Research
 

Current Research Interests:

·        Symplectic Geometry

·        Poisson Geometry

·         Integrable systems and group actions

·        Foliation theory

·         Geometric quantization

·        Groupoids and Algebroids

·        Hamiltonian Dynamics

·      and Differential Geometry/Mathematical Physics in the large....

My research deals with geometrical and dynamical aspects of singularities. In particular, I am interested in Hamiltonian systems and their singularities. I study normal forms and equivariant geometric problems arising in Symplectic, Contact, and Poisson manifolds. I am also interested in rigidity problems for group actions on these manifolds. I also work in geometric quantization of real polarizations. Some years ago, I started to consider geometrical problems on b-manifolds (inspired by Melrose b-calculus on manifolds with boundary). This is a fascinating new subject that I am working on which lies between the Symplectic and Poisson worlds.
I am lately trying to understand possible generalizations of b-manifolds such as almost regular foliations and E-symplectic manifolds and finding (unexpected!) applications of b-theory to problems in celestial mechanics.

 

My research is financed by the Catalan Institution for Research and Advanced Studies via an ICREA Academia 2016 Prize by the MTM research project MTM2015-69135-P/FEDER, Geometría y Topología de variedades, álgebra y aplicaciones (IP: Marta Casanellas and Eva Miranda) based at UPC and by the project EXPLORA ciencia MTM2015-72876-EXP based at ICMAT-CSIC (Members: Alberto Ibort, Eva Miranda and Francisco Presas(IP)).

 

I am in the head of the Laboratory of Geometry and Dynamical Systems.

   My academical trajectory in a glimpse:

1.      Ph.D thesis "On symplectic linearization of singular Lagrangian foliations" at the University of Barcelona in September 2003.

2.      October 2004-May 31, 2007, Marie Curie EIF postdoc at Laboratoire Émile Picard, Université Paul Sabatier.

3.      June 1st, 2007 – August 31, 2009, Juan de la Cierva Fellow at the Departament de Matemàtiques at Universitat Autònoma de Barcelona.

4.      September 2009-August 2013- Professor Lector (Lecturer) at the Department of Applied Mathematics I, UPC.

5.      September 2013-September 2015- Professor Agregat Interí/Interim Associate Professor at Universitat Politècnica de Catalunya.

6.      Currently (since October 2015), Associate Professor (Professor Agregat) at UPC.

7.  Since January 2016, I am principal investigator of the MTM Research project MTM2015-69135-P, Geometría y Topología de variedades, álgebra y aplicaciones with 24 members.

8. Since December 2016, I am in charge of the Laboratory of Geometry and Dynamical Systems at EPSEB.

9. Since January 2017 and for 5 years I am an ICREA Academia Professor.

 

 

 

Affiliations and memberships:

1.      Associate Professor at UPC, member at ICMAT (via Ginzburg lab) and member of Barcelona Graduate School of Mathematics BGSMath.

2.      I am a member of the European Mathematical Society and a member of Societat Catalana de Matemàtiques.

3.      Since 2011 I am the corresponding member for the SCM and EMS.

 Accreditations:

  • I am qualified as Full Professor in the area of Geometry and Topology by AQU since October 2016.

  • Since February 2010, I am qualified as Professeur des Universités en Section 25 (Mathématiques) by the French CNU (Conseil des Universités). I was qualified as MCF in February 2008.

  •  I am qualified as Professor agregat  by AQU since 2010.

  •   I am qualified as Profesor Titular de Universidad since April 2015

 

Distinctions:

 

 

Curriculum Vitae

      Publications:

·        My thesis:

        Eva Miranda, "On symplectic linearization of singular Lagrangian foliations" ,  June, 2003, edicions de la UB, http://www.tdx.cbuc.es/

·         My research papers (the versions posted here might differ slightly from the final published versions):

Publications (as author):

    1. Zeroth Poisson homology, foliated cohomology and perfect Poisson manifolds (with D. Martínez-Torres), to appear at Regular and Chaotic Dynamics, 2017.

    2. An invitation to Singular Symplectic Geometry (with R. Braddell, A. Delshams, C. Oms and A. Planas), arXiv:1705.03846, accepted at International Journal of Geometric Methods in Modern Physics, 2017.

    3. Desingularizing b^m-symplectic structures (with V. Guillemin and J. Weitsman), arXiv:1512.05303, accepted at IMRN, International Mathematics Research Notices.

    4. Rigidity of infinitesimal momentum maps (with C. Esposito), arXiv:1410.5202,  Israel Journal of Mathematics, Israel J. Math. 219 (2017), no. 2, 757–781.

    5. Cotangent models for integrable systems (with A. Kiesenhofer), http://arxiv.org/abs/1601.05041 Communications in Mathematical Physics,10.1007/s00220-016-2720-x Comm. Math. Phys. 350 (2017), no. 3, 1123–1145.

    6. Weakly Hamiltonian actions (with D. Martínez-Torres),http://arxiv.org/abs/1602.03542 Journal of Geometry and Physics, J. Geom. Phys. 115 (2017), 131–138.

    7. Examples of integrable and non-integrable systems on singular symplectic manifolds (with A. Delshams and A. Kiesenhofer), http://arxiv.org/abs/1512.08293 Journal of Geometry and Physics, J. Geom. Phys. 115 (2017), 89–97.
    8. Geometric Quantization of Semitoric Systems and almost toric manifolds (with F. Presas and R. Solha), preprint 2017, submitted.

    9. On geometric quantization of b-symplectic manifolds (with V. Guillemin and J. Weitsman), arXiv:1608.08667, submittted.

    10. Equivariant $b^m$-symplectic surfaces and Nambu structures (with Arnau Planas), https://arxiv.org/abs/1607.01748, submitted.

    11. Non-commutative integrable systems on b-symplectic manifolds (with A. Kiesenhofer), https://arxiv.org/abs/1606.02605, Regul. Chaotic Dyn. 21 (2016), no. 6, 643–659.

    12. Convexity for Hamiltonian torus actions on b-symplectic manifolds (with V. Guillemin, A.Pires, G. Scott) arXiv:1412.2488, Mathematical Research letters, to appear.

    13. A note on symplectic topology of b-symplectic manifolds (with P. Frejlich and D. Martinez Torres), preprint 2013, arXiv:1312.7329, accepted at  the Journal of Symplectic Geometry.
    14. Action-angle variables and a KAM theorem for b-Poisson manifolds (with A. Kiesenhofer and G. Scott) , Journal des Mathématiques Pures et Appliquées, J. Math. Pures Appl. (9) 105 (2016), no. 1, 66-85.
    15. Geometric Quantization of real polarizations via sheaves (with F. Presas) The Journal of Symplectic Geometry (2015) volume 13, number 2, 421-462.
    16. A note on symplectic and Poisson linearization of semisimple Lie algebra actions, http://arxiv.org/abs/1503.03840, submitted.
    17. Symplectic and Poisson geometry on b-manifolds (with V. Guillemin and A.Pires), Adv. Math. 264 (2014), 864–896.
    18. Toric actions on b-manifolds (with V. Guillemin, A.Pires and G. Scott), Int Math Res Notices (2015) 2015 (14): 5818-5848
    19. A Poincaré lemma in Geometric Quantisation (with R. Solha), Journal of Geometric Mechanics,  473 - 491, Volume 5, Issue 4, 2013.
    20. On a Poincaré lemma for foliations (with R. Solha), Foliations 2012: pp. 115-137 .
    21.  Integrable systems and group actions,  Cent. Eur. J. Math. 12 (2014), no. 2, 240–270.
    22.  Coupling symmetries with Poisson structures (with C. Laurent-Gengoux) Acta Math. Vietnam. 38 (2013), no. 1, 21–32.,
    23. Rigidity for Hamiltonian actions on Poisson manifolds( with P. Monnier and N. T. Zung) ,    Adv. Math. 229 (2012), no. 2, 1136–1179.
    24. Codimension one symplectic foliations and regular Poisson manifolds (with V. Guillemin and A. Pires) Bull Braz Math Soc, New Series 42(4), 603-623, 2011.
    25.     From action-angle coordinates to geometric quantization: a round trip, Oberwolfach report, Geometric quantization in the non-compact setting, 2011.
    26. Action-angle coordinates for integrable systems on Poisson manifolds, with C. Laurent-Gengoux and  P.Vanhaecke, (for previous version see math.SG arxiv.0805.1679) (31 pages),  Int. Math. Res. Not. IMRN 2011, no. 8, 1839–1869.
    27. Geometric quantization of integrable systems with hyperbolic singularities with M. Hamilton, arXiv:0808.0338, Annales de l'Institut Fourier, vol. 60 (1) p. 51-85, 2010. (36 pages)
    28.  Symmetries and singularities of Hamiltonian systems, J. Phys.: Conf. Ser. 175 012011, 2009.
    29.  Symplectic linearization of semisimple actions, preprint 2008.
    30.  Rigidity for Poisson group actions, Oberwolfach report for the meeting on Poisson Geometry and its applications, Oberwolfach Report, Report number, 25, 2007, pages 33—36 (ISSN: 1660-8933), April 2007.
    31. Some rigidity results for Symplectic and Poisson group actions,  XV International Workshop on Geometry and Physics,, Publ. R. Soc. Mat. Esp., 11,  177-183, R. Soc. Mat. Esp., Madrid, 2007.
    32.  A note on Equivariant normal forms of Poisson structures, (with N. T. Zung)   Mathematical Research Letters, vol 13-6, pages 1001-1012, 2006.
    33.  A normal form theorem for integrable systems on contact manifolds,  Proceedings of the XIII Fall Workshop on Geometry and Physics, Publ. R. Soc. Mat. Esp., 9, R. Soc. Mat. Esp.,  (2005), 240--246.
    34.  A singular Poincaré Lemma (with Vu Ngoc)  , IMRN International Mathematics Research Notices , n 1, 27-46, 2005.
    35.  Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems (with Nguyen Tien Zung) , Ann. Sci. Ecole Norm. Sup.,37 (2004), no. 6, 819--839 2004.
    36.  Symplectic linearization of singular Lagrangian foliations (with C. Curras-Bosch), Differential Geometry and its applications,  18, (2), 195-205, 2003.
    37.  "On symplectic linearization of singular Lagrangian foliations" , Ph. D. thesis, Universitat de Barcelona,  June, 2003.
    38.  On the symplectic classification of singular Lagrangian foliations, Proceedings of the IX Fall Workshop on Geometry and Physics (Vilanova i la Geltrú, 2000), 239--244, Publ. R. Soc. Mat. Esp., 3, R. Soc. Mat. Esp.,2001.

  Divulgatio:

  1. The Hirsch Conjecture has been disproved: an interview with Francisco Santos, Newsletter of the EMS,  "Newsletter of the European Mathematical Society", 2012, vol. 86, p. 31-37.
  2. Columna de la EMS al Notícies de la SCM, Març 2012, 9-11, 32, (2012).
  3. Columna de la EMS al Noticies de la SCM, Febrer 2013, (2013).
  4. The Clay Public Lecture and Conference on the Poincaré Conjecture, Paris, 7-9 June 2010,  Eur. Math. Soc. Newsl, issue 77, pages 21-23,  2010.

 Publications (as editor):

  1. Newsletter of the European Mathematical Society. (editor since 2010).
  2. Editor of a theme issue for the Philosophical transactions of the Royal Society A (editor in 2017).
  3. Editor of a Special Issue for the Journal of Geometry and Physics (deadline to submit: December 2017).
  4. Guest editor of Special Issue at Journal of Geometry and Physics (deadline to submit: December 2015).
  5. Guest editor of a Topical Issue on Integrable Systems at Journal of Geometry and Physics.
    Journal of Geometry and Physics Volume 87, Pages 1-496 (January 2015)
    Finite dimensional integrable systems: on the crossroad of algebra, geometry and physics
    Edited by Vladimir Matveev, Eva Miranda, Vladimir Roubtsov, Sergei Tabashnikov and San Vu Ngoc
  6. Guest Editor of new Special Issue at Journal of Geometry and Physics. More information available at: http://fdis2015.wfa.uz.zgora.pl/
  7. Editor (together with Vladimir Matveev) of a book in the collection CRM-Advanced Courses (Springer) entitled Geometry and Dynamics of Integrable systems http://www.springer.com/gp/book/9783319335025  ISBN 978-3-319-33503-2.
  8. Guest editor of a Special issue of Geometriae Dedicata for the conference GESTA 2011. This issue was published in 2013 (volume 1, issue 265, August).
  9. Guest editor of a Special issue of Acta Mathematica Vietnamica for the conference GEDYTO. This issue has been published in January 2013 (issue
    38).

Book reports:

Carlos Curras-Bosch i Eva Miranda,  Report sobre el llibre “Symplectic geometry of integrable Hamiltonian systems” ,Notícies de la Societat Catalana de Matemàtiques, 19, 43-44, 2003.

 

 

"La sombra de mi alma
huye por un ocaso de alfabetos,
niebla de libros
y palabras."
Federico García Lorca, excerpt from La Sombra de mi Alma, Libro de Poemas.