• Matemáticas I (Grado en Química)
• Álgebra lineal y geometría II (Grado en Matemáticas)

Información de los cursos para estudiantes  aquí.

Tutorías Miércoles de 10:00 a 13:00  y jueves de 11:00 a 14:00.

Algebraic and Differential Geometry:

• Vector bundles on Grassmannians and their subvarieties.
• Higgs bundles, principal bundles and principal Higgs bundles; their numerically properties,
• both in algebraic and differential geometry.
• Decorated bundles and their moduli spaces, also with transcendental techniques.

You can find here a list of my publications

• 1. Graña, B. Escisión de fibrados en G(1, 4) y subvariedades. Sección de publicaciones de la Universidad Complutense de Madrid (2004). ISSN:84-669-1797-7
  2. Arrondo E. and Graña, B. Vector bundles on G(1, 4) without intermediate cohomology. J. Algebra 214 (1999) 128-142.
  3. Arrondo E. and Graña Otero, B. Congruences on G(1,4) with split universal quotient bundle. J. Geom. Physics 56 (2006) 1057-1067.
  4. Bruzzo, U. and Graña Otero, B. Numerically flat Higgs vector bundles. Communications in Contemporary Mathematics 9 (2007), 437–446. 
  5. Bruzzo, U. and Graña Otero, B. Metrics on semistable and numerically effective Higgs bundles. Journal für die reine und angewandte Mathematik. 612 (2007), 59–79.
  6. Bruzzo, U. and Graña Otero, B., Numerical properties of Higgs bundles. “Vector bundles and low codimensional subvarieties: State of the art and recent developments”. Quaderni di matematica, Vol. 21, Aracne, Roma 2007.
  7. Graña Otero, B. Jordan Hölder filtration for principal Higgs bundle on curves. J. Geom. Physics 60 (2010) no. 11, 1852-1859.
  8. Bruzzo, U. and Graña Otero, B. Semistable and numerically effective principal (Higgs) bundles. Adv. in Math. 226 (2011) 3655–3676.
  9. Bruzzo, U and Graña Otero, B. Approximate Hermitian-Yang-Mills structures on principal Higgs bundles over Kähler manifolds. Annals of Global Analysis and Geometry. 47 (2015), no.1, 1–11
  10. Bruzzo, U and Graña Otero, B. Approximate Hitchin-Kobayashi correspondence for Higgs G-bundles. International Journal of Geometric Methods in Modern Physics. 11 (2014), no. 7, 1460015, 7pp.
  11. Biswas, I., Bruzzo, U., Graña Otero, B. and Lo Giudice, A. Yang-Mills-Higgs connections on Calabi-Yau manifolds. Asian Journal of Mathematics. 20 (2016), no. 5, 989-999.
  12. Biswas, I., Bruzzo, U., Graña Otero, B. and Lo Giudice, A. Yang-Mills-Higgs connections on Calabi-Yau manifolds, II. Travaux mathématiques. Vol. XXIV, 167-181, Trav. Math., 24 Fac. Sci. Technol. Commun. Univ. Luxembourg, 2016.
  13. Bruzzo, I and Graña Otero, B. Higgs varieties and fundamental groups. J. Geom. Phys. 128 (2018), 12-19.

  14. Bruzzo, U., Graña Otero, B., Derived Functors and Sheaf Cohomology, World Scientific. Book Series: Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes, Vol. 2 (2020) ISBN = 978-9-81-120728-0, ISSN= 2591-7668, DOI=https://doi.org/10.1142/11473

  15. Bruzzo, U., Graña Otero, B. and Hernández Ruipérez, D. On a conjecture about Higgs bundles and some inequalities. Mediterr. J. Math. 20 (2023), no. 6, Paper No. 296, 15pp. 

  16. Bruzzo, U., Capasso, A. and Graña Otero, B. Positivity for Higgs vector bundles: criteria and applications.  Accepted for publication in Revista Matemática Complutense (October 2025). DOI: 10.1007/s13163-025-00551-7