Geometry and Topology - CMUP

Geometry and Topology
CMUP
21/07/2008
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People – Permanent academic staff (pluri-anual 2007–2010)
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.
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People – Post doctoral researchers
Post-docs (FCT):
◮ Alexey Remizov (2006–2009) — Differential equations, singularity theory.
◮ Marina Logares (2007–2010) — 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.
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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 2005.
H. Reis: Semi-complete foliations of saddle-node type in dimension 3, Local
holomorphic dynamics, Centro di Ricerca Matematica Ennio De Giorgi, Pisa,
January 2007.
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 2008.
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Achievements 2003–today (cont.)
Graduate student training
15 MSc theses supervised (2003–2006).
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).
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Networking and Research projects
EDGE — European Differential Geometry Endeavour, (EC FP5 Contract no.
HPRN-CT-2000-00101); 2000–2004.
EAGER — European Algebraic Geometry Research Training Network (EC
FP5 Contract no. HPRN-CT-2000-00099); 2000–2004.
Singularities in Poisson structures, POCTI/MAT/36528/2000, October 2000
to September 2004.
Espaços Moduli e Teoria de Cordas, Fundação para a Ciência e a Tecnologia,
POCTI/MAT/58549/2004, 2005–2008 (coordinated at IST, Lisbon).
Three Portugal–Spain bilateral programmes (CRUP, GRICES/CSIC) in the
area of Higgs bundles and moduli spaces: 2003–2004, 2006–2007, 2008–2009.
ITGP — Interactions of Low-Dimensional Topology and Geometry with
Mathematical Physics (under evaluation by the European Science
Foundation), 2009–
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Research objectives 2007–2010
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.
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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.
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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 ?
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