Peter Gothen

Geometry and Topology

PhD (Mathematics)
14.55

Publications

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    Steven B. Bradlow · Oscar Garcia-Prada · Peter B. Gothen
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    ABSTRACT: Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this maximal Toledo case. The Morse theory also allows us to show connectedness when the Toledo invariant is zero. The correspondence between Higgs bundles and surface group representations thus allows us to count the connected components with zero and maximal Toledo invariant in the moduli space of representations of the fundamental group of the surface in SO*(2n).
    Geometriae Dedicata 04/2015; DOI:10.1007/s10711-014-0026-8 · 0.47 Impact Factor
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    Peter B. Gothen
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    ABSTRACT: These are the lecture notes from my course in the January 2011 School on Moduli Spaces at the Newton Institute. I give an introduction to Higgs bundles and their application to the study of character varieties for surface group representations.
    Moduli Spaces, Edited by Brambila-Paz L, Newstead P, Thomas RPW, Garcia-Prada O, 01/2014: pages 151-178; Cambridge University Press.
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    Sofia B.S.D. Castro · Sami F. Dakhlia · Peter B. Gothen
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    ABSTRACT: The main result of this paper states that there exists a residual subset of the set of critical economies whose associated equilibria are finite in number. We also show that this subset does not contain any open set and therefore the result is the best possible for our choice of topology (compact-open topology). The proof rests on results and concepts from singularity theory.
    Mathematical Social Sciences 09/2013; 66:169-175. DOI:10.2139/ssrn.1648426 · 0.46 Impact Factor
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    ABSTRACT: In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, i.e. representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper arXiv:0909.4487 [math.DG] to identify the space of representations with the moduli space of polystable Sp(2n,R)-Higgs bundles. A key step is provided by the discovery of new discrete invariants of maximal representations. These new invariants arise from an identification, in the maximal case, of the moduli space of Sp(2n,R)-Higgs bundles with a moduli space of twisted Higgs bundles for the group GL(n,R).
    Journal of Topology 03/2013; 6:64-118. DOI:10.1112/jtopol/jts030 · 0.86 Impact Factor
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    Peter B. Gothen · André Oliveira
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    ABSTRACT: Given any line bundle L of positive degree, on a compact Riemann surface, let $M_L^\Lambda$ be the moduli space of L-twisted Higgs pairs of rank 2 with fixed determinant isomorphic to $\Lambda$ and traceless Higgs field. We give a description of the singular fibre of the Hitchin map $H:M^L_\Lambda\to H^0(L^2)$, when the corresponding spectral curve has any singularity of type $A_{m-1}$. In particular, we prove directly that this fibre is connected.
    International Mathematics Research Notices 01/2013; 2013(5):1079-1121. DOI:10.1093/imrn/rns020 · 1.07 Impact Factor
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    Steven B. Bradlow · Oscar Garcia-Prada · Peter B. Gothen
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    ABSTRACT: We use Higgs bundles to answer the following question: When can a maximal Sp(4,R)-representation of a surface group be deformed to a representation which factors through a proper reductive subgroup of Sp(4,R)?
    The Quarterly Journal of Mathematics 12/2012; 63:795-843. DOI:10.1093/qmath/har010 · 0.59 Impact Factor
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    Peter B. Gothen · André G. Oliveira
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    ABSTRACT: Let $X$ be a compact Riemann surface. A quadratic pair on $X$ consists of a holomorphic vector bundle with a quadratic form which takes values in fixed line bundle. We show that the moduli spaces of quadratic pairs of rank 2 are connected under some constraints on their topological invariants. As an application of our results we determine the connected components of the $\mathrm{SO}_0(2,3)$-character variety of $X$.
    Geometriae Dedicata 12/2012; 161:795-843. DOI:10.1007/s10711-012-9709-1 · 0.47 Impact Factor
  • Steven B. Bradlow · Peter B. Gothen
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    ABSTRACT: We determine the number of connected components of the moduli space for representations of a surface group in the general linear group. Key words: Representations of fundamental groups of surfaces, Higgs bundles, connected components of moduli spaces. MSC 2000: 14D20, 14F45, 14H60. 1
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    Indranil Biswas · Peter B. Gothen · Marina Logares
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    ABSTRACT: Let X be a compact Riemann surface of genus at least two. Fix a holomorphic line bundle L over X. Let ℳ be the moduli space of Hitchin pairs E , φ ∈ H 0 (End 0 (E)⊗L) over X of rank r and fixed determinant of degree d. The following conditions are imposed: (i) degL≥2g-2, r≥2, and L ⊗r ≠K X ⊗r ; (ii) (r,d)=1; and (iii) if g=2, then r≥6, and if g=3, then r≥4. We prove that that the isomorphism class of the variety ℳ uniquely determines the isomorphism class of the Riemann surface X. Moreover, our analysis shows that ℳ is irreducible. (This result holds without the additional hypothesis on the rank for low genus.)
    Mathematical Proceedings of the Cambridge Philosophical Society 11/2011; 151:441-457. DOI:10.1017/S0305004111000405 · 0.83 Impact Factor
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    Peter B. Gothen
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    ABSTRACT: We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character varieties for representations of surface groups in real Lie groups G and the moduli space of G-Higgs bundles. Finally we survey recent joint work with Bradlow, Garc\'ia-Prada and Mundet i Riera on the moduli space of maximal Sp(2n,R)-Higgs bundles.
    XIX International Fall Workshop on Geometry and Physics, Porto; 01/2011
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    Sofia B. S. D. Castro · Sami Dakhlia · Peter B. Gothen
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    ABSTRACT: We complement the Sonnenschein–Mantel–Debreu results by establishing that an exchange economy, i.e., preferences and endowments, that generates a given aggregate excess demand (AED) function is close to the economy that generates a perturbation of this AED. As a consequence, genericity and determinacy results obtained by direct perturbation of an AED are as strong as results obtained by perturbing preferences and endowments.
    SSRN Electronic Journal 07/2010; 46(4):562-571. DOI:10.1016/j.jmateco.2010.03.008
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    Sofia B. S. D. Castro · Sami Dakhlia · Peter B. Gothen
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    ABSTRACT: We establish that an exchange economy, i.e., preferences and endowments, that generates a giiven aggregate excess demand (AED) function is close to the economy generating the AED obtained by an arbitrary perturbation of the original one.
    Journal of Mathematical Economics 01/2010; 46:562-571. · 0.50 Impact Factor
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    ABSTRACT: We develop a complete Hitchin-Kobayashi correspondence for twisted pairs on a compact Riemann surface X. The main novelty lies in a careful study of the the notion of polystability for pairs, required for having a bijective correspondence between solutions to the Hermite-Einstein equations, on one hand, and polystable pairs, on the other. Our results allow us to establish rigorously the homemomorphism between the moduli space of polystable G-Higgs bundles on X and the character variety for representations of the fundamental group of X in G. We also study in detail several interesting examples of the correspondence for particular groups and show how to significantly simplify the general stability condition in these cases.
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    Steven B. Bradlow · Oscar García-Prada · Peter B. Gothen
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    ABSTRACT: We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups and . Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott–Morse theory on the corresponding moduli spaces.
    Topology 09/2008; 47(4):203-224. DOI:10.1016/j.top.2007.06.001 · 0.23 Impact Factor
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    Steven B. Bradlow · Oscar García-Prada · Peter B. Gothen
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    ABSTRACT: A Higgs bundle is a holomorphic vector bundle together with a Higgs field. Such objects first emerged twenty years ago in Nigel Hitchin's study of the self-duality equations on a Riemann surface and in Carlos Simp-son's Ph.D. thesis and subsequent work on nonabelian Hodge theory. Hitchin introduced the term "Higgs field" because of similarities to objects labeled this way in other equations of gauge theory. In those contexts Higgs fields describe physical particles like the Higgs boson. Simpson suggested the shorthand "Higgs bundle" for a bundle together with a Higgs field. Higgs bundles have a rich structure and play a role in many different areas including gauge theory, Kähler and hyperkähler geometry, surface group representa-tions, integrable systems, nonabelian Hodge theory, the Deligne–Simpson problem on products of matrices, and (most recently) mirror symmetry and Langlands duality. In this essay we will touch lightly on a selection of these topics. We start with the definition: A Higgs bundle is a pair (E, φ) where E is a holomorphic vector bundle and φ, the Higgs field, is a holomorphic 1-form with values in the bundle of endomorphisms of E, satisfying φ ∧ φ = 0. In the simplest examples the bundle is a complex line bundle and the Higgs field is a holomorphic 1-form. To see a nonabelian example, set E = K 1/2 ⊕ K −1/2 where K 1/2 is a complex line bundle whose square is K, the canonical bundle on a Riemann surface (i.e., the bundle of holomorphic 1-forms). , where 1 is the identity section of the trivial bundle Hom(K 1/2 , K −1/2 ⊗ K). We now look at how Higgs bundles emerge in non-abelian Hodge theory. Hodge theory uses harmonic differential forms to represent de Rham cohomolo-gy classes on Riemannian manifolds. On a hermitian manifold, say X, ¯ ∂-harmonic forms give analogous rep-resentatives for Dolbeault cohomology classes. If the metric on X is Kähler, the real and complex theories are compatible. This relates topological and holomorphic data on X and reveals additional structure on the topo-logical side, i.e., on the cohomology groups H k (X; C). For k = 1 we get (1) H 1 (X; C) ≅ H 0,1 (X) ⊕ H 1,0 (X).
    Notices of the American Mathematical Society 01/2007; 54(8).
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    O. García-Prada · P. B. Gothen · V. Muñoz
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    ABSTRACT: We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their variation with this parameter.
    Memoirs of the American Mathematical Society 01/2007; 187(879). DOI:10.1090/memo/0879 · 1.78 Impact Factor
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    Peter B Gothen · Ana C Ferreira
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    ABSTRACT: We work out the Chern–Weil theory for abelian n-gerbes and consider the problem of classifying n-gerbes with (flat) connections up to gauge equivalence.
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    Steven B. Bradlow · Oscar García-Prada · Peter B. Gothen
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    ABSTRACT: Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant
    Geometriae Dedicata 01/2006; 122(1):185-213. DOI:10.1007/s10711-007-9127-y · 0.47 Impact Factor
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    Peter B. Gothen · Alastair D. King
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    ABSTRACT: Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. We show that the category of such representations is an abelian category with enough injectives by constructing an explicit injective resolution. Using this explicit resolution, we find a long exact sequence that computes the Ext groups in this new category in terms of the Ext groups in the old category. The quiver formulation is directly reflected in the form of the long exact sequence. We also show that under suitable circumstances, the Ext groups are isomorphic to certain hypercohomology groups.
    Journal of the London Mathematical Society 01/2005; 71:85-99. DOI:10.1112/S0024610704005952 · 0.88 Impact Factor
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    Steven B. Bradlow · Oscar Garcia-Prada · Peter B. Gothen
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    ABSTRACT: We determine the number of connected components of the moduli space for representations of a surface group in the general linear group.
    XII Fall Workshop on Geometry and Physics, Coimbra 2003; 01/2004

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