Geometry and Topology

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1 Geometry and Topology CMUP 21/07/2008 CMUP (21/07/2008) Geometry and Topology 1 / 9

2 People Permanent academic staff (pluri-anual ) Inês Cruz Poisson and symplectic geometry, singularities of vector fields and differential forms, Hamiltonian mechanics. Oscar Felgueiras Higher dimensional algebraic geometry, intersection theory. José Basto Gonçalves (collaborator) Geometry of differential equations, applications in control theory. Peter Gothen Moduli spaces, Higgs bundles. Helena Mena Matos Singularity theory, control optimization, Poisson geometry. Helena Reis Complex differential equations. João Nuno Tavares Mathematical physics, differential geometry, non-holonomic systems. CMUP (21/07/2008) Geometry and Topology 2 / 9

3 People Post doctoral researchers Post-docs (FCT): Alexey Remizov ( ) Differential equations, singularity theory. Marina Logares ( ) Moduli spaces, Higgs bundles. Ciência 2007 (jointly with Physics Research Centre): Iakovos Androulidakis Groupoids and algebroids, differential and noncommutative geometry, quantization. João Martins Quantum groups and low dimensional topology. Marco Zambon Poisson and symplectic geometry, generalized complex geometry, Dirac geometry, supergeometry. Physics Centre: Óscar Dias and Carlos Herdeiro Classical and Quantum Gravity, String Theory. Ciência 2008 (jointly with Algebra/Combinatorics Area): one position. CMUP (21/07/2008) Geometry and Topology 3 / 9

4 Achievements 2003 today Publications 23 papers published in international peer reviewed journals (J. Diff. Geom., J. Geom. Phys., J. Lond. Math. Soc., J. Math. Anal. Appl., J. Symp. Geom., Lett. Math. Phys., Math. Ann., Topology,... ); one monograph (Memoirs of the AMS); 2 papers published in proceedings of international conferences. Invited Lectures Several invited lectures and minicourses in universities and international conferences. Among them: P. Gothen: Morse theory on Higgs bundle moduli spaces, Workshop on the Topology of Hyperkähler manifolds, Rényi Institute of Mathematics, Budapest, November H. Reis: Semi-complete foliations of saddle-node type in dimension 3, Local holomorphic dynamics, Centro di Ricerca Matematica Ennio De Giorgi, Pisa, January M. Logares, A Torelli type theorem for the moduli space of parabolic Higgs bundles, First CTS Conference on Bundles, Tata Institute of Fundamental Research, Mumbai, March CMUP (21/07/2008) Geometry and Topology 4 / 9

5 Achievements 2003 today (cont.) Graduate student training 15 MSc theses supervised ( ). One PhD concluded (2006): Helena Reis (supervisor: José Basto Gonçalves). Current PhD students: Sandra Bento (Centro de Matemática da UBI), Tiago Fardilha, Célia Moreira, André Gama Oliveira. Active participation in PhD programmes in mathematics of the University of Porto. Conferences, minicourses, seminars Oporto Meetings on Geometry, Topology and Physics yearly meeting, w/ Physics Research Centre and CAMGSD (IST, Lisbon), 2008: 17th edition; EuroConference Vector Bundles on Algebraic Curves (2003); Free course on Quantum Field Theory (2004, jointly with Physics Research Centre); Regular Geometry and Topology seminar, including one minicourse: Andrew du Plessis (Aarhus) Singularity Theory (2004). CMUP (21/07/2008) Geometry and Topology 5 / 9

6 Networking and Research projects EDGE European Differential Geometry Endeavour, (EC FP5 Contract no. HPRN-CT ); EAGER European Algebraic Geometry Research Training Network (EC FP5 Contract no. HPRN-CT ); Singularities in Poisson structures, POCTI/MAT/36528/2000, October 2000 to September Espaços Moduli e Teoria de Cordas, Fundação para a Ciência e a Tecnologia, POCTI/MAT/58549/2004, (coordinated at IST, Lisbon). Three Portugal Spain bilateral programmes (CRUP, GRICES/CSIC) in the area of Higgs bundles and moduli spaces: , , ITGP Interactions of Low-Dimensional Topology and Geometry with Mathematical Physics (under evaluation by the European Science Foundation), 2009 CMUP (21/07/2008) Geometry and Topology 6 / 9

7 Research objectives Applications of singularity theory to normal forms: Optimal control; Implicit differential equations; Poisson structures. Non holonomic systems: generalized space for rank greater than 2; quantization. Complex ODE s: uniformization; semi-complete vector fields; Fatou Julia components. Moduli spaces and Higgs bundles: geometry and topology of moduli spaces; representations of surface groups in real Lie groups. CMUP (21/07/2008) Geometry and Topology 7 / 9

8 Holomorphic vector fields Basic Problem: Normal form and analytic classification of vector fields near an isolated singularity. Fact: For holomorphic vector fields there is a local obstruction to completeness notion of semi-completeness. (A vector field is semi-complete if the solution of the associated differential equation admits a maximal domain of definition.) Results of H. Reis: Any foliation of strict Siegel type in (C 3, 0) admits a semi-complete representative. Normal forms for foliations of saddle-node type in (C 3, 0) admitting a semi-complete representative. Some open problems with work in progress: E. Ghys conjecture: the 2-jet at an isolated singular point of a semi-complete vector field in C 3 never vanishes. Classify pairs of 3-dimensional commuting semi-complete vector fields. Study relation between semi-complete vector fields and integrable ones. CMUP (21/07/2008) Geometry and Topology 8 / 9

9 Surface groups and Higgs bundles Objects of study: X closed oriented surface; G Lie group. π 1 (X ) = a i, b i g i=1 [a i, b i ] = 1 Character variety: R(π 1 (X ), G) = Hom(π 1 (X ), G)/G. Approach: R(π 1 (X ), G) = M(G), moduli space of Higgs bundles (algebraic/holomorphic geometry) Maximal representations Hyperbolic structure on X Fuchsian representation ρ: π 1 (X ) Isom(H 2 ); generalizes to maximal representation when G = Isom(X ) for a hermitean symmetric space X of non-compact type. Goal: Have found a correspondence M max (G) = M twisted (G ) for G canonically associated to G. Understand correspondence in terms of representations, and independently of classification of Lie groups. Relation to geometric structures on X? CMUP (21/07/2008) Geometry and Topology 9 / 9

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