Talks and speakers

WEEK 1: courses

  • Advanced basics of Riemannian geometry
    • Thomas Richard, Université Paris Est-Créteil Val de Marne
    • Abstract: We will present some of the tools used by the more advanced lectures. The topics discussed will include : Gromov Hausdorff distance, comparison theorems for sectional and Ricci curvature, the Bochner formula and basics of Ricci flow.
       
  • Compactness and Finiteness Results for Gromov-Hyperbolic Spaces
    • Gilles Courtois/Gérard Besson, Institut de Mathématiques de Jussieu/Université Grenoble Alpes
    • Abstract: This is a series of lectures on Bishop--Gromov's type inequalities adapted to metric spaces. We consider the case of Gromov-hyperbolic spaces and draw consequences of these inequalities such as compactness and finiteness Theorems. This course is intended to be elementary in the sense that the necessary background is described in detail.
    • File: https://if-summer2021.sciencesconf.org/data/program/Cours_Varanasi_08_05_2021.pdf
       
  • CAT(k)-spaces
    • Philippe Castillon/Gérard Besson, Université de Montpellier/Université Grenoble Alpes
    • Abstract: The purpose of this course is to introduce the synthetic treatment of sectional curvature upper-bound on metric spaces. The basic idea of A.D. Alexandrov was to characterize the curvature bounds on the sectional curvature of a Riemannian manifold in term of properties of its distance function, and then to consider metric spaces with these properties. This approach turned out to be very fruitful and it found many applications, bringing geometric ideas to other settings.
      In this course we will introduce the metric spaces with a curvature upper-bound in the sense of Alexandrov, and derive some of their geometric properties. The subject is very vast and it is not possible to be exhaustive in the limited time of this course. We will concentrate on some properties, both local and global, emphasizing that these metric spaces share many properties with manifolds of bounded curvature.

  • An introduction to weak mean curvature flow
    • Felix Schulze, University of Warwick
    • Abstract: It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief introduction to smooth mean curvature flow and then discuss Brakke flows, their basic properties and how to establish existence via elliptic regularization. We will furthermore discuss tangent flows and regularity, and the interaction of Brakke flows with the level set flow. Time permitting, we will give an outlook on recent developments, including the proof of the mean convex neighborhood conjecture by Choi-Haslhofer-Hershkovits/Choi-Haslhofer-Hershkovits-White as well as progress towards establishing the existence of generic mean curvature flow.
       
       

WEEK 2: courses

  • Intrinsic Flat and Gromov-Hausdorff Convergence
    • Christina Sormani, CUNYGC and Lehman College
    • Abstract: We introduce various notions of convergence of Riemannian manifolds and metric spaces.  We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with uniform lower bounds on their scalar curvature.   We close the course by presenting methods and theorems that may be applied to prove these open questions including older techniques developed with Lakzian, with Huang and Lee, and with Portegies.  I will also present key new results of Allen and Perales.   Students and postdocs interested in working on these problems will be formed into teams. For a complete list of papers about intrinsic flat convergence see: https://sites.google.com/site/intrinsicflatconvergence/
  • What is the (essential) minimal volume?
    • Antoine Song, University of California at Berkeley
    • Abstract: I will discuss the notion of minimal volume and some of its variants. The minimal volume of a manifold is defined as the infimum of the volume over all metrics with sectional curvature between -1 and 1. Such an invariant is closely related to "collapsing theory", a far reaching set of results developed by Cheeger, Gromov, Fukaya and others to describe bounded sectional curvature metrics. Most of my talks will be focused on presenting the main aspects of this theory: thick-thin decomposition, F-structures and N-structures, collapsing constructions... Relations of the minimal volume to topological invariants will be explained, and some open questions will be mentioned.
       
  • Uniqueness of Weak Solutions to the Ricci Flow and Topological Applications
    • Richard Bamler, University of California at Berkeley
    • Abstract: I will present recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This proves the Generalized Smale Conjecture and gives an alternative proof of the Smale Conjecture, which was originally due to Hatcher. Second, we show that the space of metrics with positive scalar curvature on every 3-manifold is either contractible or empty. This completes work initiated by Marques.

      At the heart of our proof is a new uniqueness and stability theorem for singular Ricci flows. Singular Ricci flows can be seen as an improvement of Ricci flows with surgery, which were used in Perelman’s proof of the Poincaré and Geometrization Conjectures. The latter flows had the drawback that they were not uniquely determined by their initial data, as their construction depended on various auxiliary surgery parameters. Perelman conjectured that there must be a canonical, weak Ricci flow that automatically "flows through its singularities” at an infinitesimal scale. Our work on the uniqueness of singular Ricci flows gives an affirmative answer to Perelman's conjecture and allows the study of continuous families of singular Ricci flows, leading to the topological applications mentioned above.

      The lectures will roughly be structured as follows:
      (1) Preliminaries of Ricci flow, Blow-up analysis of singularities, Statement of the main results
      (2) Local stability Analysis
      (3) Comparing singular Ricci flows, Proof of the uniqueness and stability result
      (4) Continuous families of singular Ricci flows, Proof of the topological theorems.

  • Metric measure spaces satisfying Ricci curvature lower bounds
    • Andrea Mondino, University of Oxford
    • Abstract: The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who investigated the structure of spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach based on Optimal Transport was proposed by Lott-Villani and Sturm around ten years ago; via this approach, one can give a precise sense of what means for a non-smooth space (more precisely for a metric measure space) to satisfy a Ricci curvature lower bound and a dimensional upper bound. This approach has been refined in the last years by a number of authors (most notably Ambrosio-Gigli- Savarè) and a number of fundamental tools have now been established, permitting to give further insights in the theory and applications which are new even for smooth Riemannian manifolds. The goal of the lectures is to give an introduction to the theory and discuss some of the applications.

       

 WEEK 3: workshop

  • Compactness and partial regularity theory of Ricci flows in higher dimensions 
    • Richard Bamler, University of California at Berkeley
    • Abstract: We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow. Under a natural non-collapsing condition, this limiting flow is smooth on the complement of a singular set of parabolic codimension at least 4. We furthermore obtain a stratification of the singular set with optimal dimensional bounds depending on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds.       As an application we obtain a description of the singularity formation at the first singular time and a long-time characterization of immortal flows, which generalizes the thick-thin decomposition in dimension 3. We also obtain a backwards pseudolocality theorem and discuss several other applications.
                                                                                                                                                                      
  • Pointwise lower scalar curvature bounds for C0 metrics via regularizing Ricci flow
    • Paula Burkhardt-Guim, University of California at Berkeley
    • Abstract: We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C0 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 C0 initial data.
  • Knots, minimal surfaces and J-holomorphic curves
    • Jöel Fine, Université Libre de Bruxelles
    • Abstract: I will describe work in progress, parts of which are joint with Marcelo Alves. Let L be a knot or link in the 3-sphere. I will explain how one can count minimal surfaces in hyperbolic 4-space which have ideal boundary equal to L, and in this way obtain a knot invariant. In other words the number of minimal surfaces doesn’t depend on the isotopy class of the link. These counts of minimal surfaces can be organised into a two-variable polynomial which is perhaps a known polynomial invariant of the link, such as HOMFLYPT.
      “Counting minimal surfaces” needs to be interpreted carefully here, similar to how Gromov-Witten invariants “count” J-holomorphic curves. Indeed I will explain how this “minimal surface polynomial" can be seen as a Gromov-Witten invariant for the twistor space of hyperbolic 4-space. This leads naturally to a new class of infinite-volume 6-dimensional symplectic manifolds with well behaved counts of J-holomorphic curves. This gives more potential knot invariants, for knots in 3-manifolds other than the 3-sphere. It also enables the counting of minimal surfaces in more general Riemannian 4-manifolds, besides hyperbolic space.
       
  • A family of 3d steady gradient Ricci solitons that are flying wings
    • Yi Lai, University of California at Berkeley
    • Abstract: We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension n ≥ 4, we find a family of Z2 × O(n − 1)-symmetric but non-rotationally symmetric n-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.
     
  • Positive Scalar Curvature on Noncompact Manifolds and the Positive Mass Theorem
    • Martin Lesourd, Harvard University
    • Abstract: The study of positive scalar curvature on noncompact manifolds has seen significant progress in the last few years. A major role has been played by Gromov's results and conjectures, and in particular the idea to use surfaces of prescribed mean curvature (as opposed to minimal surfaces). Having the classic positive mass theorem of Schoen-Yau in mind, we describe a new positive mass theorem for manifolds that allows for possibly non asymptotically flat ends, points of incompleteness, and regions negative scalar curvature. The proof is based on surfaces with prescribed mean curvature, and gives an alternative proof of the Liouville theorem conjectured by Schoen-Yau, which was recently proved by Chodosh-Li. This is joint with R.Unger and S-T. Yau.
     
  • Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions
    • Chao Li, Princeton Universty
    • Abstract: In this talk, I will discuss some recent developments on the topology of closed manifolds admitting Riemannian metrics of positive scalar curvature. In particular, we will prove if a closed PSC manifold of dimension 4 (resp. 5) has vanishing π2 (resp. vanishing π2 and π3), then a finite cover of it is homotopy equivalent to Snor connected sums of Sn-1 x S1. This extends a previous theorem on the non-existence of Riemannian metrics of positive scalar curvature on aspherical manifolds in 4 and 5 dimensions, due to Chodosh and myself and independently Gromov. A key step in the proof is a homological filling estimate in sufficiently connected PSC manifolds. This is based on joint work with Otis Chodosh and Yevgeny Liokumovich.
  • Convex subsets in generic manifolds
    • Alexander Lytchak, Universität Köln
    • Abstract: In the talk I would like to discuss some  statements and questions about convex subsets and convex hulls in generic Riemannian manifolds of dimension at least 3. The statements, obtained jointly with Anton Petrunin,  are elementary but somewhat  surprising for the Euclidean intuition. For instance, the convex hull of any finite non-collinearset  turns out to be  either   the whole manifold or non-closed.
  • Time-like Ricci curvature bounds via optimal transport in Lorentzian synthetic spaces and applications
    • Andrea Mondino, University of Oxford
    • Abstract: The goal of the talk is to present a recent work in collaboration with Cavalletti (SISSA) on optimal transport in Lorentzian synthetic spaces. The aim is to set up a “Lorentzian analog” of the celebrated Lott-Sturm-Villani theory of CD(K,N) metric measure spaces. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics (with respect to a suitable Lorentzian Wasserstein distance) of probability measures. The smooth Lorentzian setting was previously investigated by McCann and Mondino-Suhr.
      After recalling the general setting of Lorentzian synthetic spaces (including remarkable examples fitting the framework), I will discuss some basics of optimal transport theory thereof in order to define "timelike Ricci curvature bounded below and dimension bounded above''  for a (possibly non-smooth)  Lorentzian space. The notion of  "timelike Ricci curvature bounded below and dimension bounded above''  for a (possibly non-smooth)  Lorentzian space is stable under a suitable weak convergence of Lorentzian synthetic spaces, giving a glimpse on the strength of the proposed approach.
      As an application of the optimal transport approach to timelike Ricci curvature lower bounds, I will discuss an extension of the  Hawking's Singularity Theorem (in sharp form) to the synthetic setting.
  • Noncollapsed degeneration and desingularization of Einstein 4-manifolds
    • Tristan Ozuch, Massachussets Institute of Technology
    • Abstract: We study the noncollapsed singularity formation of Einstein 4-manifolds. We prove that any smooth Einstein 4-manifold close to a singular one in a mere Gromov-Hausdorff (GH) sense is the result of a gluing-perturbation procedure that we develop. This sheds light on the structure of the moduli space of Einstein 4-manifolds near its boundary and lets us show that spherical and hyperbolic orbifolds (which are synthetic Einstein spaces) cannot be GH-approximated by smooth Einstein metrics. New obstructions specific to the compact situation moreover raise the question of whether or not a sequence of Einstein 4-manifolds degenerating while bubbling out gravitational instantons has to be Kähler-Einstein.
  • Recent Intrinsic Flat Convergence Theorems
    • Raquel Perales Aguilar, Institute of Mathematics campus Oaxaca of the National Autonomous University of Mexico
    • Abstract: Given a closed and oriented manifold M and Riemannian tensors g0, g1, ... on M that satisfy g0 < gj, vol(M, gj)→vol (M, g0) and diam(M, gj)≤D we will see that (M, gj) converges to (M, g0) in the intrinsic flat sense. We also generalize this to the non-empty bundary setting. We remark that under the onditions we do not nexessarily obtain smooth, C0 or even Gromov-Hausdorff convergence. furthermore, these results can be applied to show stability of a class of tori and a class of complete and asymptotically flat manifolds. That is, any sequence of tori in the former class with almost nonnegative scalar curvature convergences to a flat tori, and any sequence of manifolds in the latter with ADM masses converging to zero converges to Euclidean space. [Based on joint work with Allen, Allen-Sormani and Cabrera Pacheco-Katterer].
  • Mean curvature flow with generic initial data
    • Felix Schulze, University of Warwick
    • Abstract: Mean curvature flow is the gradient flow of the area functional and constitutes a natural geometric heat equation on the space of hypersurfaces in an ambient Riemannian manifold. It is believed, similar to Ricci Flow in the intrinsic setting, to have the potential to serve as a tool to approach several fundamental conjectures in geometry. The obstacle for these applications is that the flow develops singularities, which one in general might not be able to classify completely. Nevertheless, a well-known conjecture of Huisken states that a generic mean curvature flow should have only spherical and cylindrical singularities. As a first step in this direction Colding-Minicozzi have shown in fundamental work that spheres and cylinders are the only linearly stable singularity models. As a second step toward Huisken's conjecture we show that mean curvature flow of generic initial closed surfaces in R^3 avoids asymptotically conical and non-spherical compact singularities. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact self-similarly shrinking solutions. This is joint work with Otis Chodosh, Kyeongsu Choi and Christos Mantoulidis.
  • Boundary regularity and stability under lower Ricci bounds
    • Daniele Semola, University of Oxford
    • Abstract: The theory of non smooth spaces with lower Ricci Curvature bounds has undergone huge developments in the last thirty years. On the one hand the impetus came from Gromov’s precompactness theorem and the Cheeger-Colding theory of Ricci limit spaces. On the other hand “synthetic” theories of lower Ricci bounds have been developed, based on semigroup tools (the Bakry-Émery theory) and on Optimal Transport (the Lott-Sturm-Villani theory).

      The Cheeger-Colding theory did not consider manifolds with boundary, while in the synthetic framework even understanding what is a good definition of boundary is a challenge.
      The aim of this talk is to present some recent results obtained in collaboration with E. Bruè (IAS, Princeton) and A. Naber (Northwestern University) about regularity and stability for boundaries of spaces with lower Ricci Curvature bounds.
  • On the essential minimal volume of Einstein 4-manifolds
    • Antoine Song, University of California at Berkeley
    • Abstract: Given a positive epsilon, a closed Einstein 4-manifold admits a natural thick-thin decomposition. I will explain how, for any delta, one can modify the Einstein metric to a bounded sectional curvature metric so that the thick part has volume linearly bounded by the Euler characteristic and the thin part has injectivity radius less than delta. I will also discuss relations to conjectural obstructions to collapsing with bounded sectional curvature or to the existence of Einstein metrics.
  • Harmonic map methods in spectral geometry
    • Daniel Stern, University of Chicago
    • Abstract: Over the last fifty years, the problem of finding sharp upper bounds for area-normalized Laplacian eigenvalues on closed surfaces has attracted the attention of many geometers, due in part to connections to the study of sphere-valued harmonic maps and minimal immersions. In this talk, I'll describe a series of results which shed new light on this problem by relating it to the variational theory of the Dirichlet energy on sphere-valued maps. Recent applications include new (H^{-1}-)stability results for the maximization of the first and second Laplacian eigenvalues, and a proof that metrics maximizing the first Steklov eigenvalue on a surface of genus g and k boundary components limit to the \lambda_1-maximizing metric on the closed surface of genus g as k becomes large (in particular, the associated free boundary minimal surfaces in B^{N+1} converge as varifolds to the associated closed minimal surface in S^N). Based on joint works with Mikhail Karpukhin, Mickael Nahon and Iosif Polterovich.
     
  • Limits of Riemannian manifolds satisfying a uniform Kato condition
    • David Tewodrose, Université Libre de Bruxelles
    • Abstract: I will present a joint work with G. Carron and I. Mondello where we study Kato limit spaces. These are metric measure spaces obtained as Gromov-Hausdorff limits of smooth n-dimensional Riemannian manifolds with Ricci curvature satisfying a uniform Kato-type condition. In this context, strictly wider than the ones of Ricci limit spaces (where the Ricci curvature satisfies a uniform lower bound) and Lp-Ricci limit spaces (where the Ricci curvature is uniformly bounded in Lp for some p>n/2), we extend classical results of Cheeger, Colding and Naber, like the fact that under a non-collapsing assumption, every tangent cone is a metric measure cone. I will present these results and explain how we rely upon a new heat-kernel based almost monotone quantity to derive them.
  • Topological rigidity and positive scalar curvature
    • Jian Wang, Universität Augsburg
    • Abstract: In this talk, we shall describe some topological rigidity and its relationship with positive scalar curvature. Precisely, we will present a proof that a complete contractible 3-manifold with positive scalar curvature is homeomorphic to the Euclidean 3-space. We will furthermore explain the interplay between, minimal surfaces, scalar curvature and the topology at infinity.
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