Full Professor in Geometry and Topology at UPC and CRM-Barcelona, Alexander Von Humboldt Friedrich Wilhelm Bessel professor at Mathematisches Institut der Universität zu Köln
I am a Full Professor distinguished with two consecutive ICREA Academia Awards (2016, 2021) at (UPC), member of CRM and IMTECH. I have been recently distinguished with the François Deruyts Prize by the Royal Academy of Belgium and with a Bessel Prize by the Alexander Von Humboldt foundation. I am the 2023 London Mathematical Society Hardy lecturer as such I have enjoyed lecturing a 9 stop-tour in the summer of 2023 which has been quite a unique experience. The picture above was taken in the middle of the tour at the University of Loughborough.
I am the director of the Laboratory of Geometry and Dynamical Systems and the group leader of GEOMVAP (Geometry of Varieties and Applications). I have been the happy advisor of 11 PhD students.
My research deals with several aspects of Differential Geometry, Mathematical Physics and Dynamical Systems such as Symplectic and Poisson Geometry, Hamiltonian Dynamics, Group actions and Geometric Quantization. Almost a decade ago I started the investigation of several facets of b-Poisson manifolds (also known as log-symplectic manifolds). These structures appear naturally in physical problems on manifolds with boundary and in Celestial mechanics such as the 3-body problem (and on its restricted versions) after regularization transformations. I recently got interested in Fluid Dynamics and the study of their complexity (computational, topological, logical, dynamical) by looking through a contact mirror unveiled two decades ago by Etnyre and Ghrist. I am working in extending Floer homology to a class of Poisson manifolds including b-Poisson manifolds and the classical Weinstein conjecture in this set-up. My motivation comes from the search of periodic orbits on regularized problems in Celestial Mechanics (more information here).
I am particularly interested in understanding connections between different areas such as Geometry, Dynamical Systems, Mathematical Physics and, more recently, Fluid Dynamics. I often get a better understanding of some phenomena by using techniques at the intersection of disciplines.
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 am fascinated by periodic orbits and different approaches to their study (including Floer homology and KAM theory). 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. Collaboration with my colleagues has recently taken me to explore fascinating new lands, including that of connection between Geometry, Dynamical systems, Fluid dynamics and computer science.
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 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 also like 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 I am interested in building bridges between different areas of mathematics and lately focusing on intersections between Dynamical Systems, Fluid Dynamics, contact geometry and computer science. The ICREA Academia prize permitted me to focus on research and attain influential results in these areas.
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. So far we could prove the existence of escape orbits and generalized singular periodic orbits in a number of cases where genericity occurs in the class of Melrose contact forms.
We are also extending the Floer apparatus to the b-world.
See more in this talk I gave at a workshop in Zurich in January 2021.
Update: We have recently disproved the singular Weinstein conjecture. More soon!
Fluid Dynamics: Universality of Euler flows, h-principle for contact geometry and Turing completeness in dimension 3
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. Just for the New Year's Eve of 2020 we finished our article on constructing Turing complete Euler flows in dimension 3, closing up an open problem since the 90's by Moore and also Tao recently.
You can read it here: arXiv:2012.12828. It has been published at PNAS.
This result captured the attention of mass media. You may want to consult, for instance, the article at El Pais or at Pour la Science (complete list of articles in the Outreach section of the webpage).
Later we obtained a generalization for t-dependent Euler flows (now published at IMRN) by compactifying a former result by Graça et al of Turing complete polynomials and embedding them as Euler flows, see below.
In all the constructions above, the metric is seen as an additional "variable" and thus the method of proof does not work if the metric is prescribed.
You can learn more about this in this video of my invited talk at the 8th European congress of mathematics.
Is it still possible to construct a Turing complete Euler flow on a 3-dimensional space with the standard metric?
The answer is YES. You can check our article with Robert Cardona and Daniel Peralta-Salas published at JMPA.
We have recently got a good understanding of the entropy of a Turing complete system. Check our recent article below.
The hybrid computer:
One of my recent endeavours is to extend the construction of the Fluid computer (Turing complete Euler flows) to a more generalized abstract design. We opened a La Caixa INPHINIT grant in that direction. More soon!
Poisson Geometry as towers of E-symplectic manifolds:
One of my recent project concerns understanding Poisson manifold and different approximations from the symplectic perspectives. I am currently working jointly with Ryszard Nest on understanding the possible desingularization of Poisson manifolds as families of E-symplectic manifolds.
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.
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
The excellence cluster SYMCREA is formed by two Research ICREA professors: Marco Gualtieri and Marta Mazzocco and the ICREA Academia professor Eva Miranda. It focuses on Symplectic Geometry, the study of symmetries and interactions with Mathematical Physics.
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.
Symplectic and Poisson Geometry Unveiled: Exploring Cohomological
and Algebraic new horizons, FPI-MDM grant, Starting date: September 2023.
Søren Dyhr (Msc. Aarhus)
Representation
theory in geometric fluid dynamics, Funding: UPC FPI grants, Currently INPHINIT La Caixa grants-MDM, Centers of excellence.
Current undergraduate and master students (3)
Juan Brieva (CFIS-Univ. of Oxford)
Symplectic implosion and desingularization (cosupervision with Andrew Dancer at the University of Oxford).
Josep Fontana-McNally
Undergraduate thesis: Singular forms in Celestial mechanics and Fluid Dynamics, currently Master Student at Oxford University.
Isaac Ramos
Topological and geometrical techniques in fluid dynamics and computational complexity, undergraduate student at UCM, currently exchange student at UCLA.
Former Ph.D. students (9)
In this video Anastasia Matveeva explains about her thesis!
Poisson structures on moduli spaces and group actions.
Funding: InPHinit La Caixa, defended in October 2022, nominated for 30 Forbes under 30 in the Science category, currently Team leader at Product Science in Los Angeles. Watch a video about her research here:
Joaquim Brugués (Msc. UPC)
Floer Homology for b-symplectic manifolds. Funding: FI-AGAUR, defended March 2024.
Mir Garcia(Msc. UPC) Singularities and symmetries in Physical models
Contact topology and Reeb dynamics with applications to ideal fluids.
Funding: FPI - MdM - BGSMath Postdoc in Strasbourg. Margarita Salas Fellow at UPC. Currently Assistant Professor at UB. Galois Prize, Vicent Caselles Prize and Extraordinary PhD award.
Global Hamiltonian Dynamics of singular symplectic manifolds, October 2, 2020.
postdoc under my supervision funded with my ICREA Academia project. Postdoc at ENS Lyon. Currently Juan de la Cierva at BCAM Bilbao.
Rose Mary Dempsey Bradel (Msc. Padova-Bordeaux)
New geometrical and dynamical techniques for problems in Celestial Mechanics. Co-supervision with Amadeu Delshams.
Funded with my ICREA research project. Defended on Feb 17, 2021. Currently postdoc at BCAM.
Arnau Planas (Msc. UPC)
Symmetries and singularities of Poisson manifolds, September 2020.
Currently, Senior Data Scientist at HP.
Geometric Quantization of Integrable systems with singularities (2013)
Currently postdoc at Universidad de Granada
Former Master and undergraduate thesis students:
Pablo Nicolás
Master thesis: Poisson Geometry: old and new
Josep Fontana-McNally
undergraduate thesis: Singular forms in Celestial mechanics and Fluid Dynamics
Lara San Martin (cosupervision with Angus Gruen and Sergei Gukov at Caltech) undergraduate thesis
Quantum knot invariants
and the extension of FK to SU (3)
Pablo Nicolás
Undergraduate thesis (joint with Kolya Reshetikhin at Yau Center in Tsinghua Univ.) On the spin Calogero-Moser systems and b-Symplectic Geometry (2021-2022)
Alberto Cavallar
Undergraduate thesis (joint with Sergei Gukov at Caltech) The Chern-Simons Topological Quantum Field Theory and q-series invariants of 3-manifolds for knot complements (2021-2022)
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)
Postdoc supervision
Jagna Wisniewska (2022-2024), funded with our COMPLEXFLUIDS project.
Alfonso Garmendia (2022-2024), funded with a MDM CRM postdoctoral grant.
Cédric Oms (2020-2021), funded with my ICREA Academia prize.
Collaborators (other than former or current PhD students)
Alexey Bolsinov
Baptiste Coquinot
Carlos Curras
Chiara Esposito
Pedro Frejlich
Angel Gonzalez-Prieto
Victor Guillemin
Mark Hamilton
Camille Laurent-Gengoux
David Martínez Torres
Vladimir Matveev
Philippe Monnier
Ryszard Nest
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 Institute of Mathematics at the UPC-IMTECH. I am currently actively implied with the following research institutes, organizations and nodes:
Chaire d'Excellence 2017 de la Fondation des Sciences Mathématiques de Paris.
ICREA Academia 2016 prize.
Giovanni Prodi Chair at Würzburg 2017-2018 (declined).
Invited address at 8ECM.
Active grants
Principal investigator of an AGAUR SGR grant 2021 SGR 00603, total amount: 65000 euros, date of award 2022.
Principal Investigator of ICREA Academia 2021 Total amount: 120000 euros for 5 years (date of award January 2022, individual project).
Co-Principal investigator of the Maria de Maeztu program CEX2020-001084-M (Principal Investigator: Marcel Guardia, Center of Award: Centre de Recerca Matemàtica): Total amount: 2M euros for 5 years (date of award July 2021).
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 AEI project Geometría, Álgebra, Topología y Aplicaciones Multidisciplinares code PID2019-103849GB-I00: Total amount:160.809,00 € .
Principal investigator of an SGR Research project 2017SGR932: 65898 euros 2017-2019 (total number of members 21).
Principal investigator of the project Geometría, Álgebra, Topología y Aplicaciones Multidisciplinares GATA-Tech with code PID2019-103849GB-I00 , total number of participants: 30. Total funding:
160.809,00 €. Start date June 2020 end date May 2024.
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 2020.
Principal investigator for an AFR-Ph.D. project 2016-2019, Total amount: 160.901,19 euros.
Symplectic and Poisson Geometry Unveiled: Exploring Cohomological
and Algebraic new horizons, FPI-MDM grant, Starting date: September 2023.
Søren Dyhr (Msc. Aarhus)
Representation
theory in geometric fluid dynamics,. Funding: UPC FPI grants, Currently INPHINIT La Caixa grants-MDM, Centers of excellence.
.
Current undergraduate and master students (3)
Juan Brieva (CFIS-Univ. of Oxford)
Symplectic implosion and desingularization (cosupervision with Andrew Dancer at the University of Oxford).
Josep Fontana-McNally
undergraduate thesis: Singular forms in Celestial mechanics and Fluid Dynamics, currently Master Student at Oxford University.
Isaac Ramos
Topological and geometrical techniques in fluid dynamics and computational complexity, undergraduate student at UCM, currently exchange student at UCLA.
Former Ph.D. students (9)
In this video Anastasia Matveeva explains the ideas of her thesis for general public!
Contact topology and Reeb dynamics with applications to ideal fluids.
Funding: FPI - MdM - BGSMath Postdoc in Strasbourg. Margarita Salas Fellow at UPC. Current position: Assistant Professor at UB.
Galois Prize, Vicent Caselles Prize and Extraordinary PhD award.
Global Hamiltonian Dynamics of singular symplectic manifolds, October 2, 2020.
Currently, postdoc under my supervision funded with my ICREA Academia project. Next position: Postdoc at ENS-Lyon
Rose Mary Dempsey Bradel (Msc. Padova-Bordeaux)
New geometrical and dynamical techniques for problems in Celestial Mechanics. Co-supervision with Amadeu Delshams.
Funded with my ICREA research project. Defended on Feb 17, 2021. Currently postdoc at BCAM.
Arnau Planas (Msc. UPC)
Symmetries and singularities of Poisson manifolds, September 2020.
Currently, Senior Data Scientist at HP.
Geometric Quantization of Integrable systems with singularities (2013)
Currently postdoc at Universidad de Granada
Former Master and undergraduate thesis students:
Pablo Nicolás
Master thesis: Poisson Geometry: old and new
Josep Fontana-McNally
undergraduate thesis: Singular forms in Celestial mechanics and Fluid Dynamics
Lara San Martin (cosupervision with Angus Gruen and Sergei Gukov at Caltech) undergraduate thesis
Quantum knot invariants
and the extension of FK to SU (3)
Pablo Nicolás
Undergraduate thesis (joint with Kolya Reshetikhin at Yau Center in Tsinghua Univ.) On the spin Calogero-Moser systems and b-Symplectic Geometry (2021-2022)
Alberto Cavallar
Undergraduate thesis (joint with Sergei Gukov at Caltech) The Chern-Simons Topological Quantum Field Theory and q-series invariants of 3-manifolds for knot complements (2021-2022)
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)
Master and undergraduate supervision (in antichronological order)
Pablo Nicolás
Master thesis: Poisson Geometry: old and new
Josep Fontana-McNally
undergraduate thesis: Singular forms in Celestial mechanics and Fluid Dynamics
Lara San Martin (cosupervision with Angus Gruen and Sergei Gukov at Caltech) undergraduate thesis
Quantum knot invariants
and the extension of FK to SU (3)
Pablo Nicolás
Undergraduate thesis (joint with Kolya Reshetikhin at Yau Center in Tsinghua Univ.) On the spin Calogero-Moser systems and b-Symplectic Geometry (2021-2022)
Alberto Cavallar
Undergraduate thesis (joint with Sergei Gukov at Caltech) The Chern-Simons Topological Quantum Field Theory and q-series invariants of 3-manifolds for knot complements (2021-2022)
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
If you are interested in doing a Master thesis under my supervision, contact me for details. I tend to organize my Master thesis supervision planning a year in advance, so if you are curious contact me ahead of time.
For more details about each work and some additional proposals, please, visit the intranet.
Master thesis proposals (not updated)
Embedding dynamical systems as Euler and Navier-Stokes systems
From Arnold conjecture to Weinstein conjecture and beyond
Embedding theorems for manifolds with additional structures
Circle actions on 4-dimensional b-symplectic manifolds
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 (not currently active proposal)
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 (not updated)
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
, FIM - Institute for Mathematical Research, Eidgenössische Technische Hochschule Zürich, ETH Zurich, Zürich.
Differentiable manifolds (UPC), Master Course. .
2024-2025
Differentiable manifolds (UPC), Master Course. .
2023-2024
Differentiable manifolds (UPC), Master Course. .
2022-2023
Differentiable manifolds (UPC), Master Course. .
2021-2022
Minicourse on The Geometry and Dynamics of Singular symplectic manifolds (online at Henan University). Webpage of the course. Videos of the course:
Minicourse on Geometric Quantization via Integrable Systems (online at the University of Freibourg during the GEOQUANT summer school).Webpage of the course. Videos of the course:
Minicourse Looking at Euler flows through a contact mirror: Universality and undecidability, Course at the Fall Workshop on Geometry and Physics.Videos of the course
Smooth manifolds (UPC), Master Course. In person (omicron and future variants permitting).
2020-2021
Smooth manifolds (UPC), Master Course. The videos of this course are on ATENEA.
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
Media
La bellesa de les matemàtiques consisteix en captar l’essència de les coses (interview by Toni Pou) Interview at ARA, May 2023...