My research focuses on the Jacobian Conjecture, characteristic classes, and Lipschitz geometry, among which the Jacobian Conjecture gains my greatest interest. I proved the two-dimensional complex Jacobian Conjecture up to degree 104, which remains the best known degree bound to date. This result was recently cited by Alexander Borisov, Ofer Gabber, and Adrian Vasiu in their 175-page paper, ''On Endomorphisms of Affine Spaces and the Jacobian Problem'', published on arXiv. I also developed a new algebraic-geometric and topological approach to the conjecture, including a method for identifying potential counterexamples in high degrees. In characteristic classes, together with collaborators, we proved the Poincaré–Hopf theorem for real singular varieties and developed a new approach to characteristic classes of complex varieties using Lipschitz stratifications. In Lipschitz geometry, we provided a complete bi-Lipschitz classification of semialgebraic surfaces. Plus, in collaboration with a coauthor, I also gave a new proof of the famous Euler formula for convex polyhedra. Details and proofs of these results can be found in my list of publications below. 

1. Some classes satisfying the 2-dimensional Jacobian conjecture and a proof of the complex conjecture until degree 104. 

 Quaestiones Mathematicae, Vol. 48(2), 1-15, 2025. 


2. Local bi-Lipschitz classification of semi-algebraic surfaces. Dalat University Journal of Science, Vol. 15(3), 76-97, 2025. 


3. Classes especiais de campos de vetores na esfera. Revista Matemática Universitária, Vol. 1, 2025. 


4. Poincaré-Hopf Theorem for singular analytic varieties. Banach Center Publications, Vol. 128, 23-43, 2024. 


5. A singular variety associated to the smallest degree Pinchuk map. Matemática Contemporânea, Vol. 53, 135-154, 2023.


6. Local Euler obstruction, old and new III. Journal of Singularities, Vol. 25, 90 – 122, 2022. 


7. An elementary proof of Euler’s formula using Cauchy’s method. Topology and its applications, Vol. 293, p.107558, 2021.


8. Teorema de Poincaré-Hopf (in Portuguese - with J.-P. Brasselet). Revista eletrônica Paulista de Matemática, Vol. 16, 134-162, 2019. 


9. On singular varieties associated to a polynomial mapping from Cn to Cn-1.  The Asian Journal of Mathematics, Vol. 22, 1157-1172, 2018. 


10.  On singular varieties associated to a polynomial mapping. Journal of Singularities, Vol. 7, p 190-204, 2013. 



Reprints: 

11. Chern-Schwartz-MacPherson classes in the view-point of Obstruction Theory and Lipschitz framework. 

ArXiv: 2502.07567.


12. Ph.D. Thesis (in French): 

Etude de certains ensembles singuliers associés a une application polynomiale. 

Rapporteurs: Mutsuo OKA, HA Huy Vui. Published on: HAL Open Science, tel-00875930.


13. Mini-courses (in Portuguese): Uma Introdução a aplicações polinomiais e Conjectura Jacobiana. 


Book project: