André Gama Oliveira's Annual Report
Year:
Brief description of the research activities:
Study of geometric, topological and algebraic structures of moduli spaces of G-Higgs bundles over Riemann surfaces (compact or not), with G being a real reductive Lie group.
More precisely, the following subjects were studied/developed:
- An intrinsic approach to the study of the connectedness and non-emptiness of the moduli space of G-Higgs bundles, over a compact Riemann surface, when G is a complex reductive (not necessarily connected) Lie group. Joint work with O. García-Prada (ICMAT). This gave a paper accepted for publication in the Asian Journal of Mathematics.
- Wrote a survey paper on the theory of Higgs bundles, focused on the study of the connected components of their moduli spaces. Aimed at mathematicians (with interests in geometry and topology), not necessarily specialists in this area. Published in the CIM Bulettin.
- Wrote a survey on the moduli spaces of rank 2 quadric bundles over a compact Riemann surface X, on the occasion of the GEOQUANT 2015 Conference. Published in Travaux Mathemàtiques.
- Studied basic results (for instance on non-emptiness) on moduli spaces of quadric bundles of arbitrary rank, on a compact Riemann surface. A paper has been submitted.
- Continuation of the study of connected components of the moduli space of SO(p,q)-Higgs bundles. We are in the final steps of the proof of the existence of exotic components in these moduli spaces. When p>q+1 and p>2, this is the first example of non-split and non-hermitian groups for which these exotic components occur. Joint work in progress with S. Bradlow (Illinois), B. Collier (Maryland), O. García-Prada (ICMAT) and P. Gothen (CMUP).
- Proof of the equality of the stringy mixed Hodge polynomials of the moduli spaces of parabolic SL(n,C) and PGL(n,C)-Higgs bundles, when n=2,3 and of any degree and proof of results which allow to conjecture that the same holds for any n. This gives an indication thatthe corresponding moduli spaces (when equipped with certain gerbes) are mirror symmetric Calabi-Yau orbifolds in the sense of Hitchin. Joint work in progress with P. Gothen (CMUP).
Talks / Seminars / Courses :
Communications in international conferences
Communications in national conferences
Seminars
Organization of regular seminars:
Organizer of the regular seminars of CMUP's Geometry Group during all year (co-organized with Agnès Gadbled until July 2016).
Work visits:
Participation on the following activities:
Conference Hitchin 70: Celebrating 30 years of Higgs bundles and 15 years of generalized geometry, ICMAT (Madrid), 12–16 Setembro 2016.
Workshop New perspectives on Higgs bundles, branes and quantization, Simons Centre for Geometry and Physics, Stony Brook, NY, USA, 13–17 Junho 2016.