Permanent Faculty: Marcos Craizer

Marcos Craizer

PhD, IMPA, Rio de Janeiro,RJ-Brasil, 1989
Office: L874
Position: Associate Professor
Phone: (21) 3527-1741
E-mail: craizer
Differential Geometry

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Marcos Craizer obtained his Ph.D. at IMPA in 1989 and is a professor at the Math Department of PUC-Rio since 1988. His research area lies between Affine Differential Geometry and Singularity Theory. He is also interested in Discrete Geometry, considering its applications to Computer Vision.

Research Results


  • Improper affine spheres. There is a strong connection between Area Distance and non-convex Improper Affine Spheres that was explored in [1] and [2]. In [5], one can find a classification of stable singularities of convex Improper Affine Spheres.
  • Volume distance to hypersurfaces. Although the Area Distance in the plane is an Improper Affine Sphere, this property does not hold in higher dimensions. Nevertheless, the volume distance have some nice affine differential properties [4].
  • Affine evolutes and symmetry sets. In [7], several properties of the Area Evolute and Center Symmetry Set are described. A discrete version of these results can be found in [6]. In [3], one can find a discrete version of the Affine Evolute, Affine Distance Symmetry Set. In the same paper there are discrete versions of the the Six Vertex Theorem and an affine isoperimetric inequality.
  1. Marcos Craizer, Moacyr Alvim, Ralph Teixeira: Area Distances of Convex Plane Curves and Improper Affine Spheres, SIAM Journal on Mathematical Imaging, 1(3), p.209-227, 2008.
  2. Marcos Craizer, Ralph Teixeira, Moacyr Alvim: Affine properties of convex equal-area polygons, Discrete and Computational Geometry, 48(3), 580-595, 2012.
  3. Marcos Craizer, Equiaffine Characterization of Lagrangian Surfaces in R^4, International Journal of Mathematics, 26(9), 1550074, 2015.
  4. Marcos Craizer, Wojtek Domitrz, Pedro Rios: Even Dimensional Improper Affine Spheres, Journal of Mathematical Analysis and Applications, 421, 1803-1826, 2015.
  5. Marcos Craizer, Ralph Teixeira, Vitor Balestro: Discrete cycloids from convex symmetric polygons, Discrete and Computational Geometry, 60, 859-884, 2018.
  6. Marcos Craizer, Marcelo Saia, Luis Sánchez: Affine focal set of codimension 2 submanifolds contained in hypersurfaces, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 148A, 995-1016, 2018.
  7. Marcos Craizer, Sinesio Pesco: Affine geometry of equal-volume polygons in 3-space, Computer Aided Geometric Design 57, 44-56, 2017.
  8. Marcos Craizer, Sinesio Pesco: Centroaffine duality for spatial polygons, aceito para publicação no Discrete and Computational Geometry, 2019
  9. Marcos Craizer, Ronaldo Garcia: Quadratic points of surfaces in projective 3-space, aceito para publicação no Quarterly Journal of Mathematics, 2019.
  10. Marcos Craizer, Ronaldo Garcia: Centroaffine duality and Loewner’s type conjectures, pré-publicação, 2019.