Seminar on Differential Geometry and Analysis
Organisers
Roger Bielawski (Hannover) |
Hans-Christoph Grunau (Magdeburg) |
Miles Simon (Magdeburg) |
Knut Smoczyk (Hannover) |
Thursday, May 15, 2025
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 |
Louis Yudowitz (KTH Stockholm): Abstract: When studying the compactness theory for solutions to geometric PDEs, one can encounter the formation of singularities. This occurs, for instance, for sequences of harmonic maps, Einstein manifolds, minimal surfaces, and gradient shrinking Ricci solitons. Provided the singular set consists of isolated points, a common strategy to study them is to construct a “bubble tree” through an iterative blow-up procedure. Moreover, supplementing this with analytic, geometric, and topological arguments allows for the recovery of information lost due to the bubbling behavior. In this talk I will discuss bubbling for gradient Ricci shrinkers, with a focus on how it affects the stability of singularity models for Ricci flow. In particular, I will give an overview of how to prove upper and lower semi-continuity of the Morse index for Ricci shrinkers by studying a certain weighted eigenvalue problem. I will also explain how the same techniques can be applied to show a quantitative relation between the stability of an asymptotically conical Ricci shrinker and that of its asymptotic cone. |
16.30-17.30 |
Fabian Rupp (Universität Wien): Abstract: The curve shortening flow arises as the L^2-gradient flow of the length functional. In this talk, we will discuss a novel energy method that reinterprets “curve shortening” as “tangent aligning”. This interpretation allows us to apply energy-based methods to study the long-time behavior of the curve shortening flow of curves of infinite length. Our method also applies to higher order flows such as the curve diffusion flow, Chen’s flow, and the elastic flow, which involves not only the curve shortening but also the curve straightening effect. This is joint work with T. Miura (Kyoto). |
Previous Sessions:
Thursday, July 4, 2024
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room B 302
15.00-16.00 | Luciano Mari (Università degli Studi di Milano): Prescribing the Lorentzian mean curvature of a spacelike hypersurface, and the Born-Infeld model |
16.30-17.30 | Boris Vertman (Oldenburg): Microlocal Analysis on manifolds with fibered boundaries |
Thursday, April 25, 2024
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Pak-Yeung Chan (University of Warwick): Gap theorem for nonnegatively curved manifolds |
16.30-17.30 | Huy The Nguyen (Queen Mary University of London): Mean Curvature Flow in the Sphere |
Thursday, July 13, 2023
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main building, Room B 302
15.00-16.00 | Oliver Schnürer (Konstanz): Unbounded solutions to mean curvature flow |
16.30-17.30 | Ernst Kuwert (Freiburg): Curvature varifolds with orthogonal boundary |
Thursday, May 4, 2023
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Sascha Eichmann (Tübingen): Positivity for the clamped plate under tension |
16.30-17.30 | Jonas Hirsch (Leipzig): On the De Giorgi-Nash-Moser theorem for hypoelliptic operators |
Thursday, January 26, 2023
Insitut für Differentialgeometrie
Leibniz-Universität Hannover
Callinstr. 23, Seminar room 016, Building 3110
15.00-16.00 |
Athanasios Chatzikaleas (Münster): Non-linear periodic waves in the Anti-de Sitter spacetime and islands of stability |
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16.15-17.15 |
Tobias Lamm (Karlruhe): Index estimates for sequences of harmonic maps |
Thursday, November 3, 2022
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.15-16.15 | Klaus Kröncke (KTH Stockholm): Local and global scalar curvature rigidity of Einstein manifolds |
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16.45-17.45 | Patrick Dondl (Freiburg): Phase field models with connectedness constraints |
Thursday, January 30, 2020
Insitut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F435
15.00-16.00 |
Carla Cederbaum (Tübingen): On CMC-foliations of asymptotically flat manifolds |
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16.30-17.30 |
Gilles Carron (Nantes): Volume growth estimates on complete Riemannian manifolds |
Thursday, November 28, 2019
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Massimiliano Morini (Parma): Nonlinear stability results for nonlocal gradient flows |
16.30-17.30 | Ovidiu Munteanu (Storrs, Connecticut): Green's function estimates and the Poisson equation |
Thursday, July 4, 2019
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Callinstr. 23, Seminar room 016, Building 3110
15.00-16.00 |
Tobias Weth (Frankfurt) : Critical domains for the first nonzero Neumann eigenvalue in Riemannian manifolds |
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16.30-17.30 |
Boris Vertman (Oldenburg): Perelman entropies on singular spaces |
Thursday, May 9, 2019
Insitut für Analysis und Numerik
Otto-von-Guericke Universität Magdeburg
Universitätsplatz 2, Building 02, Room 111
15.00-16.00 |
Matteo Novaga (Università di Pisa): Anisotropic and crystalline mean curvature flow |
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16.30-17.30 |
Alix Deruelle (Sorbonne Université Paris): Classification of asymptotically conical 2D shrinking gradient Kähler-Ricci solitons |
Thursday, January 24, 2019
Insitut für Differentialgeometrie
Leibniz-Universität Hannover
Callinstr. 23, Seminar Room 016, Building 3110
15.00-16.00 |
Andrea Mondino (Warwick, UK): Some smooth applications of non-smooth Ricci curvature lower bounds |
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16.30-17.30 |
Uwe Semmelmann (Stuttgart): The kernel of the Rarita-Schwinger operator on Riemannian spin manifolds |
Thursday, July 5, 2018
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Callinstr. 23, Seminar Room 016, Building 3110
15.00-16.00 | Fabrice Bethuel (Paris VI): A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces |
16.30-17.30 | Alexander Lytchak (Köln) : Minimal discs in singular spaces |
Thursday, April 26, 2018
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Aldo Pratelli (Erlangen): On the closure of planar diffeomorphisms and non-crossing functions |
16.30-17.30 | Jiawei Liu (Magdeburg): The conical Kähler-Ricci flow and related work |
Thursday, January 25, 2018
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Callinstr. 23, Seminar Room 016, Building 3110
15.00-16.00 | Heiko von der Mosel (Aachen): Self-avoidance in higher dimensions - facts and open questions |
16.30-17.30 | Michael Singer (London): Boundary-value problems for 4-dimensional HyperKaehler metrics |
Thursday, December 7, 2017
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg,
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Fernando Galaz-Garcia (Karlsruhe): Sufficiently collapsed Alexandrov 3-spaces |
16.30-17.30 | Thomas Schmidt (Hamburg): Plateau's problem in infinite-dimensional spaces |
Thursday, June 22, 2017
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Callinstr. 23, Seminar Room 016, Building 3110
15.00-16.00 | Jason Lotay (UC London): Laplacian flow in G_2-geometry |
16.30-17.30 | Oliver Goertsches (Marburg): On the basic Dolbeault cohomology of Sasakian manifolds |
Thursday, April 27, 2017
Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Friedrich Sauvigny (Cottbus): Multiple solutions of the nonparametric Plateau problem in the Euclidean space R^p of arbitrary dimension |
16.30-17.30 | Reto Buzano, born Müller (Queen Mary University of London): The moduli space of two-convex embedded spheres and tori |
Thursday, January 19, 2017
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.00-16.00 | Hansjörg Geiges (Köln): Traps and plugs in symplectic dynamics |
16.30-17.30 | Alberto Abbondandolo (Bochum) On the length of shortest closed geodesics and shadows of symplectic balls |
Thursday, October 27, 2016
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 05, Room 307
15.00-16.00 | Felix Schulze (University College London): Ricci flow from spaces with isolated conical singularities |
16.30-17.30 | Felix Finster (Regensburg): Lorentzian spectral geometry and the fermionic signature operator |
Thursday, July 07, 2016
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.00-16.00 | Klaus Kröncke (Hamburg): The moduli space of Ricci-flat metrics |
16.30-17.30 | Adriano Tomassini (Parma, Italy): On the cohomology of complex and symplectic manifolds |
Thursday, April 21, 2016
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Tobias Lamm (Karlsruhe): Conformal Willmore Tori |
16.30-17.30 | Heiner Olbermann (Bonn): Symmetry breaking in indented elastic cones |
Thursday, January 14, 2016
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Panagiotis Gianniotis (University College London, UK): The size of the singular set of a Type I Ricci flow |
16.30-17.30 | Lisa Beck (Augsburg): Duality, regularity and uniqueness for BV-minimizers |
Thursday, November 05, 2015
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.00-16.00 | Paul Bryan (Riemann Center for Geometry and Physics): Harnack inequalities, Alexandrov reflection and ancient solutions of curvature flows on the sphere |
16.30-17.30 | Exotic nearly Kaehler structures on the 6-sphere and the product of two 3-spheres |
Thursday, July 9, 2015
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Francisco Martin (Granada, Spain): http://www-ian.math.uni-magdeburg.de/home/grunau/abstracts/F_Martin_2015.pdf |
16.30-17.30 | Andreas Gastel (Duisburg-Essen): Minimizing polyharmonic maps |
Thursday, January 29, 2015
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.00-16.00 | Nicholas Alikakos (Athens, Greece): A maximum principle for systems with variational structure |
16.30-17.30 | Towards a theory of pde's of mixed elliptic hyperbolic type |
Thursday, November 27, 2014
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Anna Dall'Acqua (Ulm): On the L^2-flow of elastic curves |
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16.30-17.30 | Melanie Rupflin (Leipzig): Global solutions and asymptotics of Teichmüller harmonic map flow |
Thursday, July 10, 2014
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Sergey Grigorian (Edinburg, Texas, USA): Flows of G2-structures |
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16.45-17.45 | Peter Albers (Münster): Orderability -- rigidity in contact geometry |
Thursday, May 22, 2014
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Nikolai S. Nadirashvili (Aix-Marseille, z.Zt. Bielefeld): Fully nonlinear elliptic equations and minimal cones |
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16.30-17.30 | Guofang Wang (Freiburg): Mass and geometric inequalities |
Thursday, January 23, 2014
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Andrew Swann (Aarhus, Denmark): Twists and special geometries |
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16.45-17.45 | Karsten Grove (Notre Dame, USA): Reflections in non-negative curvature |
Thursday, November 14, 2013
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Matthias Röger (Dortmund): Minimization of bending energies under constraints |
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16.30-17.30 | Mohameden Ould Ahmedou (Gießen): Conformal metrics of prescribed Q-curvature on 4-manifolds |
Thursday, July 11, 2013
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room B 302
15.15-16.15 | Marcos Dajczer (IMPA, Rio de Janeiro): Deformations preserving the Gauss map |
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16.45-17.45 | Theodoros Vlachos (Ioannina, Greece): The associated family of an elliptic surface and applications to minimal submanifolds |
Thursday, April 18, 2013
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Christoph Böhm (Münster): Ricci flow, Lie theory and sphere theorems |
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16.30-17.30 | Herbert Koch (Bonn): Parabolic differential equations with rough data |
Thursday, January 31, 2013
Freie Universität Berlin
Fachbereich Mathematik und Informatik
Arnimallee 22 - Hörsaal B
14.00-15.00 | Esther Cabezas Rivas (Münster): A generalization of Gromov's almost flat manifold theorem |
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15.30-16.30 | Tom Ilmanen (ETH Zürich): Singularities of Mean Curvature Flow in R^3 |
Thursday, November 22, 2012
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Francisco Martin (Granada): Gluing constructions for complete minimal surfaces with finite total curvature in H^2xR |
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16.45-17.45 | Pieralberto Sicbaldi (Marseille): New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator |
Thursday, June 28, 2012
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
16.00-17.00 | Jan Metzger (Potsdam): On isoperimetric surfaces in Riemannian manifolds |
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17.30-18.30 | Emanuele Spadaro (MPI Leipzig): Mean-convex sets and global barriers to minimal hypersurfaces |
Thursday, April 26, 2012
Freie Universität Berlin
Fachbereich Mathematik und Informatik
Arnimallee 6, SR 032
14.00-15.00 | Alexander Grigor'yan (Universität Bielefeld): Negative eigenvalues of two-dimensional Schrödinger operators |
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15.30-16.30 | Peter Topping (University of Warwick): Mean curvature flow from cones |
Thursday, January 26, 2012
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 106
15.00-16.00 | Oliver Schnürer (Konstanz): Gauß curvature flows of entire graphs |
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16.30-17.30 | Peter Hornung (MPI Leipzig): Analysis of intrinsically flat surfaces and the Willmore functional |
Thursday, November 17, 2011
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Jost-Hinrich Eschenburg (Augsburg): Symmetric spaces and division algebras |
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16.45-17.45 | Andreas Savas-Halilaj (MPI Leipzig): Bernstein type problems |
Thursday, June 30, 2011
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Carlo Mantegazza (Scuola Normale Superiore di Pisa): Flow by mean curvature inside a moving ambient space |
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16.30-17.30 | Armin Schikorra (RWTH Aachen, z.Zt. ETH Zürich): Regularity for fractional harmonic maps |
Thursday, April 14, 2011
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Josef Dorfmeister (TU München): Willmore spheres in Sn via loop groups |
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16.45-17.45 | Peter Quast (Augsburg): Centrioles in symmetric spaces |
Thursday, January 27, 2011
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 05, Room 307
15.00-16.00 | Olivier Druet (École normale supérieure de Lyon): The notion of stability for elliptic PDEs |
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16.30-17.30 | Simon Blatt (ETH Zürich): The gradient flow of O'Hara's knot energies |
Thursday, November 11, 2010
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Bernhard Hanke (Augsburg): Homotopy groups of the moduli space of metrics of positive scalar curvature |
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16.45-17.45 | Matthias Schneider (Heidelberg): Closed magnetic geodesics |
Thursday, July 1, 2010
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 106
14.45-15.45 | Yann Bernard (Freiburg): Local Palais-Smale sequences for the Willmore functional |
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16.30-17.30 | Jan Metzger (Potsdam): Small surfaces of Willmore type in Riemannian manifolds |
Thursday, April 15, 2010
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Dmitri V. Alekseevsky (Edinburgh / Hamburg): Subdistributions of the contact distribution, associated PDE of Monge-Ampere type and geometry of Lagrangian Grassmannian |
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16.45-17.45 | Andreas Juhl (Humboldt-Universität Berlin / Uppsala): The recursive structure of Q-curvature |
Thursday, January 14, 2010
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 03, Room 214
15.00-16.00 | Ben Schweizer (Dortmund): Modelling porous media: degenerate equations and variational inequalities |
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16.15-17.15 | Felix Schulze (FU Berlin): Foliations of asymptotically flat manifolds by surfaces of Willmore type |
Thursday, November 5, 2009
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Helga Baum (Humboldt-Universität Berlin): A conformal analogon of Calabi Yau manifolds |
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16.45-17.45 | Tilmann Wurzbacher (Metz): Geometry and quantization of symplectic Howe pairs |
Thursday, June 11, 2009
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 02, Room 111
15.15-16.15 | Alberto Abbondandolo (Leipzig): Estimates and computations in Rabinowitz-Floer homology |
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16.45-17.45 | Matthias Schwarz (Leipzig): Product structures in string topology and Floer homology |
Thursday, April 16, 2009
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Lorenz Schwachhöfer (Dortmund): Extrinsic symplectic symmetric spaces |
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16.45-17.45 | Dorothee Schüth (Humboldt-Universität Berlin): Spectral isolation of bi-invariant metrics on compact Lie groups |
Thursday, January 8, 2009
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 02, Room 210
15.00-16.00 | Michael Struwe (ETH Zürich): The heat flow with a critical exponential nonlinearity |
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16.30-17.30 | Sergio Conti (Bonn): Multiscale decomposition of dislocation microstructures |
Thursday, October 16, 2008
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 02, Room 210
15.15-16.15 | Jiayu Li (ICTP Trieste, Italy): Symplectic critical surfaces in Kähler surfaces |
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16.45-17.45 | Matthias Röger (MPI Leipzig): The Allen-Cahn action functional and its sharp interface limit |
Thursday, June 12, 2008
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Vladimir Matveev (Jena): Quadratic integrals of geodesic flows and solution of S. Lie problem |
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16.45-17.45 | Klaus Mohnke (Humboldt-Universität Berlin): Symplectic hypersurfaces and transversality in Gromov-Witten theory |
Thursday, April 17, 2008
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Hans-Bert Rademacher (Leipzig): Finsler metrics and closed geodesics |
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16.45-17.45 | Wilderich Tuschmann (Kiel): Almost nonnegative curvature |
Thursday, January 17, 2008
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 05, Room 118
15.15-16.15 | Reiner Schätzle (Tübingen): Minimizers of the Willmore functional under fixed conformal class |
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16.45-17.45 | Andrea Malchiodi (Sissa Trieste, Italy): Prescribing constant curvatures in conformal geometry |
Thursday, November 8, 2007
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 309
15.15-16.15 | Christian Bär (Potsdam): Conformal structures in non-commutative geometry |
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16.45-17.45 | Burkhard Wilking (Münster): New Ricci flow invariant curvature conditions |
Thursday, June 28, 2007
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 05, Room 118
15.15-16.15 | Klaus Ecker (FU Berlin): Mean value formulas for evolving metrics |
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16.45-17.45 | Felix Finster (Regensburg): Geometry and analysis of asymptotically flat manifolds of small mass |
Thursday, April 19, 2007
Institut für Differentialgeometrie
Leibniz-Universität Hannover
Welfengarten 1, Main Building, Room F 428
15.15-16.15 | Hansjörg Geiges (Köln): A Bernstein problem for contact spheres |
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16.45-17.45 | Karl Friedrich Siburg (Dortmund): Symplectic proofs of Borsuk-Ulam type theorems |
Thursday, January 25, 2007
Institut für Analysis und Numerik
Otto-von-Guericke-Universität Magdeburg
Universitätsplatz 2, Building 02, Room 111
15.15-16.15 | Frederic Robert (University of Nice): Bubbling phenomena for fourth order equations with exponential growth arising from geometry |
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16.45-17.45 | Tobias Lamm (Max Planck Institute for Gravitational Physics, Potsdam-Golm): Conservation laws for fourth order systems in four dimensions |
Thursday, November 2, 2006
Institut für Differentialgeometrie
Leibniz-Universität Hannover Welfengarten 1,
Main Building, Room F 142
15.15-16.15 | Steffen Fröhlich (Darmstadt): Krümmungsabschätzungen und Bernstein-Sätze für zweidimensionale Immersionen in Euklidischen Räumen höherer Kodimension |
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16.45-17.45 | Karsten Große-Brauckmann (Darmstadt): Symmetric constant mean curvature surfaces |