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DTSTART;TZID=America/New_York:20220301T100000
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SUMMARY:General Relativity Program Minicourses
DESCRIPTION:Minicourses\nGeneral Relativity Program Minicourses \n\nDuring the Spring 2022 semester\, the CMSA hosted a program on General Relativity. \nThis semester-long program included four minicourses running in March\, April\, and May;  a conference April 4–8\, 2022;  and a workshop from May 2–5\, 2022. \n\n  \n\n\n\n\nSchedule\nSpeaker\nTitle\nAbstract\n\n\nMarch 1 – 3\, 2022\n10:00 am – 12:00 pm ET\, each dayLocation: Hybrid. CMSA main seminar room\, G-10.\nDr. Stefan Czimek\nCharacteristic Gluing for the Einstein Equations\nAbstract: This course serves as an introduction to characteristic gluing for the Einstein equations (developed by the lecturer in collaboration with S. Aretakis and I. Rodnianski). First we set up and analyze the characteristic gluing problem along one outgoing null hypersurface.  Then we turn to bifurcate characteristic gluing (i.e.  gluing along two null hypersurfaces bifurcating from a spacelike 2-sphere) and show how to localize characteristic initial data. Subsequently we turn to applications for spacelike initial data. Specifically\, we discuss in detail our alternative proofs of the celebrated Corvino-Schoen gluing to Kerr and the Carlotto-Schoen localization of spacelike initial data (with improved decay).\n\n\nMarch 22 – 25\, 2022\n22nd & 23rd\, 10:00 am – 11:30am ET\n24th & 25th\, 11:00 am – 12:30pm ETLocation: Hybrid. CMSA main seminar room\, G-10.\nProf. Lan-Hsuan Huang\nExistence of Static Metrics with Prescribed Bartnik Boundary Data\nAbstract: The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold\, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat\, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein\, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts\, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique\, asymptotically flat\, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course\, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An\, and the tentative plan is \n1. The conjecture and an overview of the results\n2. Static regular: a sufficient condition for existence and local uniqueness\n3. Convex boundary\, isometric embedding\, and static regular\n4. Perturbations of any hypersurface are static regular \nVideo on Youtube: March 22\, 2022\n\n\nMarch 29 – April 1\, 2022 10:00am – 12:00pm ET\, each day \nLocation: Hybrid. CMSA main seminar room\, G-10.\nProf. Martin Taylor\nThe nonlinear stability of the Schwarzschild family of black holes\nAbstract: I will present aspects of a theorem\, joint with Mihalis Dafermos\, Gustav Holzegel and Igor Rodnianski\, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.\n\n\nApril 19 & 21\, 2022\n10 am – 12 pm ET\, each dayZoom only\nProf. Håkan Andréasson\nTwo topics for the Einstein-Vlasov system: Gravitational collapse and properties of static and stationary solutions.\nAbstract: In these lectures I will discuss the Einstein-Vlasov system in the asymptotically flat case. I will focus on two topics; gravitational collapse and properties of static and stationary solutions. In the former case I will present results in the spherically symmetric case that give criteria on initial data which guarantee the formation of black holes in the evolution. I will also discuss the relation between gravitational collapse for the Einstein-Vlasov system and the Einstein-dust system. I will then discuss properties of static and stationary solutions in the spherically symmetric case and the axisymmetric case. In particular I will present a recent result on the existence of massless steady states surrounding a Schwarzschild black hole. \nVideo 4/19/2022 \nVideo 4/22/2022\n\n\nMay 16 – 17\, 2022\n10:00 am – 1:00 pm ET\, each dayLocation: Hybrid. CMSA main seminar room\, G-10.\nProf. Marcelo Disconzi\nA brief overview of recent developments in relativistic fluids\nAbstract: In this series of lectures\, we will discuss some recent developments in the field of relativistic fluids\, considering both the motion of relativistic fluids in a fixed background or coupled to Einstein’s equations. The topics to be discussed will include: the relativistic free-boundary Euler equations with a physical vacuum boundary\, a new formulation of the relativistic Euler equations tailored to applications to shock formation\, and formulations of relativistic fluids with viscosity. \n1. Set-up\, review of standard results\, physical motivation.\n2. The relativistic Euler equations: null structures and the problem of shocks.\n3. The free-boundary relativistic Euler equations with a physical vacuum boundary.\n4. Relativistic viscous fluids. \nVideo 5/16/2022 \nVideo 5/17/2022
URL:https://cmsa.fas.harvard.edu/event/grminicourses/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Workshop
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220502T090000
DTEND;TZID=America/New_York:20220505T170000
DTSTAMP:20260520T022457
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SUMMARY:General Relativity Workshop
DESCRIPTION:General Relativity Workshop on scalar curvature\, minimal surfaces\, and initial data sets \nDates: May 2–5\, 2022 \nLocation: Room G10\, CMSA\, 20 Garden Street\, Cambridge MA 02138 and via Zoom webinar.\nAdvanced registration for in-person components is required. \nOrganizers: Dan Lee (CMSA/CUNY)\, Martin Lesourd (CMSA/BHI)\, and Lan-Hsuan Huang (University of Connecticut). \nSpeakers: \n\nZhongshan An\, University of Connecticut\nPaula Burkhardt-Guim\, NYU\nHyun Chul Jang\, University of Miami\nChao Li\, NYU\nChristos Mantoulidis\, Rice University\nRobin Neumayer\, Carnegie Mellon University\nAndre Neves\, University of Chicago\nTristan Ozuch\, MIT\nAnnachiara Piubello\, University of Miami\nAntoine Song\, UC Berkeley\nTin-Yau Tsang\, UC Irvine\nRyan Unger\, Princeton\nZhizhang Xie\, Texas A & M\nXin Zhou\, Cornell University\nJonathan Zhu\, Princeton University\n\nSchedule\nMonday\, May 2\, 2022 \n\n\n\n\n9:30–10:30 am\nHyun Chul Jang\nTitle: Mass rigidity for asymptotically locally hyperbolic manifolds with boundary \nAbstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold\, the Wang-Chruściel-Herzlich mass integrals are well-defined\, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk\, I will present the result that an ALH manifold which minimize the mass integrals admits a static potential. To show this\, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. And then\, we establish the rigidity of the known positive mass theorems by studying the static uniqueness. This talk is based on joint work with L.-H. Huang.\n\n\n10:40–11:40 am\nAnnachiara Piubello\nTitle: Estimates on the Bartnik mass and their geometric implications. \nAbstract: In this talk\, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular\, if the metric is round the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0\, the estimate tends to 1/2 the area radius\, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.\n\n\nLUNCH BREAK\n\n\n\n\n1:30–2:30 pm\nRyan Unger\nTitle: Density and positive mass theorems for black holes and incomplete manifolds \nAbstract: We generalize the density theorems for the Einstein constraint equations of Corvino-Schoen and Eichmair-Huang-Lee-Schoen to allow for marginally outer trapped boundaries (which correspond physically to apparent horizons). As an application\, we resolve the spacetime positive mass theorem in the presence of MOTS boundary in the non-spin case. This also has a surprising application to the Riemannian setting\, including a non-filling result for manifolds with negative mass. This is joint work with Martin Lesourd and Dan Lee.\n\n\n2:40–3:40 pm\nZhizhang Xie\nTitle: Gromov’s dihedral extremality/rigidity conjectures and their applications I \nAbstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature\, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.\n\n\nTEA BREAK\n\n\n\n\n4:10–5:10 pm\nAntoine Song (virtual)\nTitle: The spherical Plateau problem \nAbstract: For any closed oriented manifold with fundamental group G\, or more generally any group homology class for a group G\, I will discuss an infinite codimension Plateau problem in a Hilbert classifying space for G. For instance\, for a closed oriented 3-manifold M\, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.\n\n\n\n\nTuesday\, May 3\, 2022 \n\n\n\n\n9:30–10:30 am\nChao Li\nTitle: Stable minimal hypersurfaces in 4-manifolds \nAbstract: There have been a classical theory for complete minimal surfaces in 3-manifolds\, including the stable Bernstein conjecture in R^3 and rigidity results in 3-manifolds with positive Ricci curvature. In this talk\, I will discuss how one may extend these results in four dimensions. This leads to new comparison theorems for positively curved 4-manifolds.\n\n\n10:40–11:40 am\nRobin Neumayer\nTitle: An Introduction to $d_p$ Convergence of Riemannian Manifolds I \nAbstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question\, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge\, and what the limiting objects look like. In this mini-course\, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces\, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.\n\n\nLUNCH BREAK\n\n\n\n\n1:30–2:30 pm\nZhongshan An\nTitle: Local existence and uniqueness of static vacuum extensions of Bartnik boundary data \nAbstract: The study of static vacuum Riemannian metrics arises naturally in differential geometry and general relativity. It plays an important role in scalar curvature deformation\, as well as in constructing Einstein spacetimes. Existence of static vacuum Riemannian metrics with prescribed Bartnik data — the induced metric and mean curvature of the boundary — is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss some basic properties of the nonlinear and linearized static vacuum equations and the geometric boundary conditions. Then I will present some recent progress towards the existence problem of static vacuum metrics based on joint works with Lan-Hsuan Huang.\n\n\n2:40–3:40 pm\nZhizhang Xie\nTitle: Gromov’s dihedral extremality/rigidity conjectures and their applications II \nAbstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature\, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.\n\n\nTEA BREAK\n\n\n\n\n4:10–5:10 pm\nTin-Yau Tsang\nTitle: Dihedral rigidity\, fill-in and spacetime positive mass theorem \nAbstract: For compact manifolds with boundary\, to characterise the relation between scalar curvature and boundary geometry\, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk\, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.\n\n\n\n\nSpeakers Banquet\n\n\n\n\nWednesday\, May 4\, 2022 \n\n\n\n\n9:30–10:30 am\nTristan Ozuch\nTitle: Weighted versions of scalar curvature\, mass and spin geometry for Ricci flows \nAbstract: With A. Deruelle\, we define a Perelman-like functional for ALE metrics which lets us study the (in)stability of Ricci-flat ALE metrics. With J. Baldauf\, we extend some classical objects and formulas from the study of scalar curvature\, spin geometry and general relativity to manifolds with densities. We surprisingly find that the extension of ADM mass is the opposite of the above functional introduced with A. Deruelle. Through a weighted Witten’s formula\, this functional also equals a weighted spinorial Dirichlet energy on spin manifolds. Ricci flow is the gradient flow of all of these quantities.\n\n\n10:40–11:40 am\nRobin Neumayer\nTitle: An Introduction to $d_p$ Convergence of Riemannian Manifolds II \nAbstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question\, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge\, and what the limiting objects look like. In this mini-course\, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces\, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.\n\n\nLUNCH BREAK\n\n\n\n\n1:30–2:30 pm\nChristos Mantoulidis\nTitle: Metrics with lambda_1(-Delta+kR) > 0 and applications to the Riemannian Penrose Inequality \nAbstract: On a closed n-dimensional manifold\, consider the space of all Riemannian metrics for which -Delta+kR is positive (nonnegative) definite\, where k > 0 and R is the scalar curvature. This spectral generalization of positive (nonnegative) scalar curvature arises naturally\, for different values of k\, in the study of scalar curvature in dimension n + 1 via minimal surfaces\, the Yamabe problem in dimension n\, and Perelman’s surgery for Ricci flow in dimension n = 3. We study these spaces in unison and generalize\, as appropriate\, scalar curvature results that we eventually apply to k = 1/2\, where the space above models apparent horizons in time-symmetric initial data sets to the Einstein equations and whose flexibility properties are intimately tied with the instability of the Riemannian Penrose Inequality. This is joint work with Chao Li.\n\n\n2:40–3:40 pm\nZhizhang Xie\nTitle: Gromov’s dihedral extremality/rigidity conjectures and their applications III \nAbstract: Gromov’s dihedral extremality and rigidity conjectures concern comparisons of scalar curvature\, mean curvature and dihedral angle for compact manifolds with corners. They have very interesting consequences in geometry and mathematical physics. The conjectures themselves can in some sense be viewed as “localizations” of the positive mass theorem. I will explain some recent work on positive solutions to these conjectures and some related applications (such as a positive solution to the Stoker conjecture). The talks are based on my joint works with Jinmin Wang and Guoliang Yu.\n\n\nTEA BREAK\n\n\n\n\n4:10–5:10 pm\nXin Zhou\n(Virtual)\nTitle: Min-max minimal hypersurfaces with higher multiplicity \nAbstract: It is well known that minimal hypersurfaces produced by the Almgren-Pitts min-max theory are counted with integer multiplicities. For bumpy metrics (which form a generic set)\, the multiplicities are one thanks to the resolution of the Marques-Neves Multiplicity One Conjecture. In this talk\, we will exhibit a set of non-bumpy metrics on the standard (n+1)-sphere\, in which the min-max varifold associated with the second volume spectrum is a multiplicity two n-sphere. Such non-bumpy metrics form the first set of examples where the min-max theory must produce higher multiplicity minimal hypersurfaces. The talk is based on a joint work with Zhichao Wang (UBC).\n\n\n\n\nMay 5\, 2022 \n\n\n\n\n9:00–10:00 am\nAndre Neves\nTitle: Metrics on spheres where all the equators are minimal \nAbstract: I will talk about joint work with Lucas Ambrozio and Fernando Marques where we study the space of metrics where all the equators are minimal.\n\n\n10:10–11:10 am\nRobin Neumayer\nTitle: An Introduction to $d_p$ Convergence of Riemannian Manifolds III \nAbstract: What can you say about the structure or a-priori regularity of a Riemannian manifold if you know certain bounds on its curvature? To understand this question\, it is often important to understand in what sense a sequence of Riemannian manifolds (possessing a given curvature constraint) will converge\, and what the limiting objects look like. In this mini-course\, we introduce the notions of $d_p$ convergence of Riemannian manifolds and of rectifiable Riemannian spaces\, the objects that arise as $d_p$ limits. This type of convergence can be useful in contexts when the distance functions of the Riemannian manifolds are not uniformly controlled. This course is based on joint work with Man Chun Lee and Aaron Naber.\n\n\n11:20–12:20 pm\nPaula Burkhardt-Guim\nTitle: Lower scalar curvature bounds for C^0 metrics: a Ricci flow approach \nAbstract: 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.\n\n\nLUNCH BREAK\n\n\n\n\n1:30–2:30 pm\nJonathan Zhu\nTitle: Widths\, minimal submanifolds and symplectic embeddings \nAbstract: Width or waist inequalities measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds\, as well as applications to quantitative symplectic camels.
URL:https://cmsa.fas.harvard.edu/event/grworkshop/
LOCATION:CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Event,Workshop
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/GR-Workshop-Poster.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220506T100000
DTEND;TZID=America/New_York:20220508T170000
DTSTAMP:20260520T022457
CREATED:20230706T181343Z
LAST-MODIFIED:20231227T080733Z
UID:10000104-1651831200-1652029200@cmsa.fas.harvard.edu
SUMMARY:2022 NSF FRG Workshop on Discrete Shapes
DESCRIPTION:On May 6–8\, 2022\, the CMSA  hosted a second NSF FRG Workshop. \nThis project brings together a community of researchers who develop theoretical and computational models to characterize shapes. Their combined interests span Mathematics (Geometry and Topology)\, Computer Science (Scientific Computing and Complexity Theory)\, and domain sciences\, from Data Sciences to Computational Biology. \nScientific research benefits from the development of an ever-growing number of sensors that are able to capture details of the world at increasingly fine resolutions. The seemingly unlimited breadth and depth of these sources provide the means to study complex systems in a more comprehensive way. At the same time\, however\, these sensors are generating a huge amount of data that comes with a high level of complexity and heterogeneity\, providing indirect measurements of hidden processes that provide keys to the systems under study. This has led to new challenges and opportunities in data analysis. Our focus is on image data and the shapes they represent. Advances in geometry and topology have led to powerful new tools that can be applied to geometric methods for representing\, searching\, simulating\, analyzing\, and comparing shapes. These methods and tools can be applied in a wide range of fields\, including computer vision\, biological imaging\, brain mapping\, target recognition\, and satellite image analysis. \nThis workshop is part of the NSF FRG project: Geometric and Topological Methods for Analyzing Shapes. \nThe workshop was held in room G10 of the CMSA\, located at 20 Garden Street\, Cambridge\, MA. \n\nWorkshop on Discrete Shapes\nMay 6–8\, 2022\nOrganizers: \n\nDavid Glickenstein (University of Arizona)\nJoel Hass (University of California\, Davis)\nPatrice Koehl (University of California\, Davis)\nFeng Luo (Rutgers University\, New Brunswick)\nMaria Trnkova (University of California\, Davis)\nShing-Tung Yau (Harvard)\n\nSpeakers: \n\nMiri Ben-Chen (Technion)\nAlexander Bobenko (TU Berlin)\nJohn Bowers (James Madison)\nSteven Gortler (Harvard)\nDavid Gu (Stony Brook)\nAnil Hirani (UIUC)\nYanwen Luo (Rutgers)\nPeter Schroeder (Caltech)\nJustin Solomon (MIT)\nTianqi Wu (Clark University)\n\nContributed Talk Speakers: \n\nOded Stein (MIT)\nBohan Zhou (Dartmouth)\n\nSchedule\nSchedule (PDF) \nFriday\, May 6\, 2022 \n\n\n\n\n10:00–10:05 am\n\nWelcome Opening\n\n\n10:05–10:55 am\nAnil N. Hirani\nTitle: Discrete vector bundles with connection \nAbstract: We have recently initiated a generalization of discrete exterior calculus to differential forms with values in a vector bundle. A discrete vector bundle with connection over a simplicial complex has fibers at vertices and transport maps on edges\, just as in lattice gauge theory. The first part of this work involves defining and examining properties of a combinatorial exterior covariant derivative and wedge product. We find that these operators commute with pullback under simplicial maps of the base space. From these definitions emerges a combinatorial curvature. In the second part of this work we showed that the curvature behaves as one expects: it measures failure of parallel transport to be independent of the path\, and is the local obstruction to a trivialization. For a bundle with metric\, metric compatibility of the discrete connection is equivalent to a Leibniz rule.  Vanishing curvature is indeed equivalent to an appropriately defined discrete flat connection\, and curvature obstructs trivializability. In this talk I will focus on just the first part\, and talk about naturality of the discrete exterior covariant derivative and discrete wedge product using simple examples. Joint work with Daniel Berwick-Evans (UIUC) and Mark Schubel (Apple\, Inc.).\n\n\n11:10–12:00 pm\nDavid Gu\nTitle: Surface Quadrilateral Meshing Based on Abel-Jacobi Theory \nAbstract: Surface quadrilateral meshing plays an important role in many fields. For example\, in CAD (computer-aided design)\, all shapes are represented as Spline surfaces\, which requires structured quad-meshing; in CAE (computer-aided engineering)\, the surface tessellation greatly affects the accuracy and efficiency of numerical simulations. Although the research on mesh generation has a long history\, it remains a great challenge to automatically generate structured quad-meshes with high qualities. The key is to find the governing equation for the singularities of the global structured quad-meshes. \nIn this talk\, we introduce our recent discovery:  the singularities of a quad-mesh are governed by the Abel theorem. We show that each quad-mesh determines a conformal structure and a meromorphic quadratic differential\, the configuration of the mesh singularities can be described as the divisor of the differential. The quad-mesh divisor minus four times of the divisor of a holomorphic one-form is principal and satisfies the Abel theorem: its image under the Jacobi map is zero in the Jacobi variety. \nThis leads to a rigorous and efficient algorithm for surface structured quadrilateral meshing. After determining the singularities\, the metric induced by the quad-mesh can be computed using the discrete Yambe flow\, and the meromorphic quartic differential can be constructed\, the trajectories of the differentials give the quad-mesh. The method can be applied directly for geometric modeling and computational mechanics.\n\n\n12:00–2:00 pm\nLunch Break\n\n\n\n2:00–2:50 pm\n Justin Solomon\nTitle:  Geometry Processing with Volumes \nAbstract:  Many algorithms in geometry processing are restricted to two-dimensional surfaces represented as triangle meshes.  Drawing inspiration from simulation\, medical imaging\, and other application domains\, however\, there is a substantial demand for geometry processing algorithms targeted to volumes represented as tetrahedral meshes or grids.  In this talk\, I will summarize some efforts in our group to develop a geometry processing toolkit specifically for volumes.  Specifically\, I will cover our recent work on hexahedral remeshing via cuboid decomposition\, volumetric correspondence\, and minimal surface computation via geometric measure theory.\n\n\n3:00–3:20 pm\nOded Stein\nTitle: Optimization for flip-free parametrization \nAbstract: Parametrizations without flipped elements are desirable in a variety of applications such as UV mapping and surface/volume correspondence. Computing flip-free parametrizations can be challenging\, and there are many different approaches to the problem. In this talk we will look at multiple strategies for flip-free parametrizations that are based on the optimization of continuous energies. Due to the nature of the problem\, these energies are often nonconvex and unbounded\, which is a challenge for optimization methods. We will also take a closer look at our recently developed method for computing flip-free parametrizations using the Alternating Direction Method of Multipliers (ADMM).\n\n\n3:20–4:00 pm\nBreak\n\n\n\n4:00–4:50 pm\nJohn Bowers\nTitle: Koebe-Andre’ev-Thurston Packings via Flow \nAbstract: Recently\, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation\, flip-and-flow\, that allows two tangency packings whose contact graphs differ by a combinatorial edge flip to be continuously deformed from one to the other while maintaining tangencies across all of their common edges. Starting from a canonical tangency circle packing with the desired number of circles a finite sequence of flip-and-flow operations may be applied to obtain a circle packing for any desired (proper) contact graph with the same number of circles. \nThe full Koebe-Andre’ev-Thurston theorem generalizes the circle packing theorem to allow for neighboring circles to overlap by angles up to $\pi/2$. In this talk I will show that the Connelly-Gortler method can be extended to allow for circles to overlap to angles up to $\pi/2$. This results in a new proof of the general Koebe-Andre’ev-Thurston theorem for disk patterns on $\mathbb{S}^2$ as well as a numerical algorithm for computing them. The proof involves generalizing a notion of convexity for circle polyhedra that was recently used to prove the global rigidity of certain circle packings\, which is then used to show that all convex circle polyhedra are infinitesimally rigid\, a result of independent interest.\n\n\n5:00–5:30 pm\nMovies\n “conform!” & ”Koebe polyhedra”\n\n\n\n\n  \nSaturday\, May 7\, 2022 \n\n\n\n\n9:30–10:20 am\nAlexander Bobenko\nTitle: The Bonnet problem: Is a surface characterized by its metric and curvatures? \nAbstract: We consider a classical problem in differential geometry\, known as the Bonnet problem\, whether a surface is characterized by a metric and mean curvature function. Generically\, the answer is yes. Special cases when it is not the case are classified. In particular\, we explicitly construct a pair of immersed tori that are related by a mean curvature preserving isometry. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique compact surface. Discrete differential geometry is used to find crucial geometric properties of surfaces. This is a joint work with Tim Hoffmann and Andrew Sageman-Furnas\n\n\n10:20–11:00 am\nBreak\n\n\n\n11:00–11:50 am\nMiri Ben Chen\nTitle: Surface Multigrid via Intrinsic Prolongation \nAbstract: The solution of a linear system is a required ingredient in many geometry processing applications\, and multigrid methods are among the most efficient solution techniques. However\, due to the unstructured nature of triangle meshes\, mapping functions between different multigrid levels is challenging. In this talk I will present our recent work that uses an intrinsic prolongation operator as the main building block in a multigrid solver for curved triangle meshes. Our solver can be used as a black-box in any triangle-mesh based system that requires a linear solve\, and leads to order of magnitude time-efficiency improvement compared to direct solvers.\n\n\n12:00–2:00 pm\nLunch Break\n\n\n\n2:00–2:50 pm\nSteven Gortler\nTitle: Reconstructing configurations and graphs from unlabeled distance measurements \nAbstract: Place a configuration of n  points (vertices) generically in R^d. Measure the Euclidean lengths of m point-pairs (edges). When is the underlying graph determined by these $m$ numbers (up to isomorphism)? When is the point configuration determined by these $m$ numbers (up to congruence)? This question is motivated by a number of inverse problem applications. In this talk\, I will review what is known about this question.\n\n\n3:00–3:20 pm\nBohan Zhou\nTitle: Efficient and Exact Multimarginal Optimal Transport with Pairwise Costs \nAbstract: Optimal transport has profound and wide applications since its introduction in 1781 by Monge. Thanks to the Benamou-Brenier formulation\, it provides a meaningful functional in the image science like image and shape registrations. However\, exact computation through LP or PDE is in general not practical in large scale\, while the popular entropy-regularized method introduces additional diffusion noise\, deteriorating shapes and boundaries. Until the recent work [Jacobs and Leger\, A Fast Approach to Optimal Transport: the back-and-forth method\, Numerische Mathematik\, 2020]\, solving OT in a both accurate and fast fashion finally becomes possible. Multiple marginal optimal transport is a natural extension from OT but has its own interest and is in general more computationally expensive. The entropy method suffers from both diffusion noise and high dimensional computational issues. In this work with Matthew Parno\, we extend from two marginals to multiple marginals\, on a wide class of cost functions when those marginals have a graph structure. This new method is fast and does not introduce diffusion. As a result\, the new proposed method can be used in many fields those require sharp boundaries. If time allows\, we will illustrate by examples the faithful joint recover via MMOT of images with sharp boundaries\, with applications on sea ice prediction.\n\n\n3:20–4:00 pm\nBreak\n\n\n\n4:00–4:50 pm\nPeter Schroeder\nTitle: Constrained Willmore Surfaces \nAbstract: The Willmore energy of a surface is a canonical example of a squared curvature bending energy. Its minimizers are therefore of interest both in the theory of surfaces and in practical applications from physical and geometric modeling. Minimizing the bending energy alone however is insufficient. Taking a cue from univariate splines which incorporate an isometry constraint we consider Willmore minimizers subject to a conformality constraint. In this talk I will report on a numerical algorithm to find such constrained minimizers for triangle meshes. \nJoint work with Yousuf Soliman (Caltech)\, Olga Diamanti (UGraz)\, Albert Chern (UCSD)\, Felix Knöppel (TU Berlin)\, Ulrich Pinkall (TU Berlin).\n\n\n5:00–5:50 pm\n\nProblems and Application discussions\n\n\n\n\n  \nSunday\, May 8\, 2022 \n\n\n\n\n9:00–9:50 am\nTianqi Wu\nTitle: Convergence of discrete uniformizations \nAbstract: The theory of discrete conformality\, based on the notion of vertex scaling\, has been implemented in computing conformal maps or uniformizations of surfaces. We will show that if a Delaunay triangle mesh approximates a smooth surface\, then the related discrete uniformization will converge to the smooth uniformization\, with an error bounded linearly by the size of the triangles in the mesh.\n\n\n10:10–11:00 am\nYanwen Luo\nTitle:  Recent Progress in Spaces of Geodesic Triangulations of Surfaces\n\nAbstract: Spaces of geodesic triangulations of surfaces are natural discretization of the groups of surface diffeomorphisms isotopy to the identity. It has been conjectured that these spaces have the same homotopy type as their smooth counterparts. In this talk\, we will report the recent progress in this problem. The key ingredient is the idea in Tutte’s embedding theorem. We will explain how to use it to identify the homotopy types of spaces of geodesic triangulations. This is joint work with Tianqi Wu and Xiaoping Zhu.\n\n\n11:10–12:00 pm\n\nProblems and Application discussions\n\n\n12:00–1:00 pm\nMovie\n“The Discrete Charm of Geometry”
URL:https://cmsa.fas.harvard.edu/event/2022-nsf-frg-workshop/
LOCATION:CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Event,Workshop
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/FRG-Poster-1-791x1024-1.png
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