Organizers: Brian Collier, Christian Zickert
When: Mondays @ 3:15pm
Where: Math 1313

Archives: 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017

  • The Goldman symplectic form on the Hitchin component

    Speaker: Tengren Zhang (California Institute of Technology) -

    When: Wed, August 30, 2017 - 3:15pm
    Where: Kirwan Hall 1313

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    Abstract: Let S be a closed, orientable, connected surface of genus at least 2. We prove that any ideal triangulation on S determines a symplectic trivialization (with respect to the Goldman symplectic form) of the tangent bundle of the Hitchin component. One can then consider the parallel flows with respect to the flat structure given by this trivialization. We give a geometric description of all such flows in terms of explicit deformations of Frenet curves, and prove that all such flows are Hamiltonian. Applying this to a particular ideal triangulation allows us to find a maximal family of Poisson commuting Hamiltonian flows on the Hitchin component. This generalizes the well-known fact that on Teichmüller space, the twist flows along a pants decomposition of S is a maximal family of Poisson commuting Hamiltonian flows. This is joint work with Zhe Sun and Anna Wienhard.
  • The OPER holonomy map is a holomorphic immersion

    Speaker: Andrew Sanders (University of Heidelberg) -

    When: Wed, September 6, 2017 - 3:15pm
    Where: Kirwan Hall 1311

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    Abstract: TBA
  • Exotic Connected components of the SO(p,q) character variety

    Speaker: Brian Collier (UMD) -

    When: Mon, September 18, 2017 - 3:15pm
    Where: Kirwan Hall 3206

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    Abstract: In this talk we will give a complete count of the connected components of the character variety of representations of a closed surface group into SO(p,q). In particular, we will exhibit the existence of "exotic" connected component which are not labeled by a characteristic class of SO(p,q) bundles. Each of these exotic components is parameterized by the space of K^p-twisted SO(1,q-p+1) Higgs bundles with the vector space of holomorphic differentials of degree 2,4,...,2n-2. From this parameterization, the Betti numbers for q=p+1 and q=p+2 can be computed. In the end, we will give evidence that these new connected components consist entirely of geometrically interesting (Anosov) representations.
  • Hitchin systems, Langlands duality, and spin bundles

    Speaker: Richard Wentworth (UMCP) -

    When: Mon, September 25, 2017 - 3:15pm
    Where: Kirwan Hall 3206
  • Finsler metrics of constant curvature

    Speaker: Vladimir Matveev () -

    When: Mon, October 2, 2017 - 3:15pm
    Where: Kirwan Hall 3206

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    Abstract: I will mostly speak about Finsler metrics of positive constant flag curvature (I explain what is it) on closed 2-dimensional surfaces. The main result is that the geodesic flow of such a metric is conjugate to that of a Katok metric (recall that Katok metrics is are easy and well-understood examples of two-dimensional Finsler metrics of positive constant flag curvature). In particular, either all geodesics are closed, and at most two of them have length less than the generic one, or all geodesics but two are not closed; in the latter case there exists a Killing vector field. Generalizations for the multidimensional case will be given; in particular I show that in all dimensions the topological entropy vanishes and the geodesic flow is Liouville integrable. I will also show that in all dimensions a Zermelo transformation of every metric of positive constant flag curvature has all geodesics closed. The results are part of an almost finsihed paper coauthored with R. Bryant, P. Foulon, S. Ivanov and W. Ziller.
  • Compact quotients of pseudo-Riemannian symmetric spaces

    Speaker: Nicolas Tholozan (Ecole Normal Superieure ) -

    When: Mon, October 9, 2017 - 3:00pm
    Where: Kirwan Hall 3206

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    Abstract: It follows from the celebrated theorem of Borel and Harish-Chandra that every Riemannian symmetric space admits a compact quotient. In contrast, some pseudo-Riemannian symmetric spaces do not admit any discrete group of isometries acting properly discontinuously and cocompactly.
    In this talk, I will present a new obstruction to the existence of such actions, showing in particular that the pseudo-Riemannian symmetric space of signature (p, q) and constant curvature −1 does not admit compact quotients when p is odd.
  • Hitchin components for fundamental groups of 2-orbifolds

    Speaker: Florent Schaffhauser (Universidad de Los Andes) -

    When: Mon, October 9, 2017 - 4:00pm
    Where: Kirwan Hall 3206

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    Abstract: Let Y be a compact connected 2-orbifold of negative Euler characteristic and let \Pi be its orbifold fundamental group. For n > 1, we denote by R(\Pi,n) the space of representations of \Pi into PGL(n,R). The purpose of the talk is to show that R(\Pi,n) possesses a connected component homeomorphic to an open ball whose dimension we can compute explicitly (for n=2 and 3, we find again formulae due to Thurston and to Choi and Goldman, respectively). We then give several applications of the result. This is joint work with Daniele Alessandrini and Gye-Seon Lee (University of Heidelberg).
  • Monge-Ampère Iteration (thesis defense)

    Speaker: Ryan Hunter (UMD) -

    When: Mon, October 23, 2017 - 3:15pm
    Where: Kirwan Hall 3206

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    Abstract: The Ricci iteration is a sequence of metrics solving a sequence of recursively defined prescribed curvature problems on a Riemannian manifold. On compact Kähler manifolds admitting a Kähler-Einstein metric Darvas and Rubinstein proved the Ricci iteration converges to a Kähler-Einstein metric. Each step of the Ricci iteration on compact Kähler manifolds is a complex Monge-Ampère equation. In this talk we will define the Monge-Ampère iteration to be a real Monge-Ampère analogue of those complex Monge-Ampère equations. First, we will prove sufficient conditions for the convergence of the Monge-Ampère iteration. Second, we will discuss an application to the Ricci iteration of singular metrics on toric varieties.
  • Twisted K-homology of compact Lie groups

    Speaker: Jonathan Rosenberg (UMD) -

    When: Mon, October 30, 2017 - 3:15pm
    Where: Kirwan Hall 3206

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    Abstract: Computing the twisted K-homology of compact Lie
    groups is both a good test case for methods of topological K-theory
    and a subject of interest in physics (because of its connection with the
    WZW model). This problem was previously attacked by Moore,
    Hopkins, Braun, C. Douglas, and several others. We outline a new
    approach using a theorem of Khorami and the Segal spectral
    sequence. This leads to problems of computing the Hurewicz
    homomorphism in topological K-homology, which can be solved
    by standard methods in homotopy theory.
  • Projective Anosov representations and convex cocompact actions

    Speaker: Andrew Zimmer (William and Mary ) -

    When: Mon, November 6, 2017 - 3:15pm
    Where: Kirwan Hall 3206

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    Abstract: In this talk we will describe two results which relate Anosov representations with convex cocompact actions on properly convex domains in real projective space. First, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then then any irreducible projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. Second, we describe how Anosov representations in general semisimple Lie groups can be defined in terms of the existence of a convex cocompact action on a properly convex domain in some real projective space (which depends on the semisimple Lie group and parabolic subgroup). We will then describe two applications: a rigidity result involving the regularity of the limit curve and a rigidity result involving the Hilbert entropy.
  • Asymptotic behaviour of the BCOV metric

    Speaker: Gerard Freixas (Institut de Mathématiques de Jussieu) -

    When: Mon, November 13, 2017 - 3:15pm
    Where: Kirwan Hall 3206

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    Abstract: The BCOV line bundle of a family of Calabi-Yau varieties, and its metric, were introduced by Fang-Lu-Yoshikawa in connection with a conjecture by math physicists Bershadsky-Ceccotti-Osguri-Vafa. They predict that a certain spectral invariant attached to a Calabi-Yau threefold can be computed in terms of Gromov-Witten invariants of the mirror. Fang-Lu-Yoshikawa treated the case of the Dwork pencil. Their work indicates the importance of understanding the behaviour of the BCOV metric under degeneration, in order to attack other cases of the conjecture and higher dimensional generalizations. I will report on joint work with Dennis Eriksson and Christophe Mourougane, where we obtain general formulas for the degeneration of BCOV metrics, in terms of topological invariants (involving monodromy, vanishing cycles and others). With some more work, the conjecture for Dwork pencils in dimension 4 should be accessible (this conjecture has been explicitely stated by Klemm-Pandharipande).
  • Counting curve types

    Speaker: Tarik Aougab

    When: Mon, November 20, 2017 - 3:15pm
    Where: Kirwan Hall 3206

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    Abstract: For S a closed orientable surface, let N(k,S) denote the number of mapping class group orbits of closed curves with at most k self-intersections. We give upper and lower bounds on N(k,S) that both grow exponentially in the square root of k. There are three major ingredients: statistical work of Lalley describing the behavior of a "typical" geodesic on a hyperbolic surface; the geometry of Thurston's Lipschitz metric on Teichmuller space and the corresponding mapping class group action; and circle packings in hyperbolic geometry. This represents joint work with Juan Souto.
  • Tba

    Speaker: Anton Lukyanenko (George Mason) -

    When: Mon, December 11, 2017 - 3:15pm
    Where: Kirwan Hall 3206

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    Abstract: Tba