Chair in Geometry and Topology at UPC and CRM-Barcelona,
Gauss professorin at Niedersächsische Akademie der Wissenschaften zu Göttingen,
and Nachdiplom Lecturer at ETH Eidgenössische Technische Hochschule Zürich
formerly
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. Adventure never stops: I have been appointed 2025 Gauss Professor at the University of Goettingen and Nachdiplom lecturer 2025 at ETHZ in Zurich.
I am the director of the Laboratory of Geometry and Dynamical Systems and the group leader of GEOMVAP (Geometry of Varieties and Applications). With Marta Mazzocco we have just created the SYMCREA excellence unit. We are on the third floor at EPSEB. I have been the advisor of 11 PhD students.
My research spans several areas of Differential Geometry, Mathematical Physics, and Dynamical Systems, including Symplectic and Poisson Geometry, Hamiltonian Dynamics, Group Actions, and Geometric Quantization and Mathematical aspects of Computer Science. Almost a decade ago, I began investigating various facets of b-Poisson manifolds (also known as log-symplectic manifolds). These structures naturally arise in physical problems on manifolds with boundary and in Celestial Mechanics, such as the three-body problem (and its restricted versions) after regularization transformations.
More recently, I have become interested in Fluid Dynamics and in studying its many layers of complexity—computational, topological, logical, and dynamical—through a contact mirror unveiled two decades ago by Etnyre and Ghrist. I am currently exploring the deep connections between dynamical systems and computation. Building on this understanding, we have designed a new model of computation inspired by ideas from Topological Quantum Field Theory (TQFT). This Topological Kleene Field Theory (TKFT) model allows us to investigate new facets of dynamical systems, particularly undecidable phenomena.
One of the problems I am currently working on is understanding undecidable phenomena in physical systems — in particular, in the three-body problem, where classical chaos meets logical chaos. One of the questions that interest me is: what is the shape of the undecidable? In the same way that the horseshoe is the cradle of chaos, can we identify undecidability of a physical or dynamical system in a similar manner?
I am also developing an extension of Floer homology to the Poisson setting, using simple classes of Poisson manifolds such as
b-Poisson manifolds as a proof of concept, and exploring the classical Weinstein conjecture within this framework.I am currently revisiting Witten’s original point of view in the context of Poisson manifolds. My motivation stems from the search for periodic orbits in regularized problems in Celestial Mechanics (more information here).
Ph.D. thesis "On symplectic linearization of singular Lagrangian foliations" at UB (1999-2003)
Contact Information
Laboratory of Geometry and Dynamical Systems EPSEB-UPC
Av. Dr. Marañón, 44-50
08028 Barcelona, Spain
Office: Edifici P - 3rd floor, office 331 and SYMCREA office 327C
PA: Paula Martínez Felices: paula.martinez.felices@upc.edu
I am particularly interested in understanding connections between different areas such as Geometry, Dynamical Systems, Mathematical Physics and, more recently, Fluid Dynamics and Computer Science. 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.
I also extended 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. Check it out here: here.
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:here. 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).
See also this article in Quanta Magazine: Quanta Magazine
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 here.
Navier-Stokes, undecidability and Turing completeness:
In this work. , I have shown that the Navier–Stokes equations can simulate universal Turing machines. This means that fluid flows can encode undecidable problems, revealing a profound link between fluid dynamics, computation, and the limits of predictability.
New computational model? 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 have defined a new computational model. In our article "Topological Kleene Field Theories: A New Model of Computation” we introduce Topological Kleene Field Theories (TKFTs), a new computational framework that directly realizes any computable function through the flow of a vector field on a bordism. This provides a rigorous and original alternative to Turing machines: instead of encoding computation as an iterative process, TKFTs achieve it in a single dynamical pass. Our results establish TKFTs as a full model of computation equivalent to Turing machines, relate computational complexity to topological complexity, and open perspectives towards models that may surpass both classical and quantum computation.
You can read the article : here
You can check these ideas in a talk of mine at International Congress of Basic Sciences 2025 here:
Poisson topology: Poisson manifolds 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. This will allow to extend to Poisson manifolds the techniques from Symplectic manifolds.
The shape of the undecidable:
For dynamical systems, the horseshoe stands as a metaphor for chaos. A dynamical system that is Turing complete exhibits undecidable trajectories—that is, no algorithm can determine whether certain trajectories will enter a given open set within finite time. The natural question then arises: can we detect such behaviour? What is the shape of the undecidable? What plays the role of Smale’s horseshoe map in this context?
This is a joint project with Ángel González-Prieto and Daniel Peralta-Salas
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 the 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 acronym SYMCREA beautifully combines the words symmetries, symphony, symplectic, sympathetic, symbiosis, and ICREA. This is
We are located on the third floor of EPSEB at the UPC south campus.
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.
Cohomologies of Poisson manifolds, FPI-MDM grant, Starting date: September 2023. Schedule date of defense: November 2026.
Søren Dyhr (Msc. Aarhus)
Representation
theory in geometric fluid dynamics, Funding: UPC FPI grants, Currently INPHINIT La Caixa grants-MDM, Centers of excellence.Scheduled date of Defense: October 2026.
Current undergraduate and master students (4)
Elea Isasi Theus (Upenn-Chicago)
Barcodes and bubbles: the role of asphericity in Hamiltonian persistence modules. Master thesis cosupervised with Carles Casacuberta (UB), Master thesis to be defended in January 2026.
Isaac Ramos (ETHZ)
master student at ETHZ doing a Master semester paper under my supervision on "Billiard Dynamics, Geometry, and the Limits of Computation"(co-supervised by A. Figalli). Formerly our undergraduate student at UCM cosupervised by Ángel González Prieto and Daniel Peralta Salas, did an undergraduate thesis on Plugs in Symplectic Topology. Master thesis to be defended in 2027.
Leo Costa (CFIS-Univ. of Oxford)
Exoplanet detection, escape orbits, and singular contact structures (undergraduate thesis in cosupervision with Raymond Pierrehumbert at the University of Oxford).
Juan Brieva (CFIS-Univ. of Oxford)
Symplectic implosion and desingularization (undergraduate thesis cosupervision with Andrew Dancer at the University of Oxford).
Former Ph.D. students (9)
Joaquim Brugués (Msc. UPC)
Floer Homology for b-symplectic manifolds. Funding: FI-AGAUR, defended March 2024.
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:
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. Prize of the Mathematics Section of the Institute of Catalan Studies (IEC)
Global Hamiltonian Dynamics of singular symplectic manifolds, October 2, 2020.
postdoc under my supervision funded with my ICREA Academia project. Postdoc at ENS Lyon. After that Juan de la Cierva at BCAM Bilbao. Currently assistant professor at UPV in 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
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)
Josep Fontana-McNally
undergraduate thesis: Singular forms in Celestial mechanics and Fluid Dynamics
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 students)
Alexey Bolsinov
Baptiste Coquinot
Carlos Curras
Chiara Esposito
Pedro Frejlich
Angel Gonzalez-Prieto
Victor Guillemin
Mark Hamilton
Stanislav Krymskii
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 Bilateral AEI-DFG project AQUACELL: Celestial Mechanics, Hydrodynamics, and Turing Machines
AEI-DFG: Himmelsmechanik, Hydrodynamik und Turing-Maschinen 350000 euros. PI Spanish Team: Eva Miranda and Daniel Peralta-Salas, PI German team: Kai Cieliebak and Urs Frauenfelder. This is
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.
Additional services for the mathematical community
Research expert for several (inter)national research agencies including ERC, DFG, ANEP, AEI, ANR, FNRS, NWO.
Teaching
Current Teaching
This Fall semester I am Nachdiplom Lecturer at ETHZ Zurich. The other course is offered by Eugenia Malinnikova.
Here is a link to the website
Website of my Nachdiplom lecture
I take collegiality very seriously. I expect all my current and prospective mentees to uphold the highest standards of academic integrity and adhere to the UPC Code of Ethics.
Current Ph.D. students (2)
Pablo Nicolás (Msc. UPC)
Cohomologies of Poisson manifolds, FPI-MDM grant, Starting date: September 2023. Date of the defense, scheduled for November 2026.
Søren Dyhr (Msc. Aarhus)
Cosymplectic and contact geometries in Fluids and Physics. Funding: UPC FPI grants, Currently INPHINIT La Caixa grants-MDM since 2022, Centers of excellence. Date of the defense, scheduled for October 2026.
.
Current undergraduate and master students (4)
Elea Isasi Theus (Upenn-Chicago)
Barcodes and bubbles: the role of asphericity in Hamiltonian persistence modules. Master thesis cosupervised with Carles Casacuberta (UB), Master thesis to be defended in January 2026.
Isaac Ramos (ETHZ)
master student at ETHZ doing a Master semester paper under my supervision on "Billiard Dynamics, Geometry, and the Limits of Computation"(co-supervised by A. Figalli).Formerly our undergraduate student at UCM cosupervised by Ángel González Prieto and Daniel Peralta Salas, did an undergraduate thesis on Plugs in Symplectic Topology. Master thesis to be defended in 2026-2027.
Leo Costa (CFIS-Univ. of Oxford)
Exoplanet detection, escape orbits, and singular contact structures (undergraduate thesis cosupervised with Raymond Pierrehumbert at the University of Oxford).
Juan Brieva (CFIS-Univ. of Oxford)
Symplectic implosion and desingularization (undergraduate thesis cosupervised with Andrew Dancer at the University of Oxford).
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 (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
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)
Josep Fontana-McNally
undergraduate thesis: Singular forms in Celestial mechanics and Fluid Dynamics
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
Undecidability in mathematics and physics
Undecidability and P vs NP
From shifts to generalized shifts
The generalized singular Weinstein conjecture
From Morse to Floer homology à la Witten
Singular semitoric manifolds
The three-body problem
Equivariant cohomology and symplectic geometry
Symbolic dynamics and logical chaos
From Classical to Quantum computing
From singular to regular foliations
From Arnold conjecture to Weinstein conjecture and beyond
Circle actions on 4-dimensional singular symplectic manifolds
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)
A CASIO initiative for Women in Science. Honoured to join a very distinguished team of 23 scientist since Hipatia: Hipatia, Sophie Germain, Ada Lovelace, Marie Curie, Ángela Ruiz Robles, Irène Joliot-Curie, Cecilia Payne, Chien Shiung Wu, Hedy Lamarr, Katherine Johnson, Griselda Pascual, Mª Antònia Canals, Margarita Salas, Pilar Bayer, Donna Strickland, Eva Miranda, Maria Bras-Amorós, Maryam Mirzakhani, Ana Freire, Jess Wade, Sara García Alonso and Nerea Luís.
Here you can see a picture of me with the calculator.
I am the happy main character of the children's story by ApoloKIDS, Eva Miranda: La Matemática que descubrió formas invisibles. You can download it here
Outreach videos
You can see me here in Theories of Everything with Curt Jaimungal in January 2025:
You can view the slides here:
Slides of episode 1 with TOE
You can check another episode which was out this August here:
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.
I also extended 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. Check it out here: here.
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.
