint work with Rudolf Zeidler.

**May 13: Ammann and LeBrun ****(9am NYC, 3pm Paris, 9pm China)**

**Colloquium (M****ay 13 ****9:00 am ****NYC****, 3:00 pm ****Paris****) **

**Yamabe constants, Yamabe invariants and Gromov-Lawson surgeries**

In this talk I want to study the (conformal) Yamabe constant of a closed

Riemannian (resp. conformal) manifold and how it is affected by

Gromov-Lawson type surgeries. This yields information about Yamabe

invariants and their bordism invariance. So far the talk gives an

overview over older results of mine in joint work with M. Dahl, N.

Große, E. Humbert, and N. Otoba.

A further consequence is that many results about the space of metrics

with positive scalar curvature may be generalized to spaces of metrics

with Yamabe constant above $t>0$. In particular we will present the

following Chernysh-Walsh type result which is work in progress: if $N^n$

arises from $M^n$ by a surgery of dimension $k\in\{2,3,\ldots,n-3\}$,

then a Gromov-Lawson type surgery construction defines a homotopy

equivalence from the space of metrics on $M$ with Yamabe constant above $t\in (0,\Lambda_{n,k})$ to the corresponding space on $N$.

**Seminar (May ****13 ****10:15am ****NYC****, 4:15 pm ****Paris****) **

**Yamabe Invariants, Weyl Curvature, and the Differential Topology of 4-Manifolds **

Abstract: The behavior of the Yamabe invariant, as defined in Bernd Ammann’s previous lecture, differs strangely in dimension 4 from what is seen in any other dimension. These peculiarities not only manifest themselves in the context of the usual scalar curvature, but also occur in connection with certain curvature quantities that are built out of the scalar and Weyl curvatures. In this lecture, I will explain how the Seiberg-Witten equations not only allow one compute the Yamabe invariant for many interesting 4-manifolds, but also give rise to other curvature inequalities.

I will then point out applications of these results to the theory of Einstein manifolds, while also highlighting related open questions that have so far proved impervious to these techniques.

**June 17: Breiner and Stern ****(9am NYC, 3pm Paris, 9pm China)**

**Colloquium (June 17 9:00 am ****NYC**** 3:00 pm ****Paris****) **

**Harmonic maps to metric spaces and applications**

Abstract: Harmonic maps are critical points for the energy and existence and compactness results for harmonic maps have played a major role in the advancement of geometric analysis. Gromov-Schoen and Korevaar-Schoen developed a theory of harmonic maps into metric spaces with non-positive curvature in order to address rigidity problems in geometric group theory. In this talk we consider harmonic maps into metric spaces with upper curvature bounds. We will define these objects, state some key results, and highlight their application to rigidity and uniformization problems. We finish the talk by discussing some recently determined Bochner inequalities for maps from (possibly) singular domains.

**Seminar (June 17 10:15am ****NYC****, 4:15 pm ****Paris) **

**Level set methods for scalar curvature on three-manifolds**

Abstract: We'll discuss a circle of ideas developed over the last few years relating scalar curvature lower bounds to the structure of level sets of solutions to certain geometric pdes on 3-manifolds. We'll describe applications to the study of 3-manifold geometry and initial data sets in general relativity, drawing comparisons with other pde methods for detecting the influence of scalar curvature.

**Upcoming: ****Miao and Jauregui**** (postponed to Fall 2022)**

**Colloquium (May 27 9:00 am NYC, 3:00 pm Paris) **

**Some inequalities relating mass, capacity and boundary mean curvature**

Abstract: I will discuss some inequalities that relate the capacity of the boundary of an asymptotically flat 3-manifold with nonnegative scalar curvature, the total mass of the manifold, and the geometry of the boundary 2-surface.

**Seminar (May 27 10:15am NYC, 4:15 pm Paris)**

**Capacity in low regularity, with connections to general relativity**

Abstract: The concept of capacity generalizes from Euclidean space to complete Riemannian manifolds, and even to suitable classes of metric spaces. I will discuss recent joint work with Raquel Perales and Jim Portegies on understanding capacity in local integral current spaces, describing the behavior of capacity when the background spaces converge in the pointed Sormani--Wenger intrinsic flat sense. Connections between the main results and the concept of total mass in general relativity will be discussed.

**ORGANIZERS:**

**Misha Gromov**** (CIMS and IHES)**

**Bernhard Hanke**** (Uni Augsburg)**

**Christina Sormani**** (CUNYGC and Lehman)**

**Guoliang Yu**** (Texas A&M)**

**THIS SPRING 2022 WEBPAGE MAINTAINED BY ****SORMANIC@GMAIL.COM**