Research

Current Research Interests:

·        Symplectic Geometry

·        Poisson Geometry

·         Integrable systems and group actions

·        Foliation theory

·         Geometric quantization

·        Groupoids and Algebroids

·        Hamiltonian Dynamics

·        Fluid Dynamics and Contact Geometry

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

I am particularly interested in understanding connections between different areas such as Geometry, Dynamical Systems, Mathematical Physics and, more recently, Fluid Dynamics.

My research deals with geometrical and dynamical aspects of singularities. In particular, I am interested in Hamiltonian systems, their singularities and the so-called realm of Hamiltonian Dynamics. 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). Their symplectic reincarnations are called b-symplectic manifolds (or log-symplectic manifolds) and them together with their generalizations (b^m-symplectic manifolds and alike) appear modelling several problems in Celestial Mechanics. 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. I love localization theorems, equivariant cohomology and I am a recent fan of Floer homology and the study of periodic orbits which I am trying to understand in connection to problems in Celestial Mechanics such as the three-body problem.

Many of the problems that I study are connected to Celestial mechanics in one way or another

The Weinstein conjecture on periodic orbits asserts that the Reeb vector field of a compact contact manifold always have periodic orbits. With my student Cédric Oms we have understood how to construct a counterexample to the Weinstein Conjecture if we allow singularities in the contact form. Those singularities on contact structures model some problems of Beltrami flows on manifolds with boundary (see my recent preprint with my student Robert Cardona and Daniel Peralta Salas).

This counterexample to the Weinstein conjecture is achieved via plugs and is very revealing: The orbits that give the counterexample are indeed periodic orbits which are no longer smooth but have points as marked singularities.  This opens a door to a new world.

What's next? ....maybe this.

My research is financed by the Catalan Institution for Research and Advanced Studies via an ICREA Academia 2016 Prize and partially supported by several research projects. I am the principal investigator of the AGAUR SGR research project GEOMVAP with code 2017SGR932 (26 members) and the MTM research project MTM2015-69135-P/FEDER, Geometría y Topología de variedades, álgebra y aplicaciones (together with Marta Casanellas).

From September 2017 until February 2018, I have been a Chaire d'Excellence de la Fondation Sciences Mathématiques de Paris and my work has been supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (reference: ANR-10-LABX-0098). I am also a member of the research project EXPLORA ciencia MTM2015-72876-EXP based at ICMAT-CSIC (Members: Alberto Ibort, Eva Miranda and Francisco Presas(IP)). I am a collaborator of the National Science Foundation project New Avenues in Symplectic Geometry DMS 12/11819 (PI: Jonathan Weitsman).

Since December 2016 I am in the head of the Laboratory of Geometry and Dynamical Systems which I created together with Amadeu Delshams.

Since May 2018 I am the group leader of the UPC research group GEOMVAP (Geometry of varieties and applications).

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.

10. From September 2017 until February 2018, I have been a Chaire d'Excellence de la Fondation Sciences Mathématiques de Paris and my work has been supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (reference: ANR-10-LABX-0098).

11. I have been selected under the Programa Càtedres-UPC 2017 for promotion to Full Professor.

12. I am the principal investigator of  the AGAUR SGR research project 2017SGR932 (21 members).

13. Since May 2018 I am the group leader of the UPC research group GEOMVAP (Geometry of varieties and applications).

14. Since November 2018, Full Professor (Professor Catedrático) at UPC.

Affiliations and memberships:

1.      Full Professor at UPC, Chercheur Affilié at Observatoire de Paris, Honorary Doctor Vinculado at ICMAT 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.      During the period 2011-2018 I was 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, ,  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. Universality of Euler flows and flexibility of Reeb embeddings (with R. Cardona, D. Peralta-Salas and F. Presas), preprint, 2019, https://arxiv.org/abs/1911.01963.

2. Geometric Quantization of almost toric manifolds (with F. Presas and R. Solha), arXiv:1705.06572, Journal of Symplectic Geometry, to appear, J. Symplectic Geom (2019).

3. Euler flows and singular geometric structures (with R. Cardona and D. Peralta-Salas), https://arxiv.org/abs/1902.00039, 2019, DOI 10.1098/rsta.2019.0034.

4. On the volume elements of a manifold with transverse zeroes (with R. Cardona), https://arxiv.org/abs/1812.03800 accepted for publication in Regular and Chaotic Dynamics, scheduled for Volume 24, Issue 2 of 2019.

5. Cotangent models for group actions on b-Poisson manifolds (with R. Braddell and A. Kiesenhofer), https://arxiv.org/abs/1811.11894

6. Contact structures with singularities (with C. Oms), https://arxiv.org/abs/1806.05638 (new version available on arxiv containing a study of the singular Reeb dynamics and realization of $b^m$-contact structures. For a talk announcing some of the results this november Look here.).

7. Open Problems, Questions, and Challenges in Finite-Dimensional integrable systems, (with A. Bolsinov, V. Matveev and S. Tabachnikov), arXiv:1804.03737,

8. On geometric quantization of b-symplectic manifolds (with V. Guillemin and J. Weitsman), arXiv:1608.08667, Adv. Math. 331 (2018), 941–951.

9. Volume 23, Issue 4 of 2018.

10. Integrable systems and closed one forms (with R. Cardona), arXiv:1712.08156,  J. Geom. Phys. 131 (2018), 204–209.

11. (with D. Bouloc and N.T Zung), arXiv:1803.08332,

12. The geometry of E-manifolds, (with G. Scott), arXiv:1802.02959, preprint 2018, submitted.

13. On geometric quantization of $b^m$-symplectic manifolds, (with V. Guillemin and J. Weitsman), arXiv:1801.03762 preprint, 2018, submitted.

14. (with A. Planas), C. R. Math. Acad. Sci. Paris 356 (2018), no. 1, 92–96.

15. Zeroth Poisson homology, foliated cohomology and perfect Poisson manifolds (with D. Martínez-Torres),  Regul. Chaotic Dyn. 23 (2018), no. 1, 47–53.

16. An invitation to Singular Symplectic Geometry (with R. Braddell, A. Delshams, C. Oms and A. Planas), arXiv:1705.03846,  Int. J. Geom. Methods Mod. Phys. 16 (2019), suppl. 1, 1940008, 16 pp.

17. (with V. Guillemin and J. Weitsman), arXiv:1512.05303, IMRN, International Mathematics Research Notices

rnx126, https://doi.org/10.1093/imrn/rnx126, Int. Math. Res. Not. IMRN 2019, no. 10, 2981–2998.
18. Comm. Math. Phys. 350 (2017), no. 3, 1123–1145.

19. (with D. Martínez-Torres),http://arxiv.org/abs/1602.03542, J. Geom. Phys. 115 (2017), 131–138.

20. Examples (with A. Delshams and A. Kiesenhofer), http://arxiv.org/abs/1512.08293 , J. Geom. Phys. 115 (2017), 89–97.
21. 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.

22. A note on symplectic topology of b-symplectic manifolds (with P. Frejlich and D. Martinez Torres), arXiv:1312.7329, J. Symplectic Geom. 15 (2017), no. 3, 719–739.
23. 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.
24. Geometric Quantization of real polarizations via sheaves (with F. Presas) The Journal of Symplectic Geometry (2015) volume 13, number 2, 421-462.
25. Toric actions on b-manifolds (with V. Guillemin, A.Pires and G. Scott), Int Math Res Notices (2015) 2015 (14): 5818-5848
26. A Poincaré lemma in Geometric Quantisation (with R. Solha), Journal of Geometric Mechanics,  473 - 491, Volume 5, Issue 4, 2013.
27. On a Poincaré lemma for foliations (with R. Solha), Foliations 2012: pp. 115-137 .
28.  Integrable systems and group actions,  Cent. Eur. J. Math. 12 (2014), no. 2, 240–270.
29.   (with C. Laurent-Gengoux) Acta Math. Vietnam. 38 (2013), no. 1, 21–32.,
30. Rigidity for Hamiltonian actions on Poisson manifolds( with P. Monnier and N. T. Zung) ,    Adv. Math. 229 (2012), no. 2, 1136–1179.
31. 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.
32.  From action-angle coordinates to geometric quantization: a round trip,
33. 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.
34. , , Annales de l'Institut Fourier, vol. 60 (1) p. 51-85, 2010. (36 pages)
35.  Symmetries and singularities of Hamiltonian systems, Conf. Ser. 175 012011, 2009.
36.  Symplectic linearization of semisimple actions, preprint 2008.
37.  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.
38.  XV International Workshop on Geometry and Physics,, Publ. R. Soc. Mat. Esp., 11,  177-183, R. Soc. Mat. Esp., Madrid, 2007.
39.  A note on Equivariant normal forms of Poisson structures, (with N. T. Zung)   Mathematical Research Letters, vol 13-6, pages 1001-1012, 2006.
40.  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.
41.  A singular Poincaré Lemma (with Vu Ngoc) , IMRN International Mathematics Research Notices, n 1, 2005.
42.  Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems  Ann. Sci. Ecole Norm. Sup.,37 (2004), no. 6, 819--839 2004.
43.  Symplectic linearization of singular Lagrangian foliations (with C. Curras-Bosch), Differential Geometry and its applications,  18, (2), 195-205, 2003.
44.
45. , 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.

Book (New!):

I am currently working on a book project entitled "Geometry and Dynamics of singular symplectic manifolds" (by invitation of the Societé Mathématique de France jointly with Fondation Sciences Mathématiques de Paris) for the collection Cours Spécialisées of the SMF.  By special agreement with FSMP the publication will be made available in open access.

This book aims to be an enhanced and (hopefully) comprehensive version of the course I delivered in October-November-December 2017  at Institut Henri Poincaré during my Chaire d'Excellence de la Fondation Sciences Mathématiques de Paris.  The webpages of the course are available here: https://www.sciencesmaths-paris.fr/fr/le-cours-deva-miranda-933.htm https://mat-web.upc.edu/people/eva.miranda/coursIHP.htm

Stay tuned!

Divulgatio:

1. Miranda, Eva; Muñoz, Miguel Carlos Maryam Mirzakhani, a light that will never go out. (Catalan) SCM Not. No. 42 (2017), 53–55.
2. Columnes de la EMS al Noticies de la SCM 2011-2017.
3. Maryam Mirzakhani, una luz que nunca se apagará (con M. Muñoz Lecanda), Boletín de la RSME. 544, 2017.
4. Congresos y charlas plenarias (Elisa Lorenzo García, Eva Miranda Galcerán, Teresa E. Pérez Fernández y Elena Vázquez Cendón), Boletin de la RSME, 571, 2018.
5. Faces of Women in mathematics, Boletin de la RSME, 573, 2018.
6. El efecto Matilda Columna Mujeres y Matemáticas, Boletin de la RSME, 576, 2018.
7. Entrevista a Sílvia Casacuberta (Boletin de la RSME), Mayo 2018, Boletin 577.
8. En recuerdo a Maryam Mirzakhani, la exploradora de Superfícies, El Pais, July 14, 2018.
9.  La matemática ucraniana que podría haber ganado la medalla fields, El País 14 de Agosto de 2018,
10. La matemática de los fenómenos que se repiten, El País, 9 de Noviembre de 2018.

Media:

Interview at ABC, March 2017

A video promoting the bid ICM2022 for Paris

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.