This School aims to bring together graduate students and young researchers from all regions in Brazil and countries of the South America as audience, and confirmed researchers from all over the world representing different approaches to singularities in Mathematics as speakers.

The school will offer a solid introduction to this rich topic by presenting the fundamentals of the topology and geometry of singularities of spaces and maps, and of the geometric bifurcation theory, together with the necessary algebraic and analytic tools, aiming to a bunch of current research lines.

It will be presented ongoing research topics such as determinantal varieties, bifurcation of families, fibrations structures and new classes of singularities, Lipschitz geometry of singularities, vision and other applications. All courses will be planned to provide the students with practical tools via the open source computer algebra system ‘Singular’.


How to register and ask for supports?

  • Applicants from abroad (any country) and from Brazil but outside São Paulo state, are invited to apply for support in the CIMPA-UNESCO link until March 29th, 2021. 
  • All candidates to the Research School should also register in the local link until April 30th, 2021.
  • One may apply for support on the CIMPA link and on the local link.


CIMPA Research School “Singularities and Applications”

Updated program 2021 and 2022


Due to the covid pandemic, we have decided to split our program in two parts:

Part 1, July 2021: online lectures followed by an exchange with students on questions and exercises via an IT platform

(for its address, please write an email to one of the organisers). The majority of the lectures will be uploaded before July as video and pdf files. There will be an interactive period supported by the platform during the period July 12-23.

Part 2, July 11-22  2022:  Scheduled to be held at ICMC-USP,  Sao Carlos, Sao Paulo-Brazil. 

In Part 1 we will offer the lectures listed below.

Part 2 will complement this list up to the initial program, and possibly go beyond it if time allows.


                                                                          --------- List of lectures and their links to beamer files ----------


AMaria Ruas, Juan Nuno Ballesteros, Raul Sinha, and Roberta Wik Atique

Hypersurfaces : algebraic methods and singularities.  Lectures
1. Basics on classification of singularities.
2. Singularities of germs of functions.


BMihai Tibar and Dirk Siersma
1. Milnor fibrations and their topology. Bouquet theorems. Lecture 1 Lecture 2
2. Nonisolated singularity and their topology. Lecture
3. Polar degree of projective hypersurfaces. Lecture 1 Lecture 2 Lecture 3 Lecture 4


CLaurentiu Maxim
Topology of complex projective hypersurfaces and of their complements

4 lectures. Totality of the beamers and videos


DYing Chen, Raimundo Araujo dos Santos, and Mihai Tibar
1. Real Milnor fibrations. Lecture
2. Newton polyhedra and mixed singularities. Lecture 1Lecture 2; ExercisesHint; References


ERenato Dias, Cezar Joita, and Mihai Tibar
1. Topology and bifurcation of polynomial maps in real and complex settings.

Lecture 1 Lecture 2-part1 Lecture 2-part2 Lecture 2-part3

Lecture 3-part1 Lecture 3-part2


FJim Damon
Medical Imaging, Skeletal structures and singularities. Download: References; Outline; Selected Exercises

3 lectures.