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Nodal Surfaces

A large part of my work is connected to (hyper-)surfaces with many singularities. The main question in this subject is: What is the maximum number μ(d) of isolated singularities on a surface of degree d in projective three-space?

Up to now, this question is only answered for d <= 6. In this section, we give known bounds on μ(d) and provide images and animations of known examples.

In 2004, I constructed a surface of degree 7 with 99 nodes. All these singularities are real (see picture at the left).

For details of the construction, see the section on septic surfaces or my article on arxiv.org.

For another website on this topic, see Stephan Endraß's site.

Recent & Upcoming (Selected Events)

  • July 24-31 2016: ICME Hamburg. Invited Talk and exhibition (with S. Halverscheid) on Felix Klein and mathematical models during the thematic afternoon on Wednesday.
  • Jule 20-23, 2016: Organiser of a Workshop at the "Imaginary Conference" on an Imaginary Book.
  • June 28, 2016: Invited Talk/Workshop at ICERM (Providence, USA) on algebraic surfaces, their visualisations and 3D-models
  • Oct. 27, 2015: Talk "On Alfred Clebsch and Cubic Surfaces." at the history of mathematics meeting at MFO, Oberwolfach, Germany
  • March 13, 2014: Talk at Nancy (France) on cubic surface models. See MO-Labs.com for some of them.
  • April 1, 2013: My new website MO-Labs.com on which I present some of the mathematical objects I have been producing.
  • Sept. 29, 2012: Workshop on the question: How to use the new capabilities of 3d-printing for teaching purposes. AK MUI Meeting at Soest, Germany.
  • Sept 20, 2012: Two workshops at the DMV-Tagung on: Neue mathematische 3d-Modelle für die Lehre.
  • Sept. 14-15, 2012: A small exhibition on mathematical 3d-models for teaching during the conference on geometry teaching at Saarbrücken, Germany.
  • August 2012: My new website MO-Labs.com has been released. You may now purchase some of my mathematical 3d-models online.
  • June 23, 2012: Talk at the conference CADGME 2012, at Novi Sad, Serbia.
  • June 12, 2012: Talk at the Mini-Workshop on the history of mathematical instruments, mashines, and models. Mainz, Germany.
  • May 31, 2012: Opening of the exhibition Forms and Formulas at Lisboa, Portugal, on 300 square meters. I am one of the three curators. We show more than 60 of my 3d-printed mathematical models, together with other new and historical pieces!
  • April 26, 2012: Invited talk at the IMAGINARY-Conference at Barcelona, Spain.
  • April 24, 2012: Talk at the Didactics Colloquium at the Saarland University.
  • April 4, 2012: Talk at the annual MNU meeting at Freiburg, Germany.
  • March 2012: Talk at the annual meeting of the GDM at Weingarten, Germany.
  • Nov. 10, 2011: Invited Talk at the ETH, Zürich, Switzerland.
  • Oct. 11, 2011: Invited Talk at the Maths Institute at Lisbon, Portugal: Parabolas are Useful!
  • Sept. 25, 2011: Talk on Zeros and Discriminants of Polynomials during the AkMUI Conference at Soest, Germany.
  • Sept. 20, 2011: Workshop on Understanding Chaotic Behaviour using GeoGebra (in German) at the so-called Lehrertag of the annual DMV Meeting. Cologne, Germany. Material
  • Sept. 1-4, 2011: Invited Talk at the conference Geometry from an Advanced Point of View at Aveiro, Portugal.
  • March - July 2011: Visiting Professor at the university of Cologne (in German: Vertretung einer Professur).
  • Nov. 6, 2010: Workshop Algebra in Bildern at the CASIO meeting called Teach and Talk, Nürnberg, Germany. PDF-Download.
  • Sept. 27, 2010: Talk at the Oberseminar Südwest, Karlsruhe, Germany: Algebraische Geometrie in der Sekundarstufe I? (in German).
  • Sept. 25, 2010: Talk and Workshop at AkMUI 2010, Soest, Germany: Algebraische Geometrie versus Analytische Geometrie in der Schule (in German).
  • Sept. 12, 2010: Talk at AkGeo 2010, Marktbreit, Germany: Terme in Bildern in der Sekundarstufe I (in German).
  • March 2010: Invited talk at the Saga Workshop 2010, Auron, France: Visualization of Real Algebraic Curves and Surfaces with Singularities.
  • February 2010: Talk at the Convex Algebraic Geometry Workshop at Banff, Canada: Visualizing Convex Algebraic Geometry
  • July 2009: Talk at the university of Hannover, Germany: Real surfaces with real singularities and applications (in didactics and architecture)
  • May 2009: Talk at the architecture department of the technical university at Cottbus, Germany: Algebraic Geometry in Architecture?
  • Feb. 18, 2009: Talk at the Otto-Hahn-Gymnasium at Saarbrücken (Germany): Von Pythagoras zu aktueller algebraischer Geometrie (PDF)
  • Nov. 3 - Nov. 7, 2008: at the Maths Research Institute at Oberwolfach (Germany): Organizer of a teachers' training course on the use of algebraic curves and surfaces in school (in German).
  • Oct. 24. - Nov. 16, 2008: Ausstellung IMAGINARY 2008 in der K4-Galerie in Saarbrücken.
  • Sept. 2008 at Kassel (Germany): Talk at the AKMUI meeting.
  • Sept 2008 at Erlangen (Germany): Talk at the DMV Conference. I am also one of the organizers of a Mini-Symposium there.
  • Sept 2008 at Oslo (Norway): Talk at the CMA seminar.
  • August 2008 at Saarbrücken: Talk at the Kulturmeilen-Fest: Über die Schönheit der Geometrie(in German). PDF-Download. Mathematische Animationen zum Vortrag: Barth-Fläche, Enriques-Flächen, meine 99-nodale Fläche.
  • January 2008: Our paper on surfaces with many solitary points on arxiv.org.
  • December 2007 at Paris (France): Invited Talk at MACIS 2007: Visualization Challenges in Real Algebraic Geometry
  • 2006, December 1: Together with Hans-Christian Graf von Bothmer, I present an advent calendar of geometrical animations: http://Calendar.AlgebraicSurface.net