Research

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
  • Mathematical Physics in the large

Summary

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

Singularities

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). Their symplectic reincarnations are called b-symplectic manifolds which appear to model 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. 3 body problem

Periodic orbits

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 the Weinstein Conjecture if we allow singularities in the contact form. In particular under compactness assumptions on the critical set we have been able to prove the existence of infinite periodic orbits on the critical set for 3-dimensional b^m-contact structure. This has led us to formulate the singular Weinstein conjecture about existence of singular periodic orbits on b^m-contact manifolds. Those singularities on contact structures model some problems of Beltrami flows on manifolds with boundary. This variant of the Weinstein conjecture is very revealing: The singular orbits are indeed periodic orbits which are no longer smooth but have points as marked singularities. This opens a door to a new world. In the direction of the singular Weinstein conjecture we are now trying to prove that the set of b^m-contact structures admitting singular Reeb orbits is generic in the set of b^m-contact forms. Solar system

Fluid Dynamics: Universality of Euler flows and h-principle for contact geometry

I have been recently interested in Fluid Dynamics where I entered driven by singularity theory. With Daniel Peralta Salas and Robert Cardona, we had been working on b-contact forms appearing in Fluid Dynamics using the correspondence between contact forms and Beltrami vector fields (see our paper on Phylosophical Transactions of the Royal Society below). In Febrary 17, 2019 I came across this entry (thank you Twitter!) in the blog of Terry Tao. This was a source of inspiration to work on h-principles for Reeb embeddings and proving universality properties of Euler flows and Turing completeness of Euler flows. This is the content of our paper arXiv:1911.01963. More is to come! Flow around a wing


ResearchTeam

My Research Team

I am the group leader of the research group in Geometry at UPC, GEOMVAP. I am also the director of the Laboratory of Geometry and Dynamical Systems.

My research team

GEOMVAP

The Geometry, Topology, Algebra, and Applications Group (GEOMVAP) is a group of researchers with interests in a wide range of fields, which include algebraic, differential and symplectic geometries, algebraic topology, commutative algebra and their applications. The group is composed of researchers rooted or formed at the Universitat Politècnica de Catalunya.

Our group works on topological and differentiable manifolds, algebraic varieties, and their applications, viewing problems from a variety of different perspectives. The group has a long tradition working on various different interfaces of algebra, geometry and topology. In the last decade we have become active contributors in interdisciplinary science and we are now focused on both a theoretical point of view and the transversal applications to several disciplines including Robotics, Machine Learning, Physics and Celestial Mechanics. Our research can be grouped in 8 different research lines which are closely related and interact in a dynamic manner. The 4 first lines are theoretical and the 4 last ones are interdisciplinary.

  • AGE: Algebraic Geometry.
  • CAN: Commutative Algebra and Number theory.
  • TOP: New Challenges in Algebraic Topology.
  • SYM: New trends in Differential Geometry, Symplectic Geometry and Geometric Mechanics.
  • BIO: Applications to Biology
  • ROB: Applications to Control Theory, Machine Learning and Robotics
  • CEL: Applications to Dynamical Systems and Celestial Mechanics
  • PHY: Applications to Physics
GEOMVAP in action
The Lab of Geometry and Dynamical systems

The Laboratory of Geometry and Dynamical Systems at UPC-EPSEB promotes the collaboration between Geometers and Dynamicists interested in common problems from different perspectives and with complementary techniques. The lab focuses on questions which are on the crossroads of Symplectic Geometry and Dynamical Systems.

The lab consists currently of 2 Full Professors, 1 Associate Professor, 2 postdocs, 6 Ph.D. students, 1 visiting PhD students, 1 master student.

Current Ph.D. students
  • Pau Mir (starting in September) (Msc. UPC)
    The geometry and physics of singularities.
    Funding: Personnel of Support to research (internal funding)
  • Joaquim Brugués (Msc. UPC)
    On Floer Homology of Poisson manifolds. Co-supervision with Sonja Hohloch (University of Antwerp).
    Funding: FI-AGAUR
  • Anastasia Matveeva (Msc. Higher School of Economics, Moscow)
    Poisson structures on moduli spaces and group actions.
    Funding: InPHinit La Caixa
  • Robert Cardona (Msc. UPC)
    Contact topology and Reeb dynamics with applications to ideal fluids.
    Funding: FPI - MdM - BGSMath
  • Roisin Braddell (Msc. Padova-Bordeaux)
    New geometrical and dynamical techniques for problems in Celestial Mechanics. Co-supervision with Amadeu Delshams.
    Funded with my ICREA research project.
  • Cédric Oms (Msc. ULB)
    Global Hamiltonian Dynamics of singular symplectic manifolds, thesis in deposit, defense scheduled for October 2, 2020.
    Funding: FNRS - GLADYSS
  • Arnau Planas (Msc. UPC)
    Symmetries and singularities of Poisson manifolds, thesis in deposit, defense scheduled for September 30 2020.
Former Ph.D. students
  • Anna Kiesenhofer
    Integrable systems on b-Poisson structures (2016)
    Currently postdoc at EPFL
  • Romero Solha
    Geometric Quantization of Integrable systems with singularities (2013)
    Currently postdoc at PUC
Postdoc supervision

Collaborators

Collaborators (other than former or current PhD students)
  • Alexey Bolsinov
  • Carlos Curras
  • Chiara Esposito
  • Pedro Frejlich
  • Victor Guillemin
  • Mark Hamilton
  • Camille Laurent-Gengoux
  • David Martínez Torres
  • Vladimir Matveev
  • Philippe Monnier
  • Daniel Peralta-Salas
  • Sergei Tabachnikov
  • Nguyen Tien Zung
  • Ana Rita Pires
  • Fran Presas
  • Nicolai Reshetikhin
  • Geoffrey Scott
  • Pol Vanhaecke
  • Vu Ngoc San
  • Jonathan Weitsman

My research networks

Along my academical trajectory I have been proactive in the creation of laboratories and research networks such as the Viktor Ginzburg Lab, the European Network CAST or the forthcoming Institute of Mathematics at the UPC-IMTECH. I am currently actively implied with the following research institutes, organizations and nodes:
Schedule

Schedule

Organized conferences

  • Workshop on Geometric methods in Symplectic Topology ICMAT.
  • Workshop on Women in Geometry and Topology ICMAT.
  • FDIS2017 CRM.
  • FDIS2017 CRM.
  • FDIS2017 CRM.
  • JISD2017 CRM.
  • GESTA2016.
  • JuniorGESTA.
  • Geometry and Dynamics of Foliations ICMAT.
  • GESTA 2014 ICMAT.
  • Symplectic Techniques of Dynamical Systems ICMAT.
  • Qdays in Barcelona CRM.
  • Conference on Integrability, Topological obstructions to Integrability and interplay with Geometry CRM.
  • GESTA 2013 Toulouse.
  • Advanced Course on Geometry and Dynamics of Integrable Systems CRM.
  • Research Programme on Geometry and Dynamics of Integrable Systems CRM.
  • CAST conference ICMAT.
  • GESTA 2011 UPC.
  • GEDYTO Hanoi.
  • Conference in honour of Paulette Libermannn IHP.
  • Geometric flows and equivariant problems in Symplectic Geometry CRM.
  • GESTA 2008 UAB.
  • GAP VI, Geometry and Physics VI CIM-CRM.
  • Conference on Moment Maps .
  • Weeken(oïd) Géométrie de Poisson Toulouse.
  • GESTA 2006 Toulouse.
  • Geometry of Integrable Hamiltonian Systems at CRM, Barcelona 2001. CRM.

Organized seminars

  • Organizer of 23 workshops and 2 Research programmes at CRM-Barcelona.
  • Former organizer, with Jaume Amorós, of the Teen Seminar on geometry.
  • Organizer of the regular one day intensive seminar HIS (Hamiltonian Intensive Seminar).
  • Former organizer of the special Seminar on Symplectic and Poisson Geometry in Barcelona.
boards

Boards

Governing Boards

  • Personnalité extérieure of the Conseil d'administration (Board of Directors) of the IHP-Institut Henri Poincaré (2020-).
  • Member at the Governing Board of the BGSMath (2018-).

Scientific Boards

Editorial Boards

CV

CV

Distinctions

  • Chaire d'Excellence 2017 de la Fondation des Sciences Mathématiques de Paris.
  • ICREA Academia 2016 Professor.
  • Giovanni Prodi Chair at Würzburg 2017-2018 (declined).
  • Invited address at 8ECM.

Active grants

  • Principal investigator of an ICREA Academia Project: Total amount: 200000 euros (Start date January 2017, duration of the grant 5 years).
  • Principal investigator of an SGR Research project 2017SGR932: 65898 euros 2017-2019 (total number of members 21).
  • Principal investigator of the project MTM2015-69135-P, total number of participants: 24. Total funding: 182.226,00 €. Start date January 2016 end date December 2019.
  • Principal investigator for an AFR-Ph.D. project 2016-2019, Total amount: 160.901,19 euros.

Affiliations and memberships

Additional services for the mathematical community

  • Referee for a list of journals including among others: Acta Mathematica, Nonlinearity, Annales de l’Institut Fourier, Journal of the London Mathematical Society, Ergodic Theory and Dynamical Systems, Discrete & Continuous Dynamical Systems, Journal of Geometry and Physics, Journal of Mathematical Physics, Journal of Symplectic Geometry, Quaterly Journal of Mathematics, Journal of Regular and Chaotic Dynamics, Revista Matemática Complutense, Differential Geometry and its applications, Acta Mathematica Applicandae, AIP Publications, Birkhauser Advanced CRM courses.
  • Research expert for several (inter)national research agencies including DFG, ANEP, AEI, ANR, NWO.
Teaching

Teaching

Supervision

Current Ph.D. students
  • Pau Mir (Msc. UPC)
    The geometry and physics of singularities.
    Funding: Personnel of Suport to Research (internal funding)
  • Joaquim Brugués (Msc. UPC)
    On Floer Homology of Poisson manifolds. Co-supervision with Sonja Hohloch (University of Antwerp).
    Funding: FI-AGAUR
  • Anastasia Matveeva (Msc. Higher School of Economics, Moscow)
    Poisson structures on moduli spaces and group actions.
    Funding: InPHinit La Caixa
  • Robert Cardona (Msc. UPC)
    Contact topology and Reeb dynamics with applications to ideal fluids.
    Funding: FPI - MdM - BGSMath
  • Roisin Braddell (Msc. Padova-Bordeaux)
    New geometrical and dynamical techniques for problems in Celestial Mechanics. Co-supervision with Amadeu Delshams.
    Funded with my ICREA research project
  • Cédric Oms (Msc. ULB)
    Global Hamiltonian Dynamics of singular symplectic manifolds, thesis already in diposit, defense scheduled for October 2, 2020.
    Funding: FNRS - GLADYSS
  • Arnau Planas (Msc. UPC)
    Symmetries and singularities of Poisson manifolds, thesis already in diposit, defense scheduled for September 2020.
Former Ph.D. students
  • Anna Kiesenhofer
    Integrable systems on b-Poisson structures (2016)
    Currently postdoc at EPFL
  • Romero Solha
    Geometric Quantization of Integrable systems with singularities (2013)
    Currently postdoc at PUC
Postdoc supervision
Master and undergraduate supervision
  • Pau Mir
    Master thesis: Rigidity of group actions, cotangent lifts and integrable systems (2020)
  • Joaquim Brugués
    Master thesis: Morse and Floer Homology (2019)
  • Robert Cardona
    Master thesis: Integrable Systems on Folded Symplectic manifolds (2018)
  • Robert Cardona
    Undergraduate thesis: Symplectic Toric manifolds, Delzant theorem and applications (2017)
  • Arnau Planas
    Master thesis: Symplectic surfaces with singularities (2015)
  • Alexander Thiele
    Master thesis: Transversality, old and new (2014)

New proposals for supervision

For more details about each work and some additional proposals, please, visit the intranet.
Master thesis proposals
  • Classical and Quantum integrable systems: Can we hear the shape of a drum?
  • Geometry and Physics of semitoric and almost toric manifolds
  • Quantization, symmetries and singularities in interaction
  • The mathematics of Maryam Mirzakhani
  • Vortex equations and celestial mechanics
  • The quest of periodic orbits: From Seifert to Conley and Weinstein
  • Locally conformally symplectic manifolds and Celestial Mechanics
Undergraduate thesis proposals
  • Els grups de Lie i les seves accions
  • Del teorema del punt fix de Lefschetz al teorema de Poincaré-Hopf
  • Formes diferencials i foliacions de codimensió 1
  • Les equacions de Hamilton, els grups de Lie i la geometria simplèctica
  • Teoria de Morse
  • Projecte Hévea: Construint tors plans a R^3
  • Projecte Hévea: Contruint esferes reduïdes

Teaching

2019-2020
  • Smooth manifolds (UPC), Master Course With the total fun of teaching during confinement I opened a youtube channel. Check it out here: https://www.youtube.com/channel/UC8Fzyf58s0EiZ-gdYgz2ghw?view_as=subscriber
2018-2019
  • Smooth manifolds (UPC), Master Course
2017-2018
  • Geometry and Dynamics of Singular Symplectic manifolds (IHP)
  • Smooth manifolds (UPC), Master Course
2016-2017
  • Fonaments Matemàtics d’Enginyeria de l’Edificació a Arquitectura Tècnica (UPC)
  • Master Course on Differentiable manifolds (UPC)
  • Fonaments Matemàtics d’Enginyeria de l’Edificació a Arquitectura Tècnica (UPC)
  • Geometria Diferencial al Grau de Matemàtiques (UPC)
2015-2016
  • Fonaments Matemàtics d’Enginyeria de l’Edificació a Arquitectura Tècnica (UPC)
  • Fonaments Matemàtics d’Enginyeria de l’Edificació a Arquitectura Tècnica (UPC)
  • Geometria Diferencial al Grau de Matemàtiques (UPC)
  • Symplectic Techniques in Dynamical Systems and Mathematical Physics (BGSMath)
2012-2013
  • Fonaments Matemàtics d'Enginyeria de l'Edificació (UPC)
  • Topologia al Grau de Matemàtiques (UPC)
  • Geometria Diferencial al Grau de Matemàtiques (UPC)
2011-2012
  • Fonaments Matemàtics d'Enginyeria de l'Edificació (UPC)
  • Topologia al grau en Matemàtiques (UPC)
  • Geometria Diferencial al grau en Matemàtiques (UPC)
2010-2011
  • Fonaments Matemàtics d'Enginyeria de l'Edificació (UPC)
  • Estadística Aplicada d'Enginyeria de l'Edificació (UPC)
  • Topologia al grau en Matemàtiques (UPC)
2009-2010
  • Fonaments Matemàtics d'Enginyeria de l'Edificació (UPC)
  • Estadística Aplicada d'Enginyeria de l'Edificació (UPC)
  • Topologia al grau en Matemàtiques (UPC)
2008-2009
  • Geometria Diferencial (UAB)
2007-2008
  • Geometria Riemanniana (UAB)
  • Algebra Lineal (UAB)
2005-2006
  • Geometria Diferencial (UB)
  • Introducció a l `algebra i la Geometria (UB)
2003-2004
  • Curs de doctorat: Geometria Simplèctica (UB)
  • Geometria Diferencial (UB)
  • Geometria Proyectiva (UB)
  • Grups i Algebres de Lie (UB)
2002-2003
  • Geometria Proyectiva (UB)
  • Geometria Diferencial (UB)
  • Grups i Algebres de Lie (UB)
2001-2002
  • Geometria Proyectiva (UB)
  • Geometria Diferencial (UB)
  • Geometria Diferencial (UB)
  • Grups i Algebres de Lie (UB)
  • Algebra Lineal (UB)
  • Càlcul Infinitesimal (UPC)
2000-2001
  • Grups i Algebres de Lie (UB)
  • Introducció a l´Algebra i la geometria (UB)
  • Geometria Diferencial (UB)
  • Fonaments Matemàtics I (UPC)
1999-2000
  • Grups i Algebres de Lie (UB)
  • Algebra Lineal (UB)
  • Geometria Diferencial (UB)
  • Geometria Diferencial de Corbes i Superficies (UB)
  • Fonaments Matemàtics I (UPC)
  • Algebra Lineal (UPC)
  • Càlcul (UPC)
1998-1999
  • Grups i Algebres de Lie (UB)
  • Geometria Diferencial (UB)
  • Geometria Diferencial de Corbes i Superficies (UB)
  • Algebra Lineal (UdL)
  • Matemàtiques I (UdL)
1997-1998
  • Geometria Lineal (UB)
  • Geometria Diferencial de Corbes i Superficies (UB)
  • Algebra Lineal (UB)
  • Càlcul i Algebra II (UB)
  • Algebra Lineal (UdL)
  • Ampliació d´Anàlisi Matemàtic (UdL)
1996-1997
  • Algebra Lineal (UB)
  • Algebra Lineal (UB)
  • Càlcul i Algebra II (UB)
Outreach

Outreach

Interviews and talks

  1. Interview by the CNRS. May 12, 2020. The webpage of CNRS (9 pages of interview).
  2. Interview at El ABCdario de las matemáticas. April, 2017. ABC.
  3. Interview at the Newsletter of the European Mathematical Society. ABC.
  4. Interview by the FSMP to promote the bid for ICMA2022 for Paris. ABC.
  5. Interview at La Mirada. April 22, 2019. Canal Sur.
  6. Viajando en el tiempo con las matemáticas. Pint of science.

Dissemination articles at El País

  1. La matemática de los fenómenos que se repiten. 9 de Noviembre de 2018. El País.
  2. La matemática ucraniana que podría haber ganado la medalla fields. 14 de Agosto de 2018. El País.
  3. En recuerdo a Maryam Mirzakhani, la exploradora de Superfícies. July 14, 2018. El Pais.

Other outreach writings

  1. Dones en Xarxa, Efecto Matilda, Dones en Xarxa.
  2. Entrevista a Sílvia Casacuberta. Boletin de la RSME 576, 2018.
  3. El efecto Matilda. Boletin de la RSME.
  4. Faces of Women in mathematics. Boletin de la RSME, 573, 2018.
  5. Congresos y charlas plenarias. Boletin de la RSME.
  6. Maryam Mirzakhani, una luz que nunca se apagará. Boletín de la RSME. 544, 2017.
  7. The Clay Public Lecture and Conference on the Poincaré Conjecture. News EMS Eur. Math. Soc. Newsl, issue 77, pages 21-23, 2010.
  8. Columna de la EMS al Noticies de la SCM, Febrer 2013, (2013).
  9. Columna de la EMS al Notícies de la SCM, Març 2012, 9-11, 32, (2012).
  10. Columnes de la EMS al Noticies de la SCM 2011-2017.
  11. Maryam Mirzakhani, una llum que mai no s’apagará. SCM notícies.
  12. The Hirsch Conjecture has been disproved: an interview with Francisco Santos. Newsletter of the EMS 2012, vol. 86, p. 31-37.

Gender Equality

  1. Participation in the Round Table "Mesa redonda con investigadoras" at BCNSpiracy, October 2018, CaixaForum, Barcelona.
  2. Organization of Dia internacional de la Dona i la Nena en la Ciencia, February 14 2019, iec.
  3. Participation at the round table Matemàtica i Dones: les barreres socials i les acadèmiques, organized by BGSMath and SCM, February 14th 2019.
  4. Participation at the round table STEAM-MAT-mat-es-ella Barcelona, Universidad de Barcelona, April 2019.