Sofia Castro Gothen
Professora Catedrática/Full Professor
Agrupamento de Matemática e Sistemas de Informação
(Photo by António Almodovar)
last update: May 2018
My postal address is:
Faculdade de Economia do Porto
Rua Dr. Roberto Frias
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.
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)
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)
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, Flexible 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 (to appear) (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 (to appear) (online first)
28. S.B.S.D. Castro, I.S. Labouriau and J. Oliveira, Projections of patterns and mode interactions, Dynamical Systems: an international journal (to appear) (arXiv:1703.10635v3)
29. S.B.S.D. Castro, Learning by replicator and best-response: the importance of being indifferent, Journal of Evolutionary Economics (to appear) (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 (to appear) (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 (to appear) (arXiv:1712.04270v3)
Some submitted papers or about to be submitted, as well as non-final versions of some published papers, may be found in the arXiv.
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 (arXiv:1405.6877)
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).
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
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.
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):
Liliana Garrido-da-Silva, Dynamics in Game Theory, Doutoramento em Matemática Aplicada, Universidade do 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.