Sofia Castro Gothen
Professora Catedrática/Full Professor
Agrupamento de Matemática
e Sistemas de Informação
Faculdade de Economia do
Porto
(Photo by António Almodovar)
last update: May 2018
My
postal address is:
Faculdade de
Economia do Porto
Rua Dr. Roberto
Frias
4200-464
Porto
Portugal.
The direct
phone number to my office, which is office
number 512, in the middle section of the EPG (new building) is +351 220 426
373.
For the
more technologically minded, my email address is: sdcastro(a)fep.up.pt
I have,
ever since I was hired, taught Mathematics courses for first
or second year undergraduates in Economics and/or Management.
At one
point in time, there were far too many students so Paulo Vasconcelos and I decided to
try something completely different in the ways of teaching. This received
the Prémio de Excelência
Pedagógica da Universidade do Porto 2015 (excellence in teaching prize). Parts of our
project have appeared in the Euronews program Learning World and were presented
at the TEDx University of
Porto in 2015. You can see a video of the presentation here.
Differential
and difference equations were part of Matemática para Economia, a course in the
MSc in Economics, which I shared with Paulo Vasconcelos. A change in the MSc programme in Economics ended all mathematics courses. From
2012, I teach Economic Modelling, which includes
the study of dynamic models and hence, difference and differential equations.
For a year
I taught an option course on Economic Dynamics (Elementos
de Economia Dinâmica), which first put me in touch
with applications of Mathematics to Economics. This happened in 1998 and was
done together with Álvaro Aguiar.
As a
consequence of my growing interest in Economics, I taught an option course on Game Theory for undergraduates
during the academic years of 2004/05 and 2005/06. I also taught Game Theory to PhD students in
Management, in a strategy-oriented way, from 2010/11 to 2012/13.
Since the
start of the PhD programme in Economics, the course
in Economia Matemática has been under my partial
(with Álvaro Aguiar or, more recently, João Correia da Silva or Pedro Gil) or
total responsability. This course was divided in two
modules, one on Topology and the other on Economic Dynamics. From 2013/14 the
Topology module was taken out by the scientific commission. There remains only
the Topology that is strictly necessary for the understanding of recursive
methods.
I have
taught in the MSc in Applied Mathematics of the Faculdade de Ciências da Universidade
do Porto courses in Bifurcation Theory and Singularity Theory.
I have
supervised a project, at a Maths Summer School for
High School pupils, on Game Theory and
Strategic Behaviour. You can see the
poster (in Portuguese) at the school’s website.
Research: whether useful or not remains to be seen; I thoroughly
enjoy it!
"... we should never say things like
'What's it used for?' Because all the big discoveries of really
useful things don't really come about because someone sits down and
thinks 'I want to discover something useful'. They occur because someone
discovers something interesting and it turns out to be
tremendously useful."
F. Duncan
M. Haldane, Physics Nobel 2016 (interview)
"So research that
exactly goes where you expected it to go is uninteresting on the whole. […]
Research is a trip, and you have to be attentive to all the things you see and
be able also to move away from the planned path."
Bengt Holmström, Economics Nobel 2016 (interview)
My PhD
thesis is on Mode Interactions with Spherical Symmetry (under the supervision
of Ian Stewart at the Maths Institute, University of
Warwick, UK). It concerns mostly a particular type of mode interaction where
the group action is trivial on one of the modes. This produces some
simplifications in the construction of normal forms but includes interesting
dynamics, such as secondary bifurcations to periodic solutions and the
existence of heteroclinic cycles.
Since
1998, Isabel Labouriau (Departamento
de Matemática Aplicada -
FCUP) and I have decided that Friday is Maths Day: we
ignore our bureaucratic chores and think solely about maths
(or, at least, try to). We started counting branches of steady-state solutions
to symmetric bifurcation problems - the counting of periodic solutions has been
a project ever since. We looked at how space phase can be reduced in symmetric
bifurcation problems and how this can help in the study of the dynamics (with
Stella Abreu from from Instituto Superior de Engenharia
do Porto). Inspired by a problem stated and partially studied by Mike Field in his Lectures on bifurcations, dynamics and
symmetry, we have studied symmetric vector fields with heteroclinic
networks, both in what concerns describing the networks and the dynamics along
it (with Manuela Aguiar from FEP). Manuela
and I have applied some related techniques to the study of switching along a
heteroclinic network in the Rock-Scissors-Paper game.
Dynamics
of game theory in general, and those of the Rock-Scissors-Paper game in
particular, are currently part of a project with Yuzuru Sato from Hokudai University, Japan, and Sebastian van Strien from Imperial
College London, UK.
On the subject of heteroclinic networks, since 2012, I have
been collaborating with Alexander Lohse, from the
University of Hamburg. We are establishing stability properties of cycles in
simple networks and how these properties affect the dynamics near the network.
My PhD
thesis touched also the subjects of Singularity Theory
and Convection. Several visits to, including a sabbatical year at, the Maths Department of the University of Aarhus, Denmark have
allowed me to explore the singular world a bit further, by working with Andrew du Plessis. I've also come
across mode interactions again in Boussinesq
convection via the collaboration with both Isabel Labouriau
and Olga Podvigina (from the Observatoire
de la Côte d’Azur and the International
Institute of Earthquake Prediction Theory and Mathematical Geophysics). We have studied
the existence of heteroclinic networks involving patterns in different modes, switching at
some of the nodes and eventually on the network itself.
Geographical
constraints (this is a school of Economics, after all) have taken me to the
world of Economics. Problems in oligopoly theory provide a wide range of
variations in game theory - static, dynamic, repeated, with or without complete
information - and have been a source of pleasure since 1999. This would not
have been possible without the assistance and collaboration of António Brandão.
During the
academic year of 2002/03, with António Brandão, I
have set up a project for a BIC-grant (Bolsa de Iniciação Científica) in Game
Theory applied to Oligopoly Theory. It is meant for students of the last couple
of years of an undergraduate degree. The money for this project comes from the FCT via the research centre I belong to: CMUP.
Symmetry
in economics came to my attention through a paper by Bala
called A pitchfork bifurcation in the
tatonnement process, Economic Theory, where it is claimed that this
bifurcation is robust under suitable
perturbations of the economy. As people familiar with symmetric bifurcation
have already realized, suitable
means preserving the symmetry of the
problem. If you perturb the economy while breaking the initial symmetry you
observe hysteresis in the tatonnement
process, that is a delay in response to a same input. This also means that the
economy is sensitive not only to variation of parameters but also to the
direction of this variation. This study was part of a joint project with Carlos
Hervés-Beloso from the University of Vigo. It became joint work with João Correia da Silva from FEP. A
presentation of part of this can be found on the web page of the Newton Institute.
More
recently, I’ve become interested in New Economic Geography (NEG). Together with
João Correia da Silva and Pascal Mossay (University of Reading, UK). We have studied
all possible equilibria in a core-periphery model with 3 regions. In 1991, P.
Krugman [Increasing returns and economics
geography, Journal of Political Economy 99(3)] showed that when there are
two identical possible locations for industry then whether all workers
concentrate in one of the two locations or whether they are equally distributed
by the two depends on parameters such as transportation costs, share of
spending in manufactures and taste for diversity. A dynamic model of population
movements among three possible identical locations was set up not long
afterwards but so far, only numerical results exist concerning possible
equilibrium configurations. The fact that the regions are identical introduces
symmetry in the problem and that provides my main interest in it. A fair amount
can be said by viewing the regions as cells, identically coupled. Another open
issue in the core-periphery model is that of the existence of more than two
sectors in the economy. Together with João Correia-da-Silva and Vasco Leite (a former PhD student of ours) we have constructed
and studied a model with three sectors. João Correia-da-Silva and I are now
trying to generalize these issues to a finite number of regions with a PhD
student, José Gaspar. We also looked at the impact of using a different utility
function, with an MSc student, Alexandre Freitas. Together with Paulo Vasconcelos, from FEP, and an
MSc student, Liliana Garrido-da-Silva, we looked at the discrete version of a
two-regions core-periphery model. The existing model presents chaotic behaviour which we showed can be avoided by careful
modelling.
Collaboration
with Sami Dakhlia (ESCE International Business School, Paris)
has provided another connection to the world of singularities, without symmetry
this time, through the problem of general equilibrium in economics. The problem
is to complete
the description of the set of equilibrium price vectors for an exchange economy
with a finite number of goods and consumers. We also look at the relation
between preferences and utility functions, together with Peter Gothen (Departamento de Matemática -
FCUP).
I have
also decided to look at problems in macroeconomics, endogenous growth in particular. This was a challenge placed by a
student of Economic Modelling who had a preference for
macroeconomics. Together with Pedro Gil, from FEP, I
supervised André Almeida's MSc dissertation on Multidimensional Dynamics of
Endogenous Growth.
Publications: the links provided
are to previous preprint versions; should you want a reprint, please email me.
Published papers in
journals:
1.
S. B. S. D. Castro, Mode interactions with spherical symmetry, International Journal of
Bifurcation and Chaos, Vol. 4, No. 4,885--904 (1994)
2.
S. B. S. D. Castro, Mode interactions with symmetry,
Dynamics and Stability of Systems, Vol. 10, No. 1, 13--31 (1995)
3.
S. B. S. D. Castro, The disappearance of the limit cycle in a
mode interaction problem with Z2-symmetry, Nonlinearity, Vol.
10, No. 2, 425--432 (1997)
4.
S. B. S. D. Castro and António Brandão, Existence of a Markov Perfect Equilibrium in a third
market model, Economics Letters, 66, 297--301 (2000)
5.
M.A.D. Aguiar, S.B.S.D. Castro and
I.S. Labouriau, Path
formulation for a modal family, Portugaliae
Mathematica 58 (4), 407—424 (2001)
6.
M.A.D. Aguiar, S.B.S.D. Castro and
I.S. Labouriau, Dynamics
near a heteroclinic network, Nonlinearity 18(1), 391—414 (2005) (link
to working paper)
7.
M.A.D. Aguiar, S.B.S.D. Castro and
I.S. Labouriau, Simple
vector fields with complex behaviour,
International Journal of Bifurcation and Chaos in Applied Sciences and
Engineering 16 (2), 369—381 (2006) (link
to working paper)
8.
Brandão
and S.B.S.D. Castro, State-owned
enterprises as indirect instruments of entry regulation, Journal of
Economics, vol. 92, no. 3, 263 — 274 (2007) (link
to working paper)
9.
S.B.S.D. Castro, Symmetry and Bifurcation of Periodic Solutions in Neumann Boundary
Value Problems, Portugaliae Mathematica, Vol. 65,
Fasc. 3, 373 – 385 (2008) (link to
working paper)
10. V.
Leite, S. B. S. D. Castro and J. Correia-da-Silva, The core periphery model with asymmetric
inter-regional and intra-regional trade costs, Portuguese Economic Journal,
Vol.8, nº 1, pp. 37-- 44,
2009 (link
to working paper)
11. S.
B. S. D. Castro, S. Dakhlia and P. B. Gothen, Direct
Perturbations of Aggregate Excess Demand, Journal of Mathematical
Economics, vol. 46, 562--571 (2010) (link to
online version)
12. M.
A. D. Aguiar and S. B. S. D. Castro, Chaotic
switching in a two-person game, Physica D:
Nonlinear Phenomena, vol. 239, issue 16, 1598-1609 (2010) (link
to online version)
13. S.
B. S. D. Castro, I. S. Labouriau and O. Podvigina, A heteroclinic network in mode interaction
with symmetry, Dynamical Systems: an international journal, vol. 25 (3),
359--396 (2010) (link
to working paper)
14. S.
B. S. D. Castro, J.
Correia-da-Silva and P. Mossay, The core-periphery model with three regions and more, Papers in Regional Science, vol. 91 (2),
401--419 (2012) (link
to online version)
15. B.
Alarcón, S. B. S. D. Castro
and I. S. Labouriau, The Discrete Markus-Yamabe Conjecture for
symmetric planar polynomial maps, Indagationes Mathematicae, vol. 23, 603-608 (2012) (link to
working paper)
16. B.
Alarcón, S. B. S. D. Castro and I. S. Labouriau, A local
but not global attractor for a Zn-symmetric
map, Proceedings of the 2nd Workshop on Generic Geometry and Applications,
Journal of Singularities, vol. 6, 1-14 (2012)
17. V.
Leite, S. B. S. D. Castro and J.
Correia-da-Silva, A third sector in the
core-periphery model: nontradable goods, Annals
of Regional Science, vol. 5 (1), 71-108 (2013) (link to online first
version)
18. B.
Alarcón, S. B. S. D. Castro and I. S. Labouriau, Global
Dynamics for Symmetric Planar Maps, Discrete and Continuous Dynamical
Systems - series A,Vol. 33 (6), 2241
- 2251 (2013) (link
to online version)
19. S.
B. S. D. Castro, S. Dakhlia and P. B. Gothen, From
Singularity Theory to Finiteness of Walrasian
Equilibria, Mathematical Social Sciences, Vol. 66, 169-175 (2013) (SSRN
1648426)
20. S.
B. S. D. Castro and A. Lohse, Stability
in simple heteroclinic networks in R4, Dynamical Systems: an
international journal, Vol. 29 (4), 451 – 481 (2014)
(arXiv:1401.3993v2)
21. L.
Garrido-da-Silva, S. B. S. D. Castro and P. B.
Vasconcelos, Discrete dynamics for the
core-periphery model, Spatial Economic Analysis, Vol. 10 (1), 36–51 (2015)
(link to
online version)
22. J.
Oliveira, S.B.S.D. Castro and I.S. Labouriau, Hexagonal Projected Symmetries, Acta Crystallographica Section A: Foundations and Advances, Vol.
71 (5), 549–558 (2015)
(arXiv:1502.00461v2)
23. S.B.S.D.
Castro and A. Lohse, Switching in
heteroclinic networks, SIAM Journal on Applied Dynamical Systems (SIADS),
Vol. 15 (2), 1085 -- 1103 (2016) (arXiv:1510.00178v1)
24.
S.B.S.D. Castro and A. Lohse, Construction of heteroclinic networks in R4,
Nonlinearity, Vol. 29, 3677--3695 (2016) (arXiv:1504.05459v3)
25. P.M. Gil, A. Almeida and S.B.S.D. Castro, Rich Transitional Dynamics in a Non-Scale Fully Endogenous Growth Model, Oxford Economics Papers, Vol. 69 (3), 678--706 (2017) (online first)
26.
B. Alarcón, S.B.S.D.
Castro and I.S. Labouriau, Global saddles for planar maps, Journal of Dynamics and
Differential Equations, Vol. 30 (2), 601--612 (2018) (arXiv:1605.08000v2)
27. J.M. Gaspar, S.B.S.D. Castro and J. Correia-da-Silva, Agglomeration patterns in a multi-regional
economy without income effects, Economic Theory, Vol. 66 (4), 863--899
(2018) (link
to online version)
28. S.B.S.D. Castro, I.S. Labouriau and J. Oliveira, Projections of patterns and mode interactions, Dynamical Systems: an international journal, Vol. 33 (4), 547--564 (2018) (arXiv:1703.10635v3)
29.
S.B.S.D. Castro, Learning by replicator and best-response: the importance of being indifferent,
Journal of Evolutionary Economics, Vol. 28 (4), 985--999 (2018) (arXiv:1602.00906v6)
30. L.
Garrido-da-Silva and S. B. S. D. Castro, Stability
of quasi-simple heteroclinic cycles, Dynamical Systems: an
international journal, Vol.34 (1), 14--39 (2019) (arXiv:1606.02592v6)
31. O. Podvigina, S.B.S.D.
Castro and I.S. Labouriau, Stability of a
heteroclinic network and its cycles: a case study from Boussinesq
convection, Dynamical Systems: an
international journal, Vol.34 (1), 157--193 (2019) (arXiv:1712.04270v3)
32. P. Ashwin, S.B.S.D. Castro
and A. Lohse, Almost complete and equable
heteroclinic networks, Journal of Nonlinear Science, Vol. 30 (1), 1 – 22 (2020) (arXiv:
1811.03350v2)
33. L. Garrido-da-Silva and S.B.S.D. Castro, Cyclic dominance in a two-person
rock-scissors-paper game, International Journal of Game Theory, Vol. 40,
885—912 (2020) (arXiv:1607.08748)
34. O. Podvigina,
S.B.S.D. Castro and I.S. Labouriau, Asymptotic stability of robust heteroclinic
networks, Nonlinearity, Vol. 33, 1757—1788 (2020) (arXiv:1905.06419)
35. J.M. Gaspar, S.B.S.D. Castro and J. Correia-da-Silva, The Footloose Entrepreneur model with a
finite number of equidistant regions, International Journal of Economic
Theory, Vol. 16 (4), 420 — 446
(2020) (link to online version)
37. L. Garrido-da-Silva, S.B.S.D. Castro and J. Correia-da-Silva, Location of hous- ing and industry around city centre
amenities, Regional Science and Urban Economics (2022) (link
to online version)
38. S.B.S.D. Castro, A. Ferreira,
L. Garrido-da-Silva and I.S. Labouriau, Stability
of cycles in a game of Rock-Scissors-Paper-Lizard-Spock, SIAM Journal on
Dynamical Systems (SIADS), Vol. 21 (4), 2393 — 2431 (2022) (link to online
version)
39. S.B.S.D. Castro and A. Lohse, A
hybrid heteroclinic cycle, Examples and Counterexamples, Vol. 2, 100071
(2022) (link
to online version)
40. S.B.S.D. Castro and A. Lohse, Arbitrarily
large heteroclinic networks in fixed low-dimensional state space, Chaos: An
Interdisciplinary Journal of Nonlinear Science, Vol. 33, 083156 (2023) (link
to online version)
41. S.B.S.D. Castro and L. Garrido-da-Silva, Finite
switching near heteroclinic networks, Nonlinearity, Vol. 36 (12) 6239
(2023) (link to online
version)
42. S.B.S.D. Castro, A. Ferreira
and I.S. Labouriau, Stability of cycles and survival
in a Jungle Game with four species, Dynamical Systems: an International
Journal (accepted for publication)
Some submitted papers or about to be submitted, as well
as non-final versions of some published papers, may be found in the arXiv.
Book chapters:
1. S.B.S.D.
Castro, Mixed-mode solutions in mode
interaction problems with symmetry, P. Chossat
(ed.), Dynamics, Bifurcation and Symmetry,
69 – 77, Kluwer Academic Publishers (1994)
2. S.B.S.D.
Castro and I.S. Labouriau, Counting persistent pitchforks, J.W. Bruce
and F. Tari (Eds.), Vth Workshop on Real
and Complex Singularities, 215 – 222, CRC
Research Notes in Mathematics 412 (2000)
3. M.A.D.
Aguiar, S.B.S.D. Castro and I.S. Labouriau, Switching along a network, F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede, S. Verduyn Lunel (eds.), Equadiff 2003 -- Proceedings of the International
Conference on Differential Equations, 458--463, World Scientific (2005)
4. S.B.S.D.
Castro and A.A. du Plessis, Intrinsic complete transversal
and the recognition of equivariant bifurcations, F. Dumortier,
H. Broer, J. Mawhin, A. Vanderbauwhede, S. Verduyn Lunel (eds.), Equadiff 2003 –
Proceedings of the International Conference on Differential Equations, 449 – 451,
World Scientific (2005)
5.
B. Alarcón,
S. B. S. D. Castro and I. S. Labouriau, Discrete Symmetric Planar Dynamics,
International Conference and Advanced School Mathematics
of Planet Earth, CIM Mathematical Sciences, Vol. 1, 17 – 30 (2015) (arXiv:1405.6877)
6.
S. B. S. D. Castro, A mathematician’s landing on the world
of economics, A. Teixeira, A.P. Delgado, L. Carvalho, M.I. Mota, M.M.
Castroe Silva (eds.) Estudos de Homenagem a José da Silva Costa, 324 –
334, U.Porto Press (2023)
Working papers (unpublished):
Together
with J.Buescu, A.P.S.Dias
and I.S.Labouriau, I have edited Bifurcations, symmetry and patterns. A conference in honour of
M. Golubitsky and I. Stewart, Birkhauser Verlag, Basel, Switzerland (2003)
"Being a
graduate student is like becoming all of the Seven Dwarves. In the beginning
you're DOPEY and BASHFUL. In the middle, you are usually sick (SNEEZY), tired
(SLEEPY), and irritable (GRUMPY). But at the end, they call you DOC, and then
you're HAPPY." Azuma (1997)
Here is a list of my students and their thesis (or
theme, for those still in progress).
MSc:
Paula Gaio, Estudo de um
Problema de Bifurcação com Simetria Esférica, Mestrado em Matemática
Aplicada da Faculdade de Ciências da Universidade do Porto, 1996.
Teresa Ferreira, Formulação
por Caminhos no Estudo de Problemas de Bifurcação, with
Isabel Labouriau, Mestrado em Matemática Aplicada da
Faculdade de Ciências da Universidade do Porto, 1996.
Miguel Mendes, Anti-simetrias, Mestrado em Matemática Aplicada da Faculdade
de Ciências da Universidade do Porto, 1999.
Maria da Conceição Leite, Bifurcações
de um sistema químico com controlador, with
Isabel Labouriau, Mestrado em Matemática Aplicada da
Faculdade de Ciências da Universidade do Porto, 2001.
Maria João Albuquerque, Métodos de
redução da dimensão do espaço de fase controlador, with
Isabel Labouriau, Mestrado em Matemática Aplicada da
Faculdade de Ciências da Universidade do Porto, 2005.
José Gaspar, The Footloose Entrepreneur
Model with Three Regions, with João Correia da Silva, Mestrado em Economia da
Faculdade de Economia do Porto, 2012
Ana
Ferreira, Bifurcations in the footloose enterpreneur model, with João Correia da
Silva, Mestrado em Economia da Faculdade de Economia do Porto, 2013
Liliana
Garrido-da-Silva, Modelo
Centro-Periferia: contributos em tempo discreto, with
Paulo B. Vasconcelos, Mestrado em Economia da Faculdade de Economia do Porto,
2013
André
Almeida, Dinâmica
multidimensional em crescimento endógeno, with
Pedro Gil, Mestrado em Economia da Faculdade de Economia do Porto, 2014
Alexandre
Freitas, The Core-Periphery model with a CES utility function, with João Correia da Silva, Mestrado em Economia da
Faculdade de Economia do Porto, 2015
Daniela
Rodrigues, Modelo
Centro-Periferia em tempo discreto: padrões de aglomeração entre três regiões,
Mestrado em Economia da Faculdade de Economia do Porto, 2016
Selma
Pinto, Modelo Oligopolístico Heterogéneo em tempo discreto, Mestrado
em Economia da Faculdade de Economia do Porto, 2016
Eduardo
Silva, Caos em modelos
de crescimento económico: algumas observações, Mestrado em Economia da
Faculdade de Economia do Porto, 2016
Catarina
Almeida, Equivalence of economic
models under different time representations, with Pedro Gil, Mestrado em Economia da Faculdade de
Economia do Porto, 2018
Jorge
Saraiva,
Voluntariado -- Incentivos e Reputação Social, with João Correia da Silva, Mestrado em Economia da
Faculdade de Economia do Porto, 2018
Joana
Ribeiro, A economia
quantitativa espacial, com duas e três regiões, no contexto português, with João Correia da Silva, Mestrado em Economia da
Faculdade de Economia do Porto, 2019
PhD:
Stella Abreu, Redução do
espaço de fase em campos de vectores simétricos, with Isabel Labouriau,
Doutoramento em Matemática da Faculdade de Ciências da Universidade do Porto,
2000.
Manuela Aguiar, Vector Fields with Heteroclinic
Networks, with Isabel Labouriau,
Doutoramento em Matemática da Faculdade de Ciências da Universidade do Porto,
2003.
António Marques, Three Essays on the
Interaction Among Exit, Sunk Costs and
Entry Decision by the Firms,
with António Brandão, Doutoramento em Economia,
Faculdade de Economia do Porto, Universidade do Porto, 2005.
Vasco Leite, Three essays on the
Core-Periphery model, with João Correia-da-Silva, Doutoramento em Economia,
Faculdade de Economia do Porto, Universidade do Porto, 2010.
Juliane Oliveira, Bifurcation of projected patterns,
with Isabel Labouriau,
Doutoramento em Matemática Inter-Universitário
Coimbra-Porto, Faculdade de Ciências da Universidade do Porto, 2017.
José Gaspar, New Economic Geography: perspectives, multiple regions and individual heterogeneity, with João
Correia da Silva, Doutoramento em Economia da Faculdade de Economia da
Universidade do Porto, 2017. This
thesis was awarded the Prémio António Simões Lopes 2018, for the best PhD thesis in Economics in
Portugal.
Liliana Garrido-da-Silva, Heteroclinic Dynamics
in Game Theory, Doutoramento em Matemática
Aplicada, Universidade do Porto, Minho e Aveiro, 2018.
Jorge Saraiva, Essays on Economic
Geography, with João Correia da Silva, Doutoramento em Economia da
Faculdade de Economia da Universidade do Porto, 2023.
Post-Doc:
Stella Abreu, Simetrias em
Problemas de Bifurcação e Redes de Células Acopladas, with
Ana Paula Dias and Isabel Labouriau,
2004/05.
Begoña Alarcón, O problema
discreto de Markus-Yamabe equivariante,
with Isabel Labouriau,
2010/11.
Alexander Lohse, Dynamics and
stability in heteroclinic networks, April-
October 2015.
PhD (in progress):
Ana
Margarida Ferreira, with Isabel Labouriau, MAP-PDMA (Programa
de Doutoramento em Matemática Aplicada, Minho-Aveiro-Porto).
In 2000, I
co-organized a conference and summer school on Bifurcations, Symmetry and
Patterns which was part of a thematic term on Dynamics, Bifurcation and Biology.
The conference was in honour of Marty Golubitsky and Ian Stewart.
In 2005, I
co-organized the EARIE2005 conference in FEP.
In 2014, I
co-organized the Workshop on Symmetry and Dynamics: on
the occasion of Isabel Labouriau’s 60th
In 2015, I co-organized two
special sessions, one in the annual meeting of the Portuguese Mathematical
Society (Heteroclinic Cycles and Networks: Stability and
Dynamics) and the
other in the annual meeting of the German Mathematical Society (Heteroclinic Dynamics:
theory and applications).
In 2016, I co-organized a
one-day Workshop on Heteroclinic Phenomena: the role of
symmetry.