Apresentei uma reclamação à FCT. Nesse sentido, este blogue fica desactivado temporariamente.

Adenda a 21/04/2017: recebi hoje a notificação por parte da FCT que a minha reclamação foi indeferida. Nesse sentido, passa a estar disponível o conteúdo da candidatura e, em particular, o projecto de investigação.

A. General description of the application

• A.1. Position levelStarting grant
• A.2. Title of the projectNew trends in quantization
• A.3. ORCID0000-0003-0736-8302
• A.4. GenderMale
• A.5. Current institutionUniversidade de Lisboa, CAMGSD
• A.6. Country of current institutionPortugal
• A.7. NationalityPortugal
• A.8. Main scientific areaExact Sciences
• A.8. Secondary scientific areaMathematics
  A.9. (omitido - conflito de interesses)

• A.10. Date of PhD completion2011-01-28
  Files uploaded:

  Certificado de Doutoramento

• A.10.1. Justification for deviations, Maternity
  Files uploaded:No files uploaded.
• A.10.2. Justification for deviations, Paternity
  Files uploaded:No files uploaded.
• A.10.3. Justification for deviations, Long-term illness
  Files uploaded:No files uploaded.
• A.11. KeywordsMeasure theory;Graphs;Generalized functions;Feynman Integral;Hopf Algebras
• A.12. Eligible for a exploratory research project?Yes
• A.12.1. Budget20000.00 €
• A.13. Degree of disability-
  Files uploaded:No files uploaded.

B. Synopsis of the application

• B.1. Major contributions / highlightsJ. N. Esteves, M. Hirsch, J. C. Romão, W. Porod, J. W. F. Valle and A. Villavona del Moral
  Flavour violation at the LHC: type I versus type II seesaw in minimal supergravity,
  JHEP 05 (2009) 003, (made evident how constrained are susy parameters in type I seesw to give correct data for Dark Matter and neutrino masses. Collaboration in the theoretical framework; numerical computations; collaboration in analysing data)
  
  J. N. Esteves, S. Kaneko, M. Hirsch, J. C. Romão and W. Porod,
  Dark Matter in minimal supergravity with type- II seesaw mechanism,
  Phys Rev. D 80, 095003 (2009), (made evident how less constrained are susy parameters in type II seesaw to give correct data for Dark Matter and neutrino masses; collaboration in the theoretical framework; numerical computations; collaboration in analysing data)
  
  J. N. Esteves, F. R. Joaquim, A. S. Joshipura, J. C. Romão, M. A. Tórtola and J. W. F. Valle,
  A4-based neutrino masses with Majoron decaying dark matter, (2010) Phys.Rev. D 82 073008 (2010), (alternative description of Dark Matter with a seesaw model based on a discrete symmetry group; computation of the scalar potential; numerical computations; collaboration in analysing data)
  
  J. N. Esteves, M. Hirsch, W. Porod, J. C. Romão, F. Staub and A. Vicente, LHC and lepton flavour violation phenomenology of a left-right extension of the MSSM, JHEP 1012 (2010) 077
  (numerical computations for a new sugra setup for seesaw. collaboration in the theoretical framework; numerical computations; collaboration in analysing data)
  
  J. N. Esteves, M. Hirsch, W. Porod, J. C. Romão, F. Staub and A. Vicente, Dark matter and LHC phenomenology in a left-right Supersymmetric model, JHEP01(2012)095,
  (numerical computations for a new sugra setup for seesaw. collaboration in the theoretical framework; numerical computations; collaboration in analysing data)
  
  J. N Esteves, J. M. Mourão, and J. P. Nunes, Quantization in singular real polarizations: Kähler regularization, Maslov correction and pairings, Journal of Physics A: Mathematical and Theoretical 48 (2015), no. 22,. (Shows how to get singular real polarizations as the limit of regular complex ones; computes the Maslov phases for the harmonic oscillator in terms of the limit of BKS pairing between two regular complex polarizations; collaboration in the theoretical framework; explicit computation of the BKS pairing and Maslov phases for the harmonic oscillator)
  
  J. N. Esteves, Hopf Algebras and Topological Recursion, arXiv:1503.02993 [math-ph] http://arxiv.org/abs/1503.02993. Accepted for publication in JPhys A as stated on email of 14th Sept 2015 from the Journal (obtains explicit solutions for the Eynard-Orantin topological recursion formula in terms of the Hopf-Algebra of Loday-Ronco and a new set of graphs with loops obtained from this Hopf-Algebra; makes contact with the framework of topological quantum field theory)
• B.2. Synopsis of the CVJoão N. Esteves, graduated in Physics in 1993 at the FCUL, Lisbon, completed the curriculum of a MSc in Mathematics in 2000 after some experience as a High School teacher in Physics and Chemistry. In this curriculum he strengthened his knowledge in Pure Mathematics in disciplines like Differential Geometry, Algebraic Topology, Ordinary Differential Equations and others. Through his advisor's initiative and as a member of the European Network "Geometric Analysis" (http://www.dipmat.univpm.it/rtn/) he had the chance to stay at the Lab. Emile Picard, Univ. Paul Sabatier, Toulouse, France, in 2001 and 2002 studying the Hopf-Connes-Kreimer Algebra of Renormalization of QFT and working on singular spaces with Prof. Andr\'e Legrand and also in 2003 at the Math Dep. of the Universita degli Studi di Ancona, Italy, working with Prof. Nicolae Teleman. Due to personal and financial motives, he interrupted this scientific track and returned to Portugal, where he started a MSc in Particle Physics at IST, Lisbon (lecturing simultaneously at a High School until 2005), which he concluded in 2007 with a thesis on neutrino masses and supersymmetric models with R-parity violation. Meanwhile, he started in 2006 a PhD in Physics with a grant from FCT at the CFTP research center of the Physics Dept. of IST, under the supervision of Prof. Jorge C. Romão, which he concluded in Jan. 2011 with the thesis "Supersymmetric Models and Neutrino Masses". In this thesis some supersymmetric extensions of the Standard Model are addressed in order to answer to the open problem of how neutrinos get their tiny masses and how this is related to Lepton Flavour violation (LFV) process which could be observed in current or near future experiments, like MEG, Xenia or at the LHC. For this thesis he acquired a strong background in several aspects of Quantum Field Theory that he intends to apply to his present research. Moreover during this period and afterwards six papers were published in international peer reviewed journals (3 in JHEP and 3 in PRD) that emphasize the relation between neutrino masses, supersymmetry LFV and Baryogenesis through Leptogenesis. This last process is deeply connected with topological configurations in gauge field theory and was explored in his Ph.D. thesis. In the final stage of his Ph.D. he stayed at CERN for 3 months. Since Nov. 2011 he has a post-doc position at CAMGSD, after Fev. 2012 with the FCT grant SFRH/BPD/77123/2011, working on Geom. Quant. with José Cidade Mourão and João Pimentel Nunes with whom he published a paper on Quantization with mixed polarizations in Journal of Physics A. During this period he also worked on the topological recursion formula of Eynard and Orantin which has its origins in Matrix Models and applications in various fields of mathematics and physics and submitted to Journal of Physics A the paper "Topological Recursion and Hopf Algebras" that has been accepted for publication and is under final revision.
• B.3. Synopsis of the research project and career development planQuantization is one of the major achievements of Mathematical Physics (Math. Phys) of the 20th century. Nevertheless a comprehensive and fully abstract description of how it should be implemented is still lacking. Gravitation is an example of the difficulties that still remain in understanding completely the applicability of Quantum Mechanics (QM) to a sensible physical model and in deciding if this quantization process can in fact be carried out. This is a multidisciplinary challenge that involves Complex, Symplectic, Differential and Algebraic Geometry or Functional Analysis and Operator Algebras on the mathematical side, and QM, Quantum Field Theory (QFT), Topological Quantum Field Theory (TQFT) and Superstring Theory on the physical side, and can only implemented by the collaborative work of several people in interaction as is the case at the CAMGSD Research Centre. This research project addresses the subject of quantization from an algebraic perspective on the one hand, and in the sense of measure theory on the other hand. In the first case the purpose is to complete and deepen the research started in [17] inspired by matrix models [7,9], TQFT [2], cobordism theory and the so-called Lego-Teichmuller game [19] and in the second case to investigate in detail the distributional nature of the fields in QFT [20] that seem to impose less restrictive conditions than in QM on the nature of the path or Feynman integral. In particular, in the first case the candidate aims to clarify the nature of the map presented in [17] that is conjectured to be a representation map from the abstract space of correlation functions to the Hopf Algebra of Loday-Ronco or to the algebra of graphs with loops explicitly obtained from it in [17] and to clarify the connection of these algebras of graphs with TQFT along the lines of [2] and [19]. In the second case the purpose is to investigate if the distributional nature of QFT allows to soften the conditions that impose the non-existence of a translation invariant measure on the space of continuous paths in the formulation of QM with the path-integral [18] clarifying in this way if the path integral in QFT is less problematic than in QM.
  Both of these goals require deepening the candidate's knowledge of (and making contact with the latest developments in) Symplectic and Complex Geometry, Functional Analysis and Operator Theory and their connections with Math. Phys.
  This grant will allow financial independence in order to visit top research centers and to participate in major conferences and workshops; it will permit to build a solid bibliographical resource with top publications by experts in several areas. This in turn will contribute deeply to a solid supervision of graduate students and to the quality of lecturing that are two major objectives for the career besides the production of interesting research work to be published in top peer-reviewed journals.

C. Full description of the application

C.1. Research project

• C.1.1. BackgroundRelated to the first part of this project, that is, the relation of Hopf Algebras with Eynard-Orantin recursion formula, one can say that the use of graphs, in particular of trees, binary trees and planar binary trees, in mathematical physics has a long tradition. The canonical examples are perhaps Feynman diagrams but the connection with Hopf Algebras of trees started with the works of Connes and Kreimer [6, 7, 8] that describe the combinatorics of the procedure of extracting sub-divergences in Quantum Field Theory known as the BPHZ renormalization procedure [5]. Another approach to the use of graphs in QFT and in particular in QED, considering binary trees, planar or not, was followed by Brouder and Frabetti [3, 4, 12]. Later it was understood that these two approaches are very similar and in some cases equivalent and are related to quasisymmetric and noncommutative quasi-symmetric functions, see for instance [1, 10-13].In the paper [17] we showed how the Hopf Algebra of planar binary trees of Loday and Ronco [15] can be seen as a representation of the vector space generated by correlation functions that obey the Eynard-Orantin recursion formula. These correlation functions are graded by the Euler characteristic and we can consider for each degree the vector space over Q generated by them and then take the direct sum of these vector spaces for all degrees. First we considered the Hopf Algebra of Loday-Ronco [15] whose underlying vector space space $\sum_ n k[Y^n]$ is the direct sum of the vector spaces over a field k of planar binary trees of order n, that is with n vertices and n + 1 leaves, as a representation of the space of genus g = 0 correlation function $W^0_k(p, p_1,\dots, p_{k-1})$ of Euler characteristic $\chi = 2 - 2g - k$ (do not confuse with the field k) equal to $-n$. Here the Euler characteristic is the one of Riemann or topological surfaces of genus $g$ and $k$ punctures or borders to which the correlation functions $W^g_k(p,p_1,\dots, p_{k-1})$ are usually related in some concrete problems. For g = 0 we label the root with p and the n + 1 leaves with the $p_1,\dots, p_{n+1}$ variables. Then by connecting the nearest neighbors leaves with a single edge and reducing the number of pairs of labels in the same way as increasing the genus we obtain graphs with loops that we see as a representation of higher genus correlation functions with the same Euler characteristic. We showed that both types of graphs, the planar binary trees and the graphs with loops obtained from them, are explicit solutions of the Eynard-Orantin recursion formula. On the other hand in topological field theories as described for instance in [2] cobordism gives a natural product for topological surfaces and it is well known that 2-dimensional topological field theory are related with Frobenius Algebras. It would be interesting to see how this two different approaches fit together. In particular the conjectured representation map $\psi$ obtained in [17] could give at least a ring structure to the vector space of correlation functions if considered in the reversed direction. Then topological quantum field theories would be a particular case of this setup when correlation function $W^g_k(p,p_1,...\p_{k-1})$ are identified with topological surfaces of genus g, with k punctures and with Euler characteristic $\chi=2-2g-k$.
  Related to the second part of this project we would like to investigate if the conditions for a well defined Path Integral are less stringent in Quantum Field Theory than in Quantum Mechanics given the fact that distributions or generalized functions are intrinsic to Quantum Field Theory, as it is shown in [20], in particular in Chap. 4. More specifically, since distributions have always a well-defined derivative (obtained by duality or integration by parts on the space of test functions) contrary to the case of continuous paths in $R^n$ that are the usual domain of the path integral in Quantum Mechanics [18] there seems a priori to be no obstacle in defining the action on the space of paths of fields, that are operator valued distributions, nor in applying a generalization of the stationary phase method in order to obtain the semiclassical approximation. Moreover, Lebesgue integrability does not seem to be a mandatory condition when performing certain computations as it happens for instance in the Fourier transform of the scattering matrix form the momenta to the coordinates [20, Chap. 4]. In this way we would like to investigate if the non-existent translation invariant Lebesgue measure on the space of continuous paths in Quantum Mechanics [18] can be replaced in Quantum Field Theory by a measure in the distributional sense. Finally, noting that generalized functions are limits of functions in the $L^2$ Hilbert space, then if this question gets a positive answer it could shed light to the nature of the Path Integral in Quantum Mechanics.
• C.1.2. Research plan and methodsIn [17] we considered a map $\psi$, which is conjectured to be a representation, from the vector space of correlation functions of genus g to the vector space of graphs with loops such that, in particular, to a correlation function $W^0_{n+2}(p, p_1, . . . , p_{n+1})$ of Euler characteristic $\chi = -n$ would correspond the trees of order n. In fact we defined the representation of $W_0^{n+2}(p, p_1,\dots, p_{n+1})$ as the sum of all trees of order $n$. It is not clear if this map gives a true representation in the strict mathematical sense. It is linear by definition and it is obvious that it is surjective, since we can associate some instance of a correlation function $W^0_{n+2}(p, p_1, . . . , p_{n+1})$ to any tree of order n. Whether it is injective and a homomorphism is a more delicate issue because even if one considers $W^0_{n+2}$ as being the sum of all instances of correlation functions of Euler characteristic -n each represented by a tree $t \in Y^n$ in the same way as in Particle Physics, where different Green functions contribute to the same scattering amplitude, it is not evident that the space of correlation functions has a product with an identity that would correspond to the trivial tree |. Note that this would give at least a ring structure and in the case of topological quantum field theory, where correlation functions are identified with topological surfaces with punctures, cobordism is a good candidate for such a product. In fact it is a consequence of the axioms of topological quantum field theory, as stated by Atiyah for example in [2], that the cylinder $\Sigma \times I$, where $\Sigma$ is a topological surface without boundary and I is a interval of real numbers, can be identified with the identity map between two vector spaces. One of the main purposes of this research project is to clarify the nature of this map $\psi$ and see whether it can give at least a ring structure to the space of correlation functions if considered in the reversed direction. Also, it is well known that topological quantum field theories in 2 dimensions are in one-to-one correspondence with Frobenius Algebras, so it would be very interesting to clarify the algebraic nature of graphs with loops considered in [17] and obtained from the Hopf Algebra of Loday-Ronco and see how they fit between Hopf Algebras and Frobenius Algebras. Related to this we would like to study the decomposition of complex surfaces into spheres with punctures and cylinders, which is known as the Lego-Teichm\''{u}ller game [19] and see how the graph description of this procedure shown in [19] could be merged with our own in [17].
  With respect to the second part of this research project it is well-known that the fields in quantum field theory have a distributional nature. This starts already with the canonical commutation relations of the scalar field and its conjugate momentum field which are equal to the distributional delta functions with the momenta as arguments. Also the description that physicists give of the would-be Hilbert space that is the domain of the theory are in terms of eigenfunctions of the momentum operator which in the coordinate representation are plane-waves that do not belong to the Hilbert space $L^2$. This is typical of a theory that describes scattering where the kinematic observables have a continuum spectrum and that is compatible with special relativity, contrary to quantum mechanics that describes mostly non-relativistic bound states. More important is the argument presented in [20, Chap. 4] based on the cluster decomposition principle and the Riemann-Lebesgue lemma showing that the Fourier transform of the connected scattering matrix from the momenta to the coordinates cannot be integrable in the Lebesgue sense but must contain a factor of the distributional delta function. Then we would like to investigate if the distributional character of the theory allows less stringent conditions than in quantum mechanics in obtaining a completely rigorous path integral in the mathematical sense. There are hints that this may be so. In fact Theorem 4.6.1 in [18] shows that there does not exist a countably additive complex measure on the $\sigma$-algebra of continuous paths in $R^n$ that could be seen as an analytic continuation of the Wiener measure because such a measure would have an infinite total variation in the limit $n\rightarrow\infty$ where $n$ is the typical time and paths discretization of the Wiener measure. But if one considers instead a family of measures not depending on a single complex constant $\lambda$ but on a sequence of complex constants $\lambda_n$ with the imaginary part of $\lambda_n$ $Im(\lambda_n)=(1/n)^{1/2}$ then it is clear that $\lim_{n\rightarrow \infty} (\frac{|\lambda_n|}{Re(\lambda_n)})^{n/2}$ is a finite real number which implies that the total variation is no longer infinite. On the other hand such a sequence of measures would converge to a Dirac delta function when properly normalized. We would like to clarify if this simple observation is in fact pertinent in the sense that it would open the possibility of obtaining a measure in the distributional sense on the space of generalized functions, that hopefully would be translation invariant. As is well-known and explicit in the concept of rigged Hilbert spaces of Gelfand, this space is the closure of the Hilbert space $L^2$, so this would also be of relevance in the framework of Quantum Mechanics.
• C.1.3. Expected outcomes / impactFundamental research is always surprising and unexpected, so it is hard to say in what future directions this project will lead. A better understanding of the mathematical framework of the path integral can hardly be overestimated although we won't make too much for granted at this point either. Also the link between Hopf algebras, combinatorics, graphs and topological field theory has far reaching consequences, from algebraic geometry and moduli spaces to the quantum theory of gravity and the literature on these subjects is immense. In any case what can be said is that this project will be invaluable to the candidate in providing the basis for an independent research trajectory, because it will allow him to make contact with the state of the art on all of these subjects, to interact with several international research centres and to establish connections with different people.
  These research topics requires mastering distinct areas of Mathematics, from algebraic geometry to functional analysis, and its connections to Physics like the relation of measure theory with quantization. Doing this will be of enormous benefit both from the personal and professional perspective with very positive consequences for the candidate's future career. In any case, the clarification of the quantization process in some as yet unknown respects will certainly be of relevance in understanding the general framework.
  This project is also of relevance within the scientific strategy of the CAMGSD Research Centre. The areas of activity of CAMGSD are central in mathematics itself and are fundamental for many applications in science, engineering and in particular in Mathematical Physics. This last subject is in fact one of the strategic areas of interest of CAMGSD. One of its main objectives is to promote research and graduate studies in its areas of activity with excellence of high international quality. Along the years, the CAMGSD developed an intense activity, with objectives and organization close to the best research centers of mathematics in the world. The activities include a postdoctoral program, a program of invited researchers, and several regular seminars, some of which are held every week. The CAMGSD organizes on a regular basis international colloquia, conferences, and short courses, some of them within series of European or international conferences and workshops. It regularly hosts postdoctoral fellows and also invited researchers for long periods, as well as PhD students. It also promotes concentration periods and Summer schools on specific subjects, involving also PhD students and postdoctoral fellows. In this way this research project fits completely into the scientific strategy of CAMGSD and it will be a significant value to the Center to be able to count on an FCT Researcher.
• C.1.4. References1. Marcelo Aguiar and Frank Sottile, Structure of the Loday-Ronco Hopf algebra of trees, J. Algebra 295 (2006), no. 2, 473–511. MR 2194965 (2006k:16078)
  2. Michael F Atiyah, Topological quantum field theory, Publications Mathématiques de l’IHÉS 68 (1988), 175–186.
  3. C. Brouder, Runge-Kutta methods and renormalization, Eur.Phys.J. C12 (2000), 521–534.
  4. Christian Brouder, On the trees of quantum fields, Eur.Phys.J. C12 (2000), 535–549.
  5. J.C. Collins, Renormalization: An introduction to renormalization, the renormalization group and the operator-product expansion, Cambridge University Press, 1984.
  6. A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Quantum field theory: perspective and prospective (Les Houches, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 530, Kluwer Acad. Publ., Dordrecht, 1999, pp. 59–108. MR 1725011 (2002e:81092)
  7. P. Di Francesco, Paul H. Ginsparg, and Jean Zinn-Justin, 2-D Gravity and random matrices, Phys.Rept. 254 (1995), 1–133.
  8. Olivia Dumitrescu, Motohico Mulase, Brad Safnuk, and Adam Sorkin, The spectral curve of the Eynard-Orantin recursion via the Laplace transform, Algebraic and geometric aspects of integrable systems and random matrices, Contemp. Math., vol. 593, Amer. Math. Soc., Providence, RI, 2013, pp. 263–315.
  9. B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007), no. 2, 347–452.
  10. L. Foissy, Les algèbres de Hopf des arbres enracinés décorés. I, Bull. Sci. Math. 126 (2002), no. 3, 193–239.
  11. , Les algèbres de Hopf des arbres enracinés décorés. II, Bull. Sci. Math. 126 (2002), no. 4, 249–288.
  12. Alessandra Frabetti, Simplicial properties of the set of planar binary trees, J. Algebraic Combin. 13 (2001), no. 1, 41–65.
  13. Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S. Retakh, and Jean-Yves Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995), no. 2, 218–348.
  14. Brandon Humpert and Jeremy L. Martin, The incidence Hopf algebra of graphs, SIAM J. Discrete Math. 26 (2012), no. 2, 555–570.
  15. Jean-Louis Loday and María O. Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139 (1998), no. 2, 293–309.
  16. William R. Schmitt, Incidence Hopf algebras, J. Pure Appl. Algebra 96 (1994), no. 3, 299–330.
  17. J. N. Esteves, Hopf Algebras and Topological Recursion, arXiv:1503.02993 [math-ph] http://arxiv.org/abs/1503.02993. Accepted for publication on JPhys A.
  18. G.W. Johnson and M.L. Lapidus, The feynman integral and feynman's operational calculus, Oxford mathematical monographs, Clarendon Press, 2000.
  19. B. Bakalov and A.A. Kirillov, Lectures on tensor categories and modular functors, Translations of Mathematical Monographs, American Mathematical Soc.,2001.
  20. S. Weinberg, The Quantum Theory of Fields vol. 1, Cambridge University Press, 1995.

C.2. Career development plan

• C.2.1. Career objectives / Development and consolidation of an independent career / Networking and Internationalisation plansSince the beginning of his university career the candidate has been very interested in fundamental issues in Particle and Quantum Physics. There are still major mathematical problems to solve concerning the foundations of Quantum Field Theory, like a rigorous mathematical description of the path integral as it is considered by the physicists in their formulation of interactions and scattering processes, or a completely satisfactory treatment of the process of quantization, since at present the several approaches to the problem namely Geometric Quantization, Deformation Quantization or Noncommutative geometry, are incomplete and/or not based on first principles. As is well known, General Relativity has always been elusive to quantization and it is crucial to give a definitive answer to the problem of quantization of gravity, one way or the other. The question is multidisciplinary and it may be addressed with different tools, such as Complex, Differential, Symplectic and Algebraic Geometry, Functional Analysis and Operator Algebras, or from a more physical perspective with Quantum Mechanics, the Path Integral Method, Quantum Field Theory, Topological Quantum Field Theory or Superstring Theories.
  All of these branches of Mathematics and Physics are represented at the CAMGSD Research Centre where the candidate currently works after completing his Ph.D. at the CFTP Research Centre, also at IST, and to which he is applying through this grant call. Working in this environment is of great significance for the consolidation of an independent scientific path because it allows to have contact with different areas and different approaches that contribute to the subject at hand. The periodic multidisciplinary seminars that take place in CAMGSD are also of great significance in this respect and contribute decisively to a smooth transition from the Ph.D. subject to these more fundamental areas of Mathematical Physics and to establish the foundations of an independent career.
  The continuation of this transition is the goal for the next 3 years during the implementation of the current research project and in the course of which the candidate intends to strengthen and complement his knowledge on Algebraic, Complex and Symplectic Geometries and Functional Analysis and Operator Algebras and its applications to quantization. Certainly some interesting research work will come out of this, which will be published in peer-reviewed journals. Visits to international Research Centres will be very important in pursuing this goal, as well as making direct contact with other researchers. In this way he intends to start or strengthen the contacts with top researchers in the present areas of mathematics and physics, and possibly to invite them to visit CAMGSD for lectures, seminars or conferences. The exploratory research project will provide partial funding for these activities but the candidate also intends in the near future to apply to the regular FCT calls for project funding. Working closely with Professor Roger Picken at CAMGSD who is an experienced researcher, in particular in topological quantum field theory, the candidate counts also on his advice to start building a network of contacts based on the candidate's present and future work in these fields of research. He also has the intention of initiating the supervision of students, starting with graduate and MSc students, and later, after acquiring some experience, passing to PhD students.
  The candidate would like very much to pursue his career in Portugal and in particular at CAMGSD where he already has a network of contacts and is well integrated, although he would like also to visit other Research Centers for short periods, in Portugal or abroad.

D. Ethical and legal issues

• D.1. Does your research project / application needs to disclose any statement regarding ethical or legal issues?No

E. Host institution

• E.1. Host InstituitionCentro de Análise Matemática, Geometria e Sistemas Dinâmicos (CAMGSD/IST/ULisboa)
• E.2. Description of the host conditions (omitido)