Biographical Information

Dr. Craizer received a B. degree in mathematics in 1983 from Federal University of Rio de Janeiro (UFRJ), a M.Sc degree in 1985 and a Ph.D. degree in 1989,  both in mathematics at the Institute for Pure and Applied Mathematics (IMPA).

Dr. Craizer is a professor at the Department of Mathematics of Pontifical Catholic University of Rio de Janeiro (PUC-Rio) since 1988.

Research Interests

• Affine and Projective Differential Geometry

• Discrete Geometry

• Applications of Singularity Theory to Geometry

Former Students

PhD Students:

1. Décio Ângelo Fonini Jr.- Ph.D. Thesis (co-advised with E.A.B.da Silva): Successive approximation quantization, 25/3/2003.

2. Lhaylla dos Santos Cryssaff- Ph.D. Thesis: Geometric energies on triangulated surfaces, 17/02/2009.

3. Ady Cambraia Junior- Ph.D.Thesis: Envelope of mid-planes of surfaces in R^3, 23/01/2015.

4. Rafael Segadas dos Santos- Ph.D. Thesis: Cáusticas de Wigner e conjuntos de medida de largura constante em planos normados com bolas unitárias suaves ou poligonais, 2019.

5. Anderson Reis de Vargas- Ph.D.Thesis: Asymptotic nets with constant affine mean curvature, 2021.

6. Fernanda Py Silva Cordeiro- Ph.D.Thesis: Quádricas de Moutard em superfícies, 2023.

7. Samuel Pacitti Gentil- Ph.D.Thesis: Discretization of ”four-vertex type” theorems for spatial and spherical polygons, 2024.

8. Filipe Bellio da Nóbrega- Ph.D.Thesis: On osculating conics in the real projective plane, 2025.

Master Students:

1. Bruno Coelho Maciel - M.Sc. Dissertation: Multi-resolution edge detection and image reconstruction from the edges, 29/02/2000.

2. Marcilene Fátima Dianin- M.Sc. Dissertation: Mathematical morphology, segmentation and curve evolution, 30/7/2001.

3. João Domingos Gomes da Silva Jr. - M.Sc. Dissertation (co-advised with H.Lopes): Curvature estimators based on parametric curve fitting, 31/01/2005.

4. Cinthia de Oliveira Lage Ferreira- M.Sc. Dissertation: Evolution of a union of balls from its medial axis, 16/02/2005.

5. Paulo César Alves Araujo- M.Sc.Dissertation: Algumas desigualdades isoperimétricas para curvas planas, 28/04/2011.

6. Marcelo Chaves Nascimento- M.Sc.Dissertation: Problemas isoperimétricos no plano de Minkowski, 23/03/2015.

7. Edison Fausto Cuba Huamani- M.Sc.Dissertation: Affine minimal surfaces with singularities, 12/09/2017.

8. Filipe Bellio da Nóbrega- M.Sc.Dissertation: Representation of generic curves by their singularities, 17/07/2018.

9. Fernanda Py Silva Cordeiro- M.Sc.Dissertation: Comportamento das linhas de curvatura de uma superfície no 3-espaço euclidiano próximo a um ponto umbílico, 12/04/2019.

10. Samuel Pacitti Gentil- M.Sc.Dissertation: Aspectos geométricos de poligonais genéricas: Curvatura total e convexidade, 2020.

11. Joel Albertacci Marques da Silva- M.Sc.Dissertation: Envoltórias de retas bissetoras de polígonos planos, 2023.

12. Fabricio Barbosa dos Santos- M.Sc.Dissertation: Singularidades genéricas de superfícies pseudo-esféricas, 2024.

13. Márcio dos Guimarães Peixoto- M.Sc.Dissertation: Transformações de Bäcklund entre superfícies hiperbólicas, 2024.

ProfMat:

1. Lucio Sebastião Coelho da Silva- TCC: O Teorema de Morley, 2014.

2. Marcelo do Nascimento Lório- TCC: Aproximações de números reais por números racionais : Por que as convergentes de frações contínuas fornecem as melhores aproximações?, 2014.

3. Thais de Sales Ribeiro- TCC: Poliedros de Arquimedes: um estudo enriquecedor para as aulas de geometria espacial na rede pública, 2015.

4. Sergio Ferreira Silva- TCC:Geometria Analítica: caminhos para aprendizagem, 2015.

5. Paula Burkardt Moreira- TCC: Proposta para o Ensino da Matemática através da Construção e Aplicação do Tangram- da Educação infantil ao Ensino Fundamental II, 2016.

6. Bruno de Assis Xarifa- TCC: Lugares geométricos e pontos notáveis do triângulo: Uma proposta de atividades na perspectiva do modelo Van Hiele, 2020.

7. Vanessa Rodrigues Tinoco- TCC: Educação Financeira uma abordagem no Ensino Fundamental – anos finais, 2020.

8. Felipe Assis da Costa- TCC: Uma Visão Interdisciplinar do Teorema de Desargues, 2024.

9. Ricardo Pedrosa Freitas- TCC: Aplicações de matemática nos métodos de busca a vítimas perdidas em matas, 2025.

10. Alexandre de Araújo França- TCC: Cálculo de áreas de figuras poligonais usando o Teorema de Pick, 2025.

Publications

• Papers
  
  

  1. M.Craizer, Entropy of inner functions, Israel Journal of Mathematics, vol.74, 2-3, pp.129-168, 1991.

  2. M.Craizer, The Bernoulli property of inner functions. Ergodic Theory and Dynamical Systems, 12, p.209-215, 1992.

  3. J.L.Arraut, M. Craizer, Stability of blocks of compact orbits of an action of R^2 on M^3. Advanced series in non-linear dynamics, 4,p.15-21,1993.

  4. M.Craizer, Homogenization of codimension 1 actions of R^n near a compact orbit, Annales de L'Institut Fourier, vol.44, n.5, pp.1435-1448, 1994.

  5. J.L.Arraut, M.Craizer, Foliations of M^3 defined by R^2-actions,  Annales de L'Institut Fourier, vol.45, n.4, pp.1091-1118, 1995.

  6. M.Craizer, E.A.B.da Silva and E.G.Ramos, New results on successive approximation vector quantization, Electronics Letters IEE, vol.34, n.1., pp.59-60, 1998.

  7. M.Craizer, E.A.B.da Silva and E.G.Ramos, Convergent algorithms for successive approximation vector quantization with applications to wavelet image compression,  IEE Proceedings - Vision, Image and Signal Processing, vol. 146, n.3, pp 159-164, 1999.

  8. M.Craizer, A.O.Lopes, The capacity-cost function of hard-constrained channels, International Journal of Applied Mathematics, vol.2, n.10, pp. 1165-1180, 2000.

  9. M.Craizer, D.A.Fonini Jr. and E.A.B.da Silva, Alpha-expansions: A class of frame decompositions, Applied and Computational Harmonic Analysis, vol.13, pp.103-115, 2002.

  10. M.Craizer, D.A.Fonini Jr. and E.A.B.da Silva, Successive approximation for image compression, IEEE Circuits and Systems Magazine, vol.2,n.3, 3rd quarter, 2002.

  11. M.Craizer, Evolution of critical points in curvature and affine morphological scale spaces, IV International Conference on Scale-Space theories in Computer Vision, Isle of Skye, Scotland, UK, 2003.

  12. M.Craizer and R.Teixeira, Evolution of an extremum by curvature motion, Journal of Mathematical Analysis and Applications, 293, p.721-737, 2004.

  13. M.Craizer, S.Pesco and R.Teixeira, A numerical scheme for the curvature equation near the singularities, Journal of Mathematical Imaging and Vision, 22, p.89-95, 2005.

  14. M.Craizer, T.Lewiner, J.D.G.Silva Jr, H.C.V.Lopes. Curvature and Torsion Estimators based on Parametric Curve Fitting, Computer & Graphics, 29, n.5, 2005.

  15. T.Lewiner, M.Craizer, H.Lopes, S.Pesco,L.Velho, E.Medeiros, GEncode: Geometry-driven compression for general  meshes. Computer Graphics Forum, 25, p.685-695, 2006.

  16. T.Lewiner, C.Ferreira, M.Craizer, R.Teixeira, Curvature Motion for Union of Balls, XVIII Sibgrapi, Natal, 2005.

  17. M.Craizer, T.Lewiner, J.M.Morvan, Combining points and tangents into parabolic polygons, Journal of Mathematical Imaging and Vision, 29, p.131-140, 2007.

  18. M.A.H.B. da Silva, R.C.Teixeira, S.Pesco, M.Craizer, A Fast Marching Method for the Area based Affine Distance, Journal of Mathematical Imaging and Vision, 22(1), p.89-95, 2008.

  19. M.Craizer, M.A.H.B. da Silva, R.C.Teixeira, Area Distances of Convex Plane Curves and Improper Affine Spheres, SIAM Journal on Mathematical Imaging, 1(3), p.209-227, 2008.

  20. T.Lewiner, M.Craizer, Projective splines and estimators for planar curves, Journal of Mathematical Imaging and Vision, 36(1), p.81-89, 2010.

  21. M.Craizer, H.Anciaux, T.Lewiner, Discrete affine minimal surfaces with indefinite metric, Differential Geometry and its Applications, 28, 158-169, 2010.

  22. M.Craizer, T.Lewiner, R. Teixeira, Cauchy problems for discrete affine minimal surfaces, Archivum Mathematicum, 48(1), 1-14, 2012.

  23. M.Craizer, R.Teixeira, M.da Silva, A geometric representation of improper indefinite affine spheres with singularities, Journal of Geometry, 100(1), 65-78, 2011.

  24. M.Craizer, R.Teixeira, M.da Silva, Affine properties of convex equal-area polygons, Discrete and Computational Geometry, 48(3), 580-595, 2012.

  25. M.Craizer, R.C.Teixeira, Some properties of the volume distance to hypersurfaces, Differential Geometry and its Applications, 31, 510-516, 2013.

  26. M.Craizer, Singularities of Convex Improper Affine Maps, Journal of Geometry, 103(2), 207-217, 2012.

  27. M.Craizer, R.Teixeira, M.da Silva, Polygons with Parallel Opposite Sides, Discrete and Computational Geometry, 50(2), 474-490, 2013.

  28. M. Craizer, W.Domitrz, Pedro de M. Rios, Even Dimensional Improper Affine Spheres, Journal of Mathematical Analysis and Applications, 421, 1803-1826, 2015.

  29. M.Craizer, Iterates of Involutes of Constant Width Curves in the Mikowski Plane, Beiträge zur Algebra und Geometrie, 55(2), 479-496, 2014.

  30. M.Craizer and H.Martini, Involutes of Constant Width Polygons in the Minkowski Plane, Ars Mathematica Contemporanea, 11(1), 107-125, 2016.

  31. M.Craizer, Equiaffine Characterization of Lagrangian Surfaces in R^4, International Journal of Mathematics, 26(9), 1550074, 2015.

  32. M.Craizer, M.J.Saia and L.F.Sánchez, Equiaffine Darboux frames for codimension 2 submanifolds contained in hypersurfaces, J.Math.Soc.Japan, 69, n.4, p. 1331-1352, October 2017.

  33. Ady Cambraia Jr. and M.Craizer, Envelope of mid-planes of a surface and some classical notions of affine differential geometry, Results in Mathematics 72(4), 1865-1880, 2017.

  34. Ady Cambraia Jr. and M.Craizer, Envelope of Mid-Hyperplanes of a Hypersurface, Journal of Geometry 108(3), 899-911, 2017.

  35. M.Craizer, R.C.Teixeira and V.Balestro, Closed cycloids in a normed plane, Monatshefte fur Mathematik, 185(1), 43-60, 2017.

  36. M.Craizer, M.J.Saia and L.F.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.

  37. M.Craizer and S.Pesco, Affine geometry of equal-volume polygons in 3-space, Computer Aided Geometric Design 57, 44-56, 2017.

  38. M.Craizer, R.C.Teixeira and V.Balestro, Discrete cycloids from convex symmetric polygons, Discrete and Computational Geometry, 60, 859-884, 2018.

  39. M.Craizer and R.A.Garcia, Quadratic points of surfaces in projective 3-space, Quarterly Journal of Mathematics 70(3), 1105-1134, 2019.

  40. M.Barajas, M.Craizer and R.A.Garcia, Lines of affine principal curvature lines of surfaces in 3-space, Results in Mathematics, v.75, 32, 2020.

  41. M.Craizer, W.Domitrz, Pedro de M.Rios, Singular improper affine spheres from a given Lagrangian submanifold, Advances in Mathematics, v.374, 107326, 2020.

  42. M.Craizer and R.A.Garcia, Curvature lines of a transversal equiaffine vector field along a surface in 3-space, Journal of Singularities, v.25, 134-143, 2022.

  43. M.Craizer and S.Pesco, Centroaffine duality for spatial polygons, Discrete and Computational Geometry, v.19, 00136, 2019.

  44. F.Oliveira, A.Lins, A.Barreto, M.Craizer, H.Lopes, S.Pesco, Smoothing Tidal Effects in Well Test Pressure Data, IEEE Geoscience and Remote Sensing Letters, v.17, 899-902, 2019.

  45. R.S. dos Santos and M.Craizer, Isoperimetric Inequalities in Normed Planes, Journal of Convex Analysis, v.29, 321-331, 2022.

  46. M.Craizer and R.A.Garcia, Singularities of Generic Line Congruences, Journal of the Mathematical Society of Japan, 1-25, September 2022.

  47. A.R.de Vargas and M.Craizer, Discrete asymptotic nets with constant affine mean curvature. Contributions to Algebra and Geometry/ Beitrage zur Algebra und Geometrie, v.65, p.601-621, 2023.

  48. M.Craizer and R.A.Garcia, Dual relations between line congruences in R^3 and surfaces in R^4. Research in the Mathematical Sciences, v.11, p.30, 2024.

  49. A.R.de Vargas and M.Craizer, Singularities of discrete improper indefinite affine spheres. Jounal of Geometry, v.114, n.34, 2023.

  50. J.A.M.da Silva and M.Craizer,  Envelope of bisection lines of polygons. International Electronic Journal of Geometry, v.17(2), 421-436, 2024.

  51. M.Craizer,  Singularities of discrete indefinite affine minimal surfaces, Differential Geometry and its Applications, vol.97, 102206, 2024.
      
      

• Books and Course Notes
  
  

  1. M.Craizer and G.Tavares, Cálculo Integral de Várias Variáveis (in portuguese), Editora PUC, 2002.

  2. M.Craizer and G.Tavares, Equações de Evolução em Processamento de Imagens (in portuguese), CNMAC2001.

  3.  M.Craizer, Equações de Evolução em Processamento de Imagens (in portuguese), Sibgrapi2001. Revista de Informática Teórica e Aplicada, volume 8, número 2, 2001.

  4.  M.Craizer, Evolutas de Curvas e Superfícies (in portuguese), Colóquio de Matemática da Região Sudeste, 2015.