Talks A. Rendall

8/11/23 Mathematics Colloquium of the Universities of Marburg and Giessen, Giessen, Multistability in biochemical reaction networks.

11/9/23 Workshop on mathematical aspects of immunobiology, Blagoevgrad, Bulgaria, Monotonicity of the response function for the initiation of T cell activation.

5/7/23 Open Day. JGU Mainz, Macht Mathematik gesund?

16/9/22 Annual conference of the DMV, FU Berlin, Phenomenology of an in-host model for hepatitis C.

12/11/21 Applied Mathematics Seminar, St. Andrews, Scotland, online. Regulation of enzymatic activity of Lck by autophosphorylation.

16/6/21 SMB Annual Meeting, online, Global convergence to steady states in a model for the in-host dynamics of hepatitis C

20/4/21 Oberseminar Nonlinear Dynamics (FU Berlin/WIAS), online. Bogdanov-Takens bifurcations and the regulation of enzymatic activity by autophosphorylation.

8/4/21 Online seminar Mathematics of Reaction Networks (Torino/Copenhagen/Wien), Using Bogdanov-Takens bifurcations to study existence and stability of periodic solutions.

17/8/20 SMB Annual Meeting, online, Autophosphorylation as a source of multistability

26/9/19 Annual conference of the DMV, Karlsruhe, A proof of unlimited multistability for phosphorylation cycles

9/9/19 Conference, Mathematical Biology on the Mediterranean, Samos, Greece, Mathematical models for immune checkpoint therapy of cancer

23/7/19 SMB annual conference, Montreal, Canada, A proof of unlimited multistability for phosphorylation cycles

12/3/19 KTH Stockholm, Sweden, Using mathematics to improve understanding of T cell activation

30/1/19 Open Day. JGU Mainz, Macht Mathematik gesund?

10/9/18 RWTH Aachen, Periodic solutions of the MAPK cascade

8/8/18 SIAM Life Science Conference, Minneapolis, USA, Dynamics of the Selkov model of glycolysis

3/8/18 Northeastern University 2018 Systems Biology Day, Boston, USA, Models of T cell activation

2/8/18 Harvard Medical School, Boston, USA, Models for the initiation of T cell activation

23/7/18 Conference, ECMTB 2018, Lisbon, Portugal, Bounded and unbounded oscillations in the Selkov model of glycolysis

5/7/18 SYMBIONT Kick-off Meeting, Bonn, Examples of how mathematics can be used to identify qualitative features of solutions of reaction networks

2/3/18 Conference, The Mathematics of Mechanobiology and Cell Signalling, Oberwolfach. Initiation of T cell signalling

30/1/18 Open Day. JGU Mainz, Macht Mathematik gesund?

28/7/17 University of Heidelberg, Models for the initiation of T cell activation

21/6/17 Conference, Reaction Networks and Population Dynamics, Oberwolfach, Mathematical models for T cell activation

14/6/17, Mainzer Mathe Tag, Mathematik und das Immunsystem

8/6/17 Conference,  Mathematical Analysis of Biological Interaction Networks, Banff International Research Station, Canada, Multiple steady states in models for the Calvin cycle of photosynthesis

13/2/17, Winter Excursion, Elite Master Course TMP, Bayrischzell, Relations between mathematics and its applications to biology and medicine

26/7/16 Conference, Patterns of Dynamics, Berlin, Dynamics of the Calvin cycle

15/7/16 Conference ECMTB 2016, Nottingham, UK, Energy distribution and endothermy in dinosaur and modern birds

10/6/16 Theoretical Biology Laboratory, RIKEN, Wako, Japan, Mathematical models for the activation of T-cells

7/6/16 Theoretical Biology Laboratory, RIKEN, Wako, Japan, Dynamics of the Calvin cycle

2/6/16 FU Berlin, Dynamical systems arising from the Calvin cycle

11/5/16 Niels Bohr Institute, Copenhagen, Dynamics of the Calvin cycle

3/10/15 Central Fall Sectional Meeting, American Mathematical Society, Chicago, USA, Overload breakdown in models for photosynthesis

29/9/15 Colloquium, University of West Virginia, Sustained oscillations in phosphorylation cascades.

3/7/15 Conference, Mathematical Trends in Reaction Network Theory, Copenhagen, Dynamics of phosphorylation systems

4/3/15 Conference, Flowers and Friends in Frankfurt, Qualitative properties of dynamical systems arising in biology.

4/2/15 Hausdorff Colloquium, University of Bonn, The MAPK cascade and the mathematics of signalling networks in cell biology.

3/2/15  Open Day. JGU Mainz, Als Hasen Luchse frassen

1/10/14 International BIOMS Workshop Modelling Cellular Systems, BioQuant, Heidelberg, Dynamical features of the MAP kinase cascade: phenomenology and functional meaning

5/8/14 SIAM conference on the life sciences, Charlotte, USA, Bistability in the dual futile cyle and periodic Periodic solutions of the MAPK
cascadesolutions of the MAPK cascade

8/7/14 10th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, Madrid, Dynamical properties of the MAP kinase cascade

27/2/14 Max Planck Institut für Dynamik komplexer technischer Systeme, Magdeburg, Dynamische Eigenschaften der MAPK-Kaskade

30/1/14 Open Day. JGU Mainz, Als Hasen Luchse frassen

8/11/13 Mainzer Abend der Mathematiker plus Freunde, JGU Mainz, Als Hasen Luchse frassen: Mathematik und Wirklichkeit

7/11/13 Theory Colloquium, Institute for Physics, JGU Mainz, Die Verwendung struktureller Eigenschaften bei der Analyse mathematischer Modelle in der Molekularbiologie

6/11/13 Colloquium, University of Würzburg, Analysis of some dynamical systems arising in molecular biology

26/9/13 Annual conference of the ÖMG and DMV, Innsbruck, Dynamical properties of the Calvin cycle of photosynthesis.

28/6/13 Colloquium, Forschungsschwerpunkt Mathematische Modellierung, University of Bielefeld. Chemical reaction network theory and immunological signalling pathways.

14/6/13 Forschungsseminar Darmstadt-Mainz. Ein Modell für die Photosythese und Systeme mit verschiedenen Zeitskalen.

23/5/13 Inaugural Lecture, JGU Mainz.  Mathematische Analyse dynamischer Systeme aus der Zellbiologie.

15/5/13 Institute for Molecular Biology, Mainz. NFAT and T cell activation.

8/1/13 Colloquium SFB 647, Max Planck Institute for Gravitational Physics. Introduction to the BKL picture.

30/11/12 Berlin Mathematical School Friday Colloquium. Spacetime singularities and heteroclinic chains.

19/11/12 Karlstad University. Illustrative examples of the uses of mathematical modelling.

1/11/12 FU Berlin. Relaxation oscillation in the MAP kinase cascade.

29/10/12 University of Münster. Oscillatory singularities in solutions of the Einstein equations.

3-7/9/12 Spanish Relativity Meeting in Portugal, Guimarães. Talk: Construction of oscillatory singularities.

30/7-3/8/12 Conference, Mathematical Aspects of General Relativity, Oberwolfach, Talk: Stability of heteroclinic cycles and construction of oscillatory singularities.

25-28/7/12 Conference, Society for Mathematical Biology Annual Conference, Knoxville, USA. Talk: Mathematics of the NFAT signalling pathway.

21/5/12 KTH Stockholm. Future stability for the Einstein-matter equations and the cosmic no hair theorem.

11/5/12 FU Berlin. Signalling pathways in T cells and chemical reaction network theory.

9/2/12 Université de Paris VI. Singularity formation in solutions of the Einstein-Vlasov system.

20/1/12 University of Erlangen. Globale Dynamik von Lösungen des Einstein-Vlasov-Systems.

15/12/11 FU Berlin. Multiple steady states in a dynamical system arising in immunology.

4/11/11 University of Vienna. Analyse eines mathematischen Modells aus der Immunologie.

3/11/11 University of Vienna. Late-time behaviour of spatially homogeneous solutions of the Einstein equations.

25/9/11 Conference, Vlasov models in kinetic theory, Brown University, USA. Vortrag, A self-similar solution of the Einstein-Vlasov system.

22-24/6/11 Conference, Kinetic theory and related fields, Pohang University, Korea. Talk: Late-time behaviour of spatially homogeneous solutions of the Einstein-Vlasov system.

20/6/11 Seoul National University, Korea. A self-similar solution of the Einstein-Vlasov system.

17/6/11 Seoul National University, Korea. Introduction to the Einstein-Vlasov system.

18/5/11 Research Center ImmunoSciences, Charité, Berlin. Analysis of a model for interactions between T cells and macrophages.

9/2/11 University of Heidelberg. Dynamik stossfreier Materie.

28/1/11 University of Mainz. Dynamik stossfreier Materie.

27/1/11 Erwin Schrödinger Institute, Vienna. Self-similar collapse of collisionless matter.

22/12/10 RWTH Aachen. Dynamik stossfreier Materie.

4/12/10 Swedish GR meeting, Karlstad. Self-similar collapse of collisionless matter.

21-25/6/10 Conference, Geometric Analysis and General Relativity, Banff, Canada. Talk: Higher dimensional cosmological models.

5-9/4/10 Conference, PDEs, Relativity and Nonlinear Waves, Granada, Spanien. Talk: Some wave maps related to the Einstein equations.

16/3/10 University of Bonn. Anwendungen von hyperbolischen Gleichungen in der Kosmologie.

3/3/10 Kompaktkurs Relativitätstheorie, Graduiertenkolleg des SFB 647, Berlin. Open problems in general relativity.

18-29/1/10 Research program, 'Quantitative Studies of Nonlinear Wave Phenomena', Erwin Schrödinger Institute, Vienna. Talk: Relations between Gowdy and Bianchi spacetimes.

18/12/09 FU Berlin. Introduction to black holes.

4-6/11/09 Conference, Mathematical methods in general relativity and quantum field theories, Paris. Talk: Relations between Gowdy and Bianchi spacetimes

20/10/09 University of Oxford. Relations between Gowdy and Bianchi spacetimes.

14-18/9/09 Conference, Hot Topics: Black Holes in Relativity, MSRI, Berkeley. Talk: The characteristic initial value problem in general relativity.

20-22/7/09 Conference, Lorentzian Geometry, Greifswald. Talk: The Einstein-Maxwell equations and the complex hyperbolic plane.

18-19/6/09 Conference, Mathematical Relativity in Lisbon. Talk: Dynamics of linearized cosmological perturbations.

12/6/09 University of Jena. Loss of regularity in solutions of the Einstein-Euler system.

30/4/09 FU Berlin. Introduction to the Gowdy equations.

17/4/09 FU Berlin. Localization at the singularity in Bianchi models.

2/4/09 University of Edinburgh. Cosmic censorship: an introduction and status report

9/3/09 University of Cambridge. Dynamics of linearized cosmological perturbations.

26/11/08 MPIMIS Leipzig. Analysis of large scale structure formation.

15/10/08 Linköping University. Loss of regularity in solutions of the Einstein-Euler system.

9/10/08 Mittag-Leffler Institute, Stockholm. Cosmological weak null singularities

25-29/8/08 Conference, Geometry and Analysis, KTH Stockholm. Talk: Cosmological perturbation theory.

16-19/6/08 Conference, Hyperbolic Equations in Relativity. University of Bordeaux. Talk: Loss of regularity in solutions of the Einstein-Euler system.

7-17/4/08 Conference, Mathematical aspects of general relativity. Niels Bohr Institute, Copenhagen. Talk: The interface between mathematics and astrophysics in the study of cosmic acceleration.

22/4/08 HU Berlin. Spatially homogeneous solutions of the Einstein equations.

13/3/08 Jagiellonian University, Krakow. Strong cosmic censorship from the point of view of Lorentzian geometry.

12/3/08 Jagiellonian University, Krakow. Dynamics of accelerated cosmological expansion.

7/11/07 Universidad Complutense, Madrid. Dynamics of scalar field models in cosmology.

1/11/07 FU Berlin.  Dynamical systems arising from Bianchi models.

16-20/7/07 6th International Congress on Industrial and Applied Mathematics, Zurich. Talk: Asymptotics of expanding cosmological models.

7/6/07 FU Berlin. Dynamics of Bianchi models.

4/4/07 Université de Paris VI. Stabilité de l'espace de de Sitter.

2/4/07 Université de Paris VI. Structure globale des espace-temps avec matière cinétique.

24/1/07 MPI für Mathematik in den Naturwissenschaften, Leipzig. Some ODE and PDE problems arising in cosmology.

14/11/06 Colloquium of the SFB 647, HU Berlin. Exotic matter models in cosmology.

6-10/11/06 Workshop on Global Problems in Relativity, Universidad de Córdoba, Argentinien. Talk: Late-time oscillatory behaviour in cosmological spacetimes.

9-11/10/06 Conference on Multiscale Models, TU München. Talk: Theoretical approaches to gravitational radiation.

22-30/8/06 International Congress of Mathematicians, Madrid. Talk: Models for accelerated cosmological expansion.

25/5/06 Kings College London, UK. Mathematics of accelerated cosmological expansion.

27/4/06 University of Göttingen,  Solutions of the Einstein equations with accelerated expansion.

10/3/06 University of Cambridge, UK.  Dynamics of $k$-essence.

13/1/06 Oberwolfach. Accelerated cosmological expansion and $k$-essence.

12/12/05 University of Duisburg-Essen. Fuchs'sche Systeme und ihre Anwendungen auf die Allgemeine Relativitätstheorie.

22-26/8/05 Conference, Global general relativity, Isaac Newton Institute, Cambridge, UK. Talk: Asymptotic expansions for cosmological solutions of the Einstein equations.

18/8/05 Université de Yaounde I, Cameroon. Black holes and trapped surfaces in solutions of the Einstein-Vlasov system.

16/8/05 Université de Yaounde I, Cameroon. New results for Vlasov systems.

18-21/7/05 Conference, Asymptotic Analysis and Singularity, Sendai, Japan. Plenary talk: Solutions of the Einstein equations with accelerated expansion.

29/6/05 Colloquium of the SFB 647, FU Berlin. Cosmic acceleration and dark energy.

23/6/05 Colloquium, FU Berlin. Mathematik der kosmischen Beschleunigung.

9/6/05 Colloquium, University of Bayreuth. Die Allgemeine Relativitätstheorie und die kosmische Beschleunigung.

18/1/05 ETH Zürich, Switzerland. Mathematics of cosmic acceleration.

22/11/04 Colloquium, Chalmers University of Technology, Göteborg, Sweden. Mathematics of cosmic acceleration.

18/11/04 Chalmers University of Technology, Göteborg, Sweden, The plane symmetric Einstein-Vlasov system.

4/10/04 IMPRS seminar, Beetzsee. Talk: Global asymptotics of solutions of the Einstein equations.

19/7/04 GR17, Dublin, Ireland. Talk: Mathematical results on cosmological models with accelerated expansion.

14/7/04 Erwin Schrödinger Institute, Vienna, Austria. Analogies between spacetime singularities and inflationary late-time asymptotics.

12/7/04 HYKE conference, Vienna, Austria. Talk: Solutions of the Einstein equations with accelerated expansion.

9/7/04 HYKE conference, Vienna, Austria. Talk: Introduction to the Einstein equations.

8/6/04 Two-city geometry seminar, MPI for Mathematics in the Sciences, Leipzig. Dynamics of cosmological models with accelerated expansion.

24/4/04 Weierstrass-Institut, Berlin. Das Einstein-Vlasov-System.

8/3/04 IHES, France. Accelerated expansion in cosmology.

4/3/04 Bad Honnef. Mathematical properties of cosmological models with accelerated expansion.

05/01/04 Conference Miami Waves, Miami, USA. Vortrag: Asymptotics of solutions of the Einstein equations with positive cosmological constant.

5/12/03 Université de Tours, France. Asymptotics of solutions of the Einstein equations with positive cosmological constant.

17/9/03 Annual conference, Deutsche Mathematiker-Vereinigung, Rostock. Talk: Simplification of the Einstein equations due to a cosmological constant.

23/6/03 Isaac Newton Institute, Cambridge, UK. Late-time asymptotics of expanding cosmological models.

23/5/03 University of Freiburg. A hyperbolic system modelling the dynamics of spacetime geometry.

12/11/02 FU Berlin. Monotone and oscillatory behaviour near
spacetime singularities.

21/10/02 Astrophysics Institute, Potsdam. Gravity, geometry, analysis.

29/7/02, 30/7/02, 1/8/02 Lecture series, Cargèse, Corsica, France. The
Einstein-Vlasov system.

24/6/02 IMA, Minneapolis, USA. The Cauchy Problem for the Einstein equations.

21/5/02 Stanford, USA. The Einstein-Vlasov system.

7/5/02 Karlstad, Sweden,  Mathematics of self-gravitating collisionless matter.

15/2/02 Colloquium, University of Waterloo, Canada. The Einstein equations from a PDE perspective.

14/2/02 University of Waterloo, Canada. Spatially homogeneous cosmological models with collisionless matter.

8/1/02 ETH Zurich, Switzerland. When do the Einstein equations lead to simple singularities?

13/12/01 Erwin Schrödinger Institut, Vienna, Austria. The non-relativistic limit of the Einstein-Vlasov system.

20/11/01 University of Essen. Ein System von hyperbolischen Gleichungen als Modell für Raumzeit-Singularitäten.

21/7/01 GR16 Conference, Durban, South Africa. Plenary talk: Applications of the theory of evolution equations to general relativity.

3/7/01 Erwin Schrödinger Institute, Vienna, Austria. Fuchsian equations and spacetime singularities.

17-18/6/01 Conference on kinetic theory, Anogia, Crete, Greece. Two talks: The Einstein-Vlasov system.

21/5/01 Oberwolfach. Asymptotic decoupling of solutions of the Einstein equations.

29/11/00 TU Munich. Analysis of a continuum mechanics model with radiation damping,

19/10/00 MPI for Mathematics in the Sciences, Leipzig. Structure of singularities in general solutions of the Einstein equations.

13/10/00 Université Libre de Bruxelles, Belgium. New approaches to singularities in general relativity.

24-25/8/00 Asia Pacific Centre for Theoretical Physics, Seoul, Korea. Two talks: New developments in the study of spacetime singularities.

5-9/6/00 Conference, Journées Equations aux Derivées Partielles, Nantes, France. Talk: Blow-up for solutions of hyperbolic PDE and spacetime singularities.

31/05/00 Habilitation lecture, TU Berlin. Singuläre Lösungen hyperbolischer Systeme und ihre Bedeutung fuer die Gravitationstheorie.

2/3/00 Eighth International Conference on Hyperbolic Problems. Theory, Numerics, Applications, University of Magdeburg. Talk: The Einstein equations and hyperbolicity.

31/1/00 University of Zurich, Switzerland. Structure of spacetime singularities in the presence of a scalar field.

20/1/00 University of California San Diego, USA. Singularity structure of solutions of the Einstein equations.

3 weeks in 1/00. Research visit, Institute for Theoretical Physics,  Santa Barbara, USA. Talk: Hyperbolicity and numerical relativity.

5-11/11/99 Conference, Nonlinear equations in many-particle systems, Oberwolfach. Talk: Survey of results on the Einstein-Vlasov system.

2/8/99 Conference, Equadiff99, FU Berlin. Talk: Solutions of the Einstein equations with prescribed singularity structure.

2/7/99 Talk, University of Potsdam. Vlasov systems: an overview.

4/6/99 Edinburgh Mathematical Society, Aberdeen, UK. Talk: Blow-up singularities of solutions of hyperbolic PDE and their significance for general relativity.

4 weeks in 4/99. Research visit, Institute for Theoretical Physics, Santa Barbara, USA. Talk: Quiescent cosmological singularities.

11/02/99 Université de Reims, France. Solutions globales du système d'Einstein-Vlasov.

07-11/09/98 Seminar on mathematical relativity theory, Bad Honnef. Talks (4 hours): The theorem of Christodoulou and Klainerman on the global nonlinear stability of Minkowski space.

30/3-4/4/98 International Seminar on Mathematical Cosmology, University of Potsdam. Talk: Structure of singularities in inhomogeneous cosmological models.

10/3/98 Oxford Mathematical Institute, UK. Fuchsian equations and general relativity.

5/2/98 University of Potsdam. Fuchsian equations in general relativity.

6/11/97 University of Potsdam (in the framework of the 'Tag der Naturwissenschaftlichen Fakultät'). Talk: Dynamische Eigenschaften der Gravitation.

2-8/3/97 Conference, Quantenfeldtheorie und Wellenfronten, Oberwolfach. Talk: Meditation on spacetime singularities.

1-7/12/96 Conference, Nonlinear Equations in Many-Particle Systems, Oberwolfach. Talk: Solutions of the Einstein-Vlasov system with U(1) x U(1) symmetry.

7/96 2nd International Congress of Nonlinear Analysts, Athens, Greece, Talk: Crushing singularities.

28/6/96 Conference, Journées Relativistes, Ascona, Switzerland. Plenary talk: Constant mean curvature foliations in cosmological spacetimes.

8/5/96 TU Berlin. Globale symmetrische Lösungen des Einstein-Vlasov-Systems.

23/4/96 Royal Institute of Technology, Stockholm, Sweden. Global dynamics of solutions of the Einstein equations.

2 Wochen in 3/96 Research visit, Banach Centre, Warsaw, Poland. Talk: Topology of compact Cauchy horizons.

6-12/8/95 14th International Conference on General Relativity and Gravitation, Florence, Italy. Plenarvortrag: Solutions of the Einstein equations with matter.

6/4/95 Université de Bordeaux, France. Solutions cosmologiques globales du système d'Euler-Poisson.

24/3/95 Frühjahrstagung der Deutschen Physikalischen Gesellschaft, Berlin. Main talk: Die Natur kosmologischer Singularitäten.

4/11/94 Observatoire de Meudon, Paris, France. Reliability of post-Newtonian approximations.

11/10/94 Universite de Paris VI, France. Cosmologie newtonienne.

1-31/7/94 Research visit, Erwin Schroöinger Institute, Vienna, Austria. Talk: Crushing singularities in spacetimes with spherical and plane symmetry.

13-17/9/93 Conference, Canonical Gravity: From Classical to Quantum, Bad Honnef. Talk: Adjointness relations as a criterion for choosing an inner product.

2 weeks in 6/93 Research visit, Ecole Polytechnique, Palaiseau, France. Talk: La limite newtonienne des équations d'Einstein.

1-31/3/93 Research visit, University of California at Santa Barbara, USA. Talk: Global existence theorems for the spherically symmetric Vlasov-Einstein system.

12/92 Princeton University, USA. The Newtonian limit of the Vlasov-Einstein system.

9/92 Syracuse University, USA. The Newtonian limit of the Vlasov-Einstein system.

28/6-4/7/92 13th International Conference on General Relativity and Gravitation, Córdoba, Argentina. Talk: Global dynamics of solutions of the spherically symmetric Vlasov-Einstein system.

16-20/12/91 Workshop on Numerical Relativity, Southampton, UK. Talk: On the choice of matter model in general relativity.

10/91 University of Chicago, USA. The initial value problem for self-gravitating fluid bodies.

10/91 University of Michigan, USA. The initial value problem for self-gravitating fluid bodies.

9/91 Talk, University of Maryland, USA. The initial value problem for self-gravitating fluid bodies.

9/91 Talk, Princeton University, USA.  The initial value problem for self-gravitating fluid bodies

30/7-9/8/91 Xth Congress on Mathematical Physics, Leipzig. Talk: The initial value problem for self-gravitating fluid bodies

11/90 University of Linköping, Sweden. Generic properties of spacetimes in general relativity.

11/90 Royal Institute of Technology, Stockholm, Sweden. The Cauchy problem for the Einstein-Euler equations.

6/90 Conference, Nonlinear hyperbolic equations and field theory, Varenna, Italy. Talk: The characteristic initial value problem for the Einstein equations.

24/4/90 Conference, Georgenthal, DDR. Vortrag: The characteristic initial value problem for the Einstein equations.

1-31/3/90 Research visit, Institut des Hautes Etudes Scientifiques, Bures sur Yvette, France. Talk: The characteristic initial value problem for the Einstein equations.

9/89 Third Hungarian Relativity Workshop, Tihany, Hungary. Talk: Stability in the characteristic initial value problem.

2-8/7/89 12th International Conference on General Relativity and Gravitation, Boulder, USA. Talk: The characteristic initial value problem for the Einstein equations.

13/12/88 Charles University, Prague, Czechoslovakia. Global existence of solutions of the Robinson-Trautman equation.

8-13/8/88 5th Marcel Grossmann Meeting, Perth, Australia. Talk: The Robinson-Trautman equation and Lyapunov functions.

7/88 University of Sydney, Australia. Talk: Global existence of solutions of the Robinson-Trautman equation.

20-22/7/88 Conference on Mathematical Relativity, Canberra, Australia. Talk: Describing spacetimes by their curvature.