Alejandro Cabrera

Professor Titular-Livre – Área: Geometria Diferencial
(Full Professor in Mathematics – Differential Geometry)

Departamento de Matemática
Instituto de Matemática
Universidade Federal do Rio de Janeiro - UFRJ, Brasil




Sala: C – 125A
Tel: (+55 21) 2562-7036

2022-2:
Geometria Não Euclidiana
Métodos semi-clássicos em mecânica quântica

My research focuses on developments of differential geometry, both theoretical and with applications, in particular, to mathematical physics.
My topics of research include symplectic and Poisson geometry, Lie theory of algebroids and groupoids, as well as their many applications to quantization, (quantum) field theories and geometric mechanics.

Links:
[google scholar]
[curriculo Lattes (Br)]
[El País] An article about applying in gymnastics some geometric formulas I derived.
[simulações IC] Umas simulações obtidas em projetos de Iniciação Científica (some simulations obtained in student projects).
[Seminário simplético do Rio] Rio's Symplectic seminar.

  1. Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids, with M. Cueca, submitted Apr. 2022. (arxiv)
  2. Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization, accepted in Communications in Mathematical Physics (June 2022). (doi) (arxiv)
  3. Quotients of multiplicative forms and Poisson reduction, with C. Ortiz, Diff. Geom. Appl. Vol 83, August 2022, article 101898. (doi) (arxiv)
  4. Poisson double structures, with H. Bursztyn and M. del Hoyo, Journal of Geometric Mechanics, Vol. 14, No. 2, June 2022, pp. 151–178. (Published online first 2021-12-2.) (arxiv)
  5. Discrete dynamics and differentiable stacks, with M. del Hoyo and E. Pujals, Revista Matematica Iberoamericana Volume 36, Issue 7, 2020, pp. 2121-2146. (arxiv)
  6. Local formulas for multiplicative forms, with I. Marcut and M. A. Salazar, Transformation Groups volume 27, pages 371–401 (2022). (Published open access: 15 August 2020.) (arxiv)
  7. On local integration of Lie brackets, with I. Marcut and M. A. Salazar, J. Reine Angew. Math. (Crelle’s journal), Volume 2020, Issue 760, Pages 267–293. (arxiv)
  8. Obstructions to the integrability of VB-algebroids, with O. Brahic and C. Ortiz, Journal of Symplectic Geometry Vol. 16, No. 2 (2018), pp. 439-483. (arxiv)
  9. Minimal time splines on the sphere, with P. Balseiro, J. Koiller and T. Stuchi, São Paulo J. Math. Sci. (2017). (doi)
  10. Dirac Geometry of the Holonomy Fibration, with M. Gualtieri and E. Meinrenken, Communications in Mathematical Physics (2017), 355(3), 865-904. (arxiv)
  11. About simple variational splines from the Hamiltonian viewpoint, with P. Balseiro, J. Koiller and T. Stuchi, Journal of Geometric Mechanics, Volume 9, Number 3, September 2017, pp. 257–290. doi:10.3934/jgm.2017011. (arxiv)
  12. van Est isomorphism for homogeneous cochains, with T. Drummond, Pacific Journal of Mathematics 287-2 (2017), 297–336. DOI 10.2140/pjm.2017.287.297. (arxiv)
  13. Vector bundles over Lie groupoids and algebroids, with H. Bursztyn and M. del Hoyo, Advances in Mathematics, Volume 290 (2016), Pages 163-207. (arxiv)
  14. Differentiability of correlations in relativistic quantum mechanics, with E. de Faria, E. Pujals and C. Tresser, J. Math. Phys. 56, 092104 (2015); DOI 10.1063/1.4931176. (arxiv)
  15. Formal symplectic realizations, with B. Dherin, Int Math Res Notices 2016 (7): 1925-1950. doi:10.1093/imrn/rnv187. (arxiv)
  16. Multisymplectic geometry and Lie groupoids, with H. Bursztyn and D. Iglesias, D.E. Chang et al. (eds.), Geometry, Mechanics, and Dynamics, Fields Institute Communications Volume 73, 2015, pp 57--73 ; Springer New York. (arxiv)
  17. AKSZ construction from reduction data, with F. Bonechi and M. Zabzine, JHEP Volume 2012, Number 7 (2012), 68. (arxiv)
  18. Symmetries and reduction of multiplicative 2-forms, with H. Bursztyn, Journal of Geometric Mechanics, Volume 4, Issue 2, June 2012, Pages: 111 - 127. (arxiv)
  19. Multiplicative forms at the infinitesimal level, with H. Bursztyn, Mathematische Annalen Volume 353, Number 3 (2012), 663-705. (arxiv)
  20. Linear and multiplicative 2-Forms, with H. Bursztyn and C. Ortiz, L. Lett. Math. Phys. 90, (dec 2009) 59-83. (arxiv)
  21. Poisson-Lie T-Duality and non trivial monodromies, with H. Montani, M. Zuccalli, J. Geom. Phys. 59 (2009) 576-599. (arxiv)
  22. Base-controlled mechanical systems and geometric phases, J. Geom.Phys. 58 (2008) 334-367. (arxiv)
  23. A Generalized Montgomery Phase Formula for Rotating Self Deforming Bodies, J.Geom.Phys. 57 (2007), 1405-1420. (arxiv)
  24. Hamiltonian Loop group actions and T-duality for group manifolds, with H. Montani J. Geom. Phys. 56 (2006), 1116-1143. (arxiv)
  25. ------------ (other preprints:)
  26. Lie theory of vector bundles, Poisson geometry and double structures, with H. Bursztyn and M. del Hoyo, accepted in proceedings of ICMP-2015. (It seems that these proceedings are still not available.) (arxiv)
  27. Some geometric features of Berry’s phase, preprint 2007. (arxiv)