The 2020 VWRS was held in honor of Misha Gromov Organizers: Christina Sormani (CUNYGC and Lehman), Guofang Wei (University of California at Santa Barbara), Hang Chen (Northwestern Polytechnical University, P. R. China). Lan-Hsuan Huang (University of Connecticut), Pengzi Miao (University of Miami), Paolo Piazza (Sapienza Università di Roma), Blaine Lawson (Stony Brook), and Richard Schoen (University of California at Irvine). Discussions of each talk happen online in the googlegroup: 2020 Virtual Workshop on Ricci and Scalar Curvature. To participate in the discussion send your email to Professor Sormani at sormanic@gmail.com and apply to join the google group 2020 Virtual Workshop on Ricci and Scalar Curvature (you must login to google groups and then choose this group and apply to join). Below next to each talk you will see a direct link to the discussion of that talk. We encourage this discussion to continue and anyone who wishes to join the workshop late to do so. Perhaps in the future we will hold another workshop and advertise it to this participant list, so please join! Links to Existing International Videos on these Topics: (recommended for those seeking a stronger background) Introductory Lectures: Scalar Curvature: Ricci Curvature: 2020 VWRS Plenary Addresses: The abstracts are below the Invited Talks. Chao Li (Princeton University) Paula Burkhardt-Guim (University of California at Berkeley) Brian Allen (University of Hartford) Raquel Perales (UNAM Oaxaca) Martin Lesourd (Harvard University) joint with Ryan Unger (Princeton) and Shing-Tung Yau (Harvard) Thomas Richard (LAMA, UPEC, France) Yuguang Shi (Peking University) Daniel Stern (University of Toronto) Xiaochun Rong (Rutgers University) Regina Rotman (University of Toronto) Weiping Zhang (Nankai University) James Isenberg (University of Oregon) Zhongmin Shen (Indiana University Purdue University in Indianapolis) Nicola Gigli (SISSA, Trieste) Shouhei Honda (Tohoku University) Richard Bamler (University of California at Berkeley) Jintian Zhu (Peking University) Dimitri Burago (Penn State University) joint with Luis Florit (IMPA, Rio de Janiero, Brasil) Four of our initial invited plenary speakers, Sylvestre Gallot (UJF Grenoble), regret that they cannot present at this time but, like the organizers, they are grateful to Misha Gromov for all his inspiration over the years. 2020 VWRS Invited Talks: (linked below alphabetically by speaker) Invited talks are listed below with their titles and links to their videos at (youtube) or (bilibili) or elsewhere. To join a discussion about a talk go to the link marked (discuss) and login to the google group. Everyone is welcome to submit videos/papers for consideration to be included in this virtual workshop. See information at the bottom of this page regarding videos and the link at the top for submitting papers to SIGMA. The deadline was August 20, 2020. If you wish only to be included in the mailing list as a noncontributing participant, send your email to Professor Sormani at sormanic@gmail.com and apply to join the google group 2020 Virtual Workshop on Ricci and Scalar Curvature. (you must login to google groups and then choose this group and apply to join). 2020 VWRS Participants: Amir Aazami (Clark University) Mohammed Abdelmalek (ESM Tlemcen, Algeria) Ian Adelstein (Yale U) Aghil Alaee (Harvard) Luis Alberto Ake Hau (Mexico) Michael Albanese (CIRGET) H^1 cup length, positive scalar curvature, and low-dimensional aspherical manifolds (youtube) (bilibili) (discuss) Clara Aldana (Universidad del Norte, Colombia) Precompactness of conformal metrics under critical curvature estimates (youtube) (bili1, bili2, bili3) (discuss) Stephanie Alexander (Urbana, Illinois) Brian Allen (University of Hartford, Connecticut) Contrasting and Relating Notions of Convergence in Geometric Analysis (youtube) (bilibili) (discuss) Zhongshan An (U Connecticut) Michael Anderson (Stony Brook, New York) Daniele Angella (U Firenze, Italy) Gioacchino Antonelli (Scuola Normale Superiore) Paolo Antonini (SISSA) Juan-Carlos Alvarez Paiva (University of Lille) Julio Argota (Queen Mary Univ, London) Ami Aswani (SUNY Buffalo, New York) Sara Azzali (U Hamburg) Florent Balacheff (Universitat Autònoma de Barcelona) Gavin Ball (CIRGET, Canada) Richard Bamler (UC Berkeley) Lashi Bandara (U Potsdam, Germany) Jorge Eduardo Basilio (California) Sewing Sequences of Riemannian Manifolds with Positive or Nonnegative Scalar Curvature (youtube) (bili1) (bili2) (bili3) (bili4) (discuss) Igor Belegradek (Georgia Tech) Gerard Besson (Institute Fourier, France) Renato Bettiol (Lehman, CUNY) Stefano Borghini (Uppsala Univ) Boris Botvinnik (Oregon) Topology of spaces and moduli spaces of metrics with positive scalar curvature (youtube)(biblibili1 and b2bilibili2)(discuss) Elia Brue (SNS, Italy) Edward Bryden (Tubingen) Dimitri Burago (Penn State University) Bradley Burdick (U California at Riverside) Pointwise lower scalar curvature bounds for $C^0$ metrics via regularizing Ricci flow (dropbox) (bilibili) (discuss) Annegret Burtscher (Radboud University) Armando Cabrera Pacheco (Tubingen, Germany) Xiaodong Cao (Cornell, New York) Alessandro Carlotto (ETH Zurich) Florin Catrina (St. Johns Univ, NY) Explicit minimizing sequences related to the Riemannian Penrose Inequality and to Bartnik’s mass functional (youtube) (bili1, bili2, bili3) (discuss) Simone Cecchini (Goettingen, Germany) A long neck principle for Riemannian spin manifolds with positive scalar curvature (youtube) (bilibili) (discuss) Sun-Yung Alice Chang (Princeton) Nelia Charalambous (University of Cyprus) Indira Chatterji (UNICE, France) Jeff Cheeger (NYU) Lina Chen (Nanjing University, China) Xiuxiong Chen (SUNY Stony Brook) Xuezhang Chen (Nanjing University) Beomjun Choi (U Toronto) Eber Daniel Chuno Vizarreta (UFRPE, Brazil) Diego Corro (Karlsruhr U, Germany) Gilles Courtois (CNRS, France) Graham Cox (Memorial University, Newfoundland) Katy Craig (UCSB, California) Mattias Dahl (KTH, Stockholm) Xianzhe Dai (UCSB, California) Jim Davis (Indiana University) Akashdeep Dey (Princeton U, New Jersey) Eleonora Di Nezza (Sorbonne Universite) Yu Ding (CSU Long Beach, California) Jozef Dodziuk (Israel) Michael Eichmair (U Vienna) Jonathan Epstein (U Oklahoma) Juan Carlos Fernandez (UNAM, Mexico) Marisa Fernandez (U Pais Vasco, Spain) José Manuel Fernández Barroso (Extremadura, Spain)Georg-Joachim Frenck (Karlsruhr University) Luis Florit (IMPA, Brasil) work presented by Hanke Allan Freitas (Fed U of Paraiba, Brazil) Xin Fu (UC Irvine, California) Fernando Galaz-Garcia (Durham University) Greg Galloway (U Miami) Sylvestre Gallot (UJF Grenoble) Greg Galloway (U Miami) Jonathan Gloeckle (U Regensberg) David Gonzalez-Alvaro (U Poly de Madrid) Fredy Alexis Gonzalez Fonseca (UPTC Colombia) Masha Gordina (U Conn) Francisco J. Gozzi (Brasil) Melanie Graf (U Washington) Anthony Gruber (Texas Tech U) Sharmila Dhevi Gunasekaran Gnanam (Memorial U of Newfoundland) Sekit Gunsen (Adnan Menderes U, Turkey) Yifan Guo (U California at Irvine) James Heitsch (U Illinois at Chicago) Lisandra Hernandez-Vazquez (SUNY Stony Brook) Bernardo Hipolito Fernandes (KTH, Sweden) Sven Hirsch (Duke) Shouhei Honda (Tohoku U, Japan) Surena Hozoori (Georgia Tech) Xue Hu (Jinan University) Shaosai Huang (U Wisconsin) James Isenberg (U Oregon) Sergei Ivanov (Steklov Institute) Ivan Izmestiev (TU Wien, Austria) Michael Jablonski (U Oklahoma) Cooper Jacob (UC Davis) Hyun Chul Jang (U Conn) Jeff Jauregui (Union College, NY) Mustafa Kalafat (Michigan State) Dersim Kaya (Leibniz University of Hannover) Demetre Kazaras (Duke University, North Carolina) Nikolaos Kalogeropoulos (American U of Iraq) Christian Ketterer (U Toronto) Marcus Khuri (Stony Brook, New York) Seongtag Kim (INHA, Korea) James Kohout (Oxford, Great Britain) Anusha Krishnan (Syracuse U, New York) Klaus Kröncke (U Hamburg, Germany) L^p-stability and positive scalar curvature rigidity of Ricci-flat ALE manifolds (youtube) (discuss) Kwok-Kun Kwong (National Cheng-Kung University) Sajjad Lakzian (Iran) Jorge Lauret (U Nat Cordoba Argentina) Emilio Lauret (U Nat Cordoba Argentina) Dan A Lee (CUNY) Jack Lee (U Wisconsin) Kuo-Wei Lee (National Changhua University) Man-Chun Lee (Northwestern University, Illinois) Martin Lesourd (Harvard University) Ryan Unger and Shing-Tung Yau Chao Li (Princeton University) Martin Li (The Chinese Univ of Hong Kong) Yang-Yang Li (Princeton) Xiaobin Li (Southwest Jiaotong University) Alice Wu Lim (Syracuse U, NY) Codimension One Homology of Noncompact Manifolds with Nonnegative m-Bakry-Emery Ricci Curvature (talk link) (discuss) Chen-Yun Lin (Lehman CUNY) Ursula Ludwig (Uni Due) Elena Maeder-Baumdicker Martin Magid (Wellesley) Fedya Manin (UCSB, California) Christos Mantoulidis (Brown University) Stefano Marchiafava (Uni di Roma) Rafe Mazzeo (Stanford University) Steve McCormick (Uppsala University) Yashar Memarian (Switzerland) Ingrid Membrillo Solis (U Southampton, UK) Abraão Mendes (Universidade Federal de Alagoas, Brasil) Emilio Minichiello (CUNYCG) Muang Min-Oo (McMaster U, Canada) Andrea Mondino (Oxford University) Ilaria Mondello (UPEC, France) Richard Montgomery (UCSC, California) Frank Morgan (Williams College) Lawrence Mouille (Rice University) TWJ Murphy Aaron Naber (Northwestern, Illinois) Alexander Nabutovsky (U Toronto) Diego Alonso Navarro Guajardo (IMPA, Brasil) Aissatou Ndiaye (Neuchatel Universite) Cheikh Birahim Ndiaye (Howard University) Victor Nistor (U Lorraine, France) Jesús Núñez-Zimbrón (UNAM) Manuel Oliviera (U British Columbia) Luis Eduardo Osorio Acevedo (Universidad Tecnológica de Pereira, Colombia) Franco Vargas Pallete (Yale University) Jiayin Pan (UCSB) On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature (youtube) (bilibili) (discuss) Davide Parise (Cambridge) Byungdo Park (Chungbuk National University, Korea) Jiewon Park (MIT) Fabian Parsch (U Toronto) Alec Payne (California) Tracy Payne (Idaho State U) Raquel Perales (UNAM, Mexico) Javier Peraza (U de la Republica, Uruguay) Dan Pollack (UW) Jacobus Portegies (Netherlands) Roman Prosanov (TU Wien) Daniel Raede (U Augsburg) Xavier Ramos Olive (Worcester Polytech) Marcos Ranieri (UFAL-Brasil) Jesse Ratzkin (U Wuerzburg) On constant Q-curvature metrics with isolated singularities and a related fourth-order invariant (bilibili) (discuss) Martin Reiris (U de la Republica, Uruguay) Philip Reiser (KIT, Germany) Thomas Richard (LAMA UPEC, France) On the 2-systole of stretched positive scalar curvature metrics on S^2xS^2. (youtube) (bilibili) (discuss) Ernani Ribeiro Jr (Uni Fed Ceara) Chiara Rigoni (U Bonn, Germany) Alberto Roncoroni (U Firenze, Italy) Xiaochun Rong (Rutgers, New Jersey) Cesar Rosales (U Grenada, Spain) Christian Rose (Max Plank Leipzig) Jonathan M. Rosenberg (U Maryland) w Botvinnik Positive scalar curvature on Spin^c manifolds and manifolds with singularities (youtube)(bilibili1 and bilibili2) (discuss) Regina Rotman (U Toronto) Ricci Curvature and the Length of the Shortest Periodic Geodesic (youtube) (bilibili) (discuss here) Stephane Sabourau (UPEC, France) Clemens Saemann (U Toronto) Anna Sakovich (Uppsala University) Jonatan Sánchez (Polytechnic University of Madrid) A 7-dimensional nilmanifold with a non Ricci-flat Einstein pseudo-metric (mysharept) (youtube) (bilibili) (discuss) joint with Prof. Marisa Fernández (University of the Basque Country) and Prof. Marco Freibert (Christian-Albrechts-Universität zu Kiel) Bianca Santoro (City College, CUNY) Walcy Santos (Uni Fed Rio de Janiero) Thomas Schick (Universitat Gottingen) Felix Schultze Catherine Searle (Wichita State University) Danielle Semola (Pisa) Julilan Seipel (U Regensburg) Hemangi Shah (Harish-Chandra Research Institute) Krishnan Shankar (U Oklahoma) Zhongmin Shen (IUPUI) Yuguang Shi (Peking University) Fatma Muazzez Şimşir (Selçuk University, Konya/Turkey) Zahra Sinaei (U Mass) Penny Smith (Lehigh U, Pennsylvania) Pedro Solorzano (UNAM, Mexico) Nancy K Stanton (Notre Dame, Indiana) Iva Stavrov (Lewis and Clarke) Daniel Stern (U Toronto) Pablo Suarez Serrato (UNAM, Mexico) Stephan Suhr (Hamburg U, Germany) Liming Sun (JHU, Maryland) Yukai Sun (East China Normal University) Ebtsam H. Taha (Cairo University, Egypt) Chaitanya Tappu (Cornell University) Paul Tee (McGill, Canada) Wenchuan Tian (Michigan State) Bankteshwar Tiwari (CIMS, Institute of Science, Banaras Hindu University Varanasi, India) Magdelena Toda (Texas Tech) Hung Tran (Texas Tech) Andrejs Treibergs (Univ of Utah) Inan Unal (Munzar University, Turkey) Ryan Unger (Princeton) Carlos Vega (SUNY Binghamton) Luigi Verdiani (Università di Firenze, Italy) Zheyan Wan (Tsinghua University) Bing Wang (USTC) Changliang Wang (MPIM Bonn, Germany) Jian Wang (Berkeley, California) Shengwen Wang (Queen Mary Univ, London) Zhehui Wang (Notre Dame, Indiana) Shihshu Walter Wei (Oklahoma University) Fred Wilhelm (UC Riverside, California) Hollis Williams (Warwick, UK) Matthias Wink (UCLA, California) Eric Woolgar (U Alberta, Canada) Will Wylie (Syracuse U, NY) Jia-Yong Wu (Shanghai U, China) Ling Xiao (U Connecticut) Asymptotic convergence for modified scalar curvature flow (youtube) (bili1,bili2, and bili3)(discuss) Bin Xu (USTC, China) Eyup Yalcinkaya (TUBITAK) Sumio Yamada (Gakushuin University) Junrong Yan (UCSB) Ismael El Yassini (U Moulay Ismail, Morocco) Sergio Zamora (Penn State U) Lower Semicontinuty of the Fundamental Group and Convergence with Discrete Symmetry (youtube) (youku) (bilibili) (discuss) Weiping Zhang (Nankai University) Rudolf Zeidler (Muenster) Bo Zhu (University of Minnesota-Twin Cities) Jintian Zhu (Peking University, China) Xingyu Zhu (Georgia Tech) Demetre Kazaras hosted Tea Time of Scalar Curvature Topics for registered faculty, postdocs, and graduate students Fridays at 1 pm EST in August and September. Lawrence Mouille hosted Tea Time on Lower Curvature Bounds for registered postdocs and graduate students on Thursdays at 10 pm EST in August. for registered postdocs and graduate students on Wednesdays at 11 am EST in August Abstracts of the Plenary Addresses: Tuesday August 4 Plenary Addresses: Four Junior Mathematicians directly working on Gromov's Open Problems on Scalar Curvature Chao Li (Princeton University) 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: if a Riemannian polyhedron has nonnegative scalar curvature in the interior, and weakly mean convex faces, then the dihedral angle between adjacent faces cannot be everywhere less than the corresponding Euclidean model. In this talk, I will prove this conjecture for a large collection of polytopes, and extend it to metrics with negative scalar curvature lower bounds. The strategy is to relate this question with a geometric variational problem of capillary type, and apply the Schoen-Yau minimal slicing technique for manifolds with boundary. 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 $C^0$ 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 $C^0$ 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. Brian Allen (University of Hartford, Connecticut) Abstract: Results are discussed that provide natural geometric conditions which imply intrinsic flat convergence to a specified Riemannian manifold. As an application, the tori with almost non-negative scalar curvature conjecture is discussed as well as the proof in the warped product case. This proof serves to explain how one should expect to obtain the hypotheses of the main theorems in practice. Many examples are given which explain the necessity of all of the main hypotheses, show the distinction between Lebesque and intrinsic flat convergence, and further illustrate geometric considerations arising when studying geometric stability results involving scalar curvature. We end by giving proof outlines for two important main theorems. Joint work with R. Perales and C. Sormani is discussed throughout. Raquel Perales (UNAM Oaxaca, Mexico) Abstract: We go over the papers Volume Above Distance Below and Intrinsic Flat Stability of Manifolds with Boundary where Volume Converges and Distance is Bounded Below by Allen-Perales-Sormani and Allen-Perales, respectively, and mention applications of them to the stability of the positive mass theorem and tori with nonnegative scalar curvature. This last part appears in a paper by Cabrera Pacheco-Ketterer-Perales. Tuesday August 11 Plenary Addresses: Martin Lesourd (Harvard University, United States of America) joint with Ryan Unger (Princeton) and Shing-Tung Yau (Harvard) Abstract: We begin with a survey of some classics in Scalar Curvature, R, including the facts that the n-dimensional torus T^n does not admit a metric with R>0. Assuming Y^n is orientable and compact, this is also true for manifolds of the form T^n # Y^n. Here, using minimal hypersurfaces, we study the case where Y^n is non-compact. Under some dimensional restrictions and assumptions that are believed to be technical, we show that such manifolds do not admit complete metrics with R>0. This is relevant to the so-called Positive Mass Conjecture with Bends (i.e. Bad Ends) - which is open - and the Liouville Theorem of Schoen-Yau 1988, which has not been proven in full generality. Yuguang Shi (Peking University, People's Republic of China) Abstract: Let be an orientable -dimensional Riemannian manifold, be a positive function on , One of basic problems in Riemannian geometry is to ask: under what conditions is it that is induced by a Riemannian metric with nonnegative scalar curvature, for example, defined on , and is the mean curvature of in with respect to the outward unit normal vector? Recently, M.Gromov proposed several conjectures on this question. In this first part of this talk I will describe what the conjectures are and survey some known results in this direction when ; In the second part of the talk, I will present my several recent results on this which joint with Dr. Wang Wenlong, Dr. Wei Guodong and Zhu Jintian. This talk is based on my recent joint paper named “Total mean curvature of the boundary and nonnegative scalar curvature fill-ins.” Thomas Richard (LAMA, UPEC, France) Abstract: We show a new metric inequality for positive scalar curvature metrics on the product manifolds S^2xS^2. Qualitatively, it says that if one can find two surfaces homologous to S^2x{*} which are far away from each other (this is what is meant by the word "stretched" in the title), then one can find a 2-sphere homologous to S^2x{*} of controlled area (which gives a control on the 2-systole). Daniel Stern (University of Toronto, Canada) Abstract: We discuss a new family of techniques for studying the influence of scalar curvature on the large-scale structure of Riemannian three-manifolds, based on a relationship between scalar curvature and the topology of level sets of S^1-valued harmonic maps. Tuesday August 18 Plenary Addresses: Xiaochun Rong (Rutgers University) Abstract: In 1978, Gromov proved that (highly unexpected) an almost flat manifold is diffeomorphic to a nilpotent manifold up to a bounded normal covering space. This result has been a corner stone in the theory on collapsed manifolds with bounded sectional curvature by Cheeger-Fukaya-Gromov. We will discuss some recent generalization of this result to maximally collapsed manifolds with local bounded Ricci covering geometry, and to maximally collapsed Alexandrov spaces with local bounded covering geometry, as well as counterparts of the above collapsing theory in these spaces of local bounded covering geometry. Regina Rotman (University of Toronto) Abstract: In his paper Filling Riemannian Manifolds, Gromov asked the following question. If M is a closed Riemannian manifold of dimension n and volume, vol(M), does there exists a constant c(n) such that the length of a shortest closed geodesic, l(M), is bounded above by c(n)vol(M) . Similarly, one can ask if there exists a constant c ̃(n), such that l(Mn) ≤ c ̃(n)d, where d denotes the diameter of M. I will survey results in this area and then present the following theorem: If M^n is a closed Riemannian manifold of dimension n with Ricci curvature Ric \geq n-1 then the length of a shortest periodic geodesic is bounded above by 8 \pi n. Weiping Zhang (Nankai University) Abstract: A famous theorem of Lichnerowicz states that if a closed spin manifold admits a Riemannian metric of positive scalar curvature, then its Hirzebruch A-hat genus vanishes. We describe various generalizations of this result to the case of foliations. A typical example is Connes’ theorem which stated that if the A-hat genus of a closed foliated manifold with spin leaves does not vanish, then it does not admit a metric of positive scalar curvature along the leaves. James Isenberg (University of Oregon) Abstract: In this joint work with Timothy Carson, Dan Knopf, and Natasa Sesum, we study singularity formation of complete Ricci flow solutions, motivated by two applications: (a) improving the understanding of the behavior of the essential blowup sequences of Enders-Muller-Topping on noncompact manifolds, and (b) obtaining further evidence in favor of the conjectured stability of generalized cylinders as Ricci flow singularity models. Tuesday August 25 Plenary Addresses: Zhongmin Shen (Indiana University Purdue University in Indianapolis) Abstract: Ricci curvature is one of the important geometric quantities in Riemann-Finsler geometry. Together with the S-curvature, one can define a weighted Ricci curvature for a pair of Finsler metric and a volume form on a manifold. One can build up a bridge from Riemannian geometry to Finsler geometry via geodesic fields. Then one can estimate the Laplacian of a distance function and the mean curvature of a metric sphere under a lower weighted Ricci curvature simply by applying the results in the Riemannian setting. These estimates also give rise to a volume comparison of Bishop-Gromov type for Finsler metric measure manifolds. Nicola Gigli (SISSA, Trieste) Abstract: After recalling the classical synthetic approach to lower curvature bounds on non-smooth spaces, I will discuss more recent developments concerning the role that functional analysis has in this framework. I will conclude with a conjecture about the curvature of Alexandrov spaces. Abstract: In this talk I show how to obtain compactness and precompactness of metric spaces coming from conformal Riemannian metrics on a given closed manifold under critical escalar curvature estimates. The results are presented in the first part of the talk from a purely geometrical point of view. In the second part, I give the motivation in which I show how this problem relates to two problems in differential geometry: Pinching of the curvature and finding geometrical conditions under which a sequence of conformal metrics admits a convergent subsequence. In the third part I will make the connection to the analytical tools used in the proofs, I introduce $A_{\infty}$-weights and strong $A_{\infty}$-weights and present some of their properties. I show how, using these weights, we can prove compactness of conformal metrics with critical integrability conditions on the scalar curvature. The results presented here are joined work with Gilles Carron and Samuel Tapie (University of Nantes), they are included in the paper "$A_\infty$ weights and compactness of conformal metrics under $L^{n/2}$ curvature bounds", arXiv:1810.05387 and to appear in Analysis and PDE. Shouhei Honda (Tohoku University) Abstract: In this talk we show that if a closed n-dimensional Riemannian manifold has an eigenmap to the Euclidean space of dimension n+1, whose pull-back is close to the original Riemannian metric in the L1-average sense, then the manifold is diffeomorphic to a unit sphere of dimension n. The proof is based on regularity results on metric measure spaces with Ricci curvature bounded below. Tuesday September 8 Plenary Addresses: Abstract: We present results in index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a ``long neck principle'' for a compact Riemannian spin n-manifold with boundary X, stating that if scal(X) ≥ n(n-1) and there is a nonzero degree map f into the n-sphere which is area decreasing, then the distance between the support of the differential of f and the boundary of X is at most π/n. This answers, in the spin setting, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing a small n-ball from a closed spin n-manifold Y. We show that if scal(X) ≥ σ >0 and Y satisfies a certain condition expressed in terms of higher index theory, then the width of a geodesic collar neighborhood Is bounded from above from a constant depending on σ and n. Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to Nx [-1,1], with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if scal(V) ≥ n(n-1), then the distance between the boundary components of V is at most 2π/n. This last constant is sharp by an argument due to Gromov. Abstract: Following ideas by Mantoulidis and Schoen and further developments thereof by Cabrera Pacheco, McCormick, Miao, Xie (in alphabetic order) and the speaker, we construct a sequence of asymptotically flat, Riemannian 3-manifolds of non-negative scalar curvature with minimal, strictly outward minimizing inner boundary. The ADM-mass converges to the minimal value permitted by the Riemannian Penrose Inequality along this sequence, yet the manifolds themselves do not converge to the Schwarzschild manifold arising as the rigidity case of the Riemannian Penrose Inequality. Instead, they converge to an explicitly given non-smooth manifold (in a suitable topology) that extend the class of limits found by Lee-Sormani in their work on the limits of spherically symmetric manifolds approaching equality in the Riemannian Penrose Inequality. The existence of this sequence and the precise form of the limit also have consequences for Bartnik’s quasi-local mass functional. This is joint work with Armando Cabrera Pacheco. Richard Bamler (University of California at Berkeley) Abstract: We present new results concerning Ricci flows in higher dimensions after reviewing the history of the subject. This is a preexisting talk recorded originally for another seminar that works well for this workshop as well. I'd like to thank the organizers for the opportunity to present it here as well. Abstract: The starting point of this lecture is a Dirac operator approach to Gromov's question on the width of Riemannian bands: Let M be a closed spin manifold of non-trivial Rosenberg index. Then there is an a priori upper bound on the distance between the boundary components of $V = M \times [-1,1]$ in terms of a positive lower scalar curvature bound of the metric on V. Based on this, we study new variations of Gromov's largeness properties, namely $\hat{A}$-iso-enlargeability and infinite $\mathcal{KO}$-width. Several known psc-obstructions—in particular the codimension two obstruction of Hanke-Pape-Schick and obstructions to uniform psc on some open manifolds—can be understood using infinite KO-width. Finally, we give an overview on several largeness properties as well as their relation to index theory and conclude with a conjecture on KO-width and the Rosenberg index. Tuesday September 15 Plenary Addresses: Jintian Zhu (Peking University, China) Abstract: I will discuss a new idea to prove some rigidity result for complete manifolds with nonnegative scalar curvature using the foliation method. Based on this idea, I’ll show how to obtain optimal 2-systole estimates involving positive scalar curvature lower bound on several classes of complete manifolds. Abstract: We consider asymptotically Euclidean, conformally flat Riemannian manifolds of positive scalar curvature (representing the initial data of charged relativistic dust clouds). We prove these manifolds are the intrinsic flat limits of Brill-Lindquist metrics (representing the initial data of vacuum spacetimes with charged black holes). Note that Brill-Lindquest metrics without charge are scalar flat and have interior boundaries that are minimal surfaces and their intrinsic flat limits found here have strictly positive scalar curvature and no interior boundaries. Thus we have purely geometric examples where the scalar curvature jumps up upon taking an intrinsic flat limit. Dimitri Burago (Penn State University) Abstract: This plenary address in honor of Misha Gromov has been written out word for word rather than recorded and the speaker has requested that if anyone has questions to please email or skype with him rather than discussing the talk in the google group here. The talk reviews joint work with Dong Chen and with Sergei Ivanov and includes open questions. Abstract: As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive scalar curvature metrics on closed simply connected manifolds in dimensions at least five appears on spin manifolds, and is given by the non-vanishing of the $\alpha$-genus of Hitchin. When unobstructed we shall realize a positive scalar curvature metric by an immersion into Euclidean space whose dimension is uniformly close to the classical Whitney upper bound for smooth immersions. Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure for constructing positive scalar curvature metrics. At this point we use the local flexibility lemma proven by Christian Baer and the speaker in 2019. This is joint work with Luis Florit, IMPA (Rio de Janeiro). On recording your VWRS presentations for the Virtual Workshop on Ricci and Scalar Curvature in honor of Gromov: This is an asynchronous workshop. In addition to our Plenary Speakers anyone who wishes to submit a video of a talk for consideration is welcome to do so. All talks should be posted online somewhere permanently and publicly, and the submission is a title and an abstract and a link to the video of the talk to sormanic@gmail.com. Those that are selected will be shared through this google group and people will discuss the talk in replies to the message sharing the link to the talk and are also welcome to ask the speakers questions by email. The talks will also be posted on the webpage of the virtual workshop and its mirror. The deadline is August 1, 2020 but we encourage earlier submission if possible. Why asynchronous? * So that everyone can watch the videos at their convenience regardless of time zone. * So that people can pause the talk and rewind as they wish and email the speaker when they wish. * So that people can choose to watch just part of the talk, and then leave or not as they wish. * So that each talk can be completed in an hour exactly as the speaker has prepared it. * So that everyone has the opportunity to submit a talk and be considered as a speaker. Some suggestions for the format of the talks: First fifteen to twenty minutes (aimed at the level of doctoral students): * Please state your new theorem or concept within this part of the talk. * Be sure to include all relevant definitions and key background. * Explain the importance of the result and its context. * Be sure to give credit to those with related results using their last name. * Include some relevant graphics if at all possible. * Include a list of references at the end so that students can read more before watching the rest of the talk. The rest of the talk (aimed at experts) * Portray ideas in an intuitive and geometric way in the style of Gromov. * At the same time provide a rigorous and self contained presentation. * Continue to give full credit to all authors of results that are relevant. * Try to guess at the questions that might be asked and answer them. * End the talk with ideas towards the future. Note that some video archives like youtube allow you to create a playlist. In this way you can break your talk up into short videos with natural breaks. This also makes it easier for presenters to correct a talk as they need only fix the part that was incorrect. It also allows speakers to add a new video segment at the end of their talk later to address questions of interest that were emailed to them. Some speakers are asking how to post a longer video on youtube, so we recommend this video about that. We would recommend youtube as the best place for the videos however youtube is not available in some countries. One solution would be for everyone to post on youtube and at a secondary location. Some people are using Zoom, others are using YouKu, others are posting at their own university. Please seek information online about how to post long videos and playlists on youtube and elsewhere. We do not have staff available to provide technical advice. This is an unfunded workshop. All speakers were volunteers and took the time to record and upload their talks themselves. Many participants were funded by their own institutions (listed above) and by their grants. Some postdocs received stipends or part-time summer employment though Sormani's NSF Research Grant DMS - 1612049.
|