Geometry-Topology Archives for Academic Year 2018

The Goldman symplectic form on the Hitchin component

When: Wed, August 30, 2017 - 3:15pm
Where: Kirwan Hall 1313
Speaker: Tengren Zhang (California Institute of Technology) -
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

When: Wed, September 6, 2017 - 3:15pm
Where: Kirwan Hall 1311
Speaker: Andrew Sanders (University of Heidelberg) -
Abstract: TBA

Exotic Connected components of the SO(p,q) character variety

When: Mon, September 18, 2017 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Brian Collier (UMD) -
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

When: Mon, September 25, 2017 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Richard Wentworth (UMCP) -

Finsler metrics of constant curvature

When: Mon, October 2, 2017 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Vladimir Matveev () -
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

When: Mon, October 9, 2017 - 3:00pm
Where: Kirwan Hall 3206
Speaker: Nicolas Tholozan (Ecole Normal Superieure ) -
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

When: Mon, October 9, 2017 - 4:00pm
Where: Kirwan Hall 3206
Speaker: Florent Schaffhauser (Universidad de Los Andes) -
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)

When: Mon, October 23, 2017 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Ryan Hunter (UMD) -
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

When: Mon, October 30, 2017 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Jonathan Rosenberg (UMD) -
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

When: Mon, November 6, 2017 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Andrew Zimmer (William and Mary ) -
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

When: Mon, November 13, 2017 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Gerard Freixas (Institut de Mathématiques de Jussieu) -
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

When: Mon, November 20, 2017 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Tarik Aougab
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.

Higher-dimensional continued fractions and ergodicity

When: Mon, January 29, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Anton Lukyanenko (George Mason) -
Abstract: Continued fractions in one real dimension (in their various forms) are well-studied from the point of view of both dynamical systems and hyperbolic geometry. Less is known about natural generalizations to higher dimensions: the Hurwitz complex continued fractions and Heisenberg continued fractions. I will discuss these higher-dimensional fractions and their connection to (complex) hyperbolic geometry, and then show how to prove the ergodicity of the associated Gauss map under an additional assumption of lattice completeness. This is work in progress with Vandehey.

Characteristic classes as complete obstructions

When: Mon, February 5, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Martina Rovelli (Johns Hopkins) -
Abstract: We propose a uniform interpretation of characteristic classes as complete obstructions to the reduction of the structure group of a principal bundle, and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber. By plugging in the correct parameters, we recover several classical theorems. We then define a family of invariants for principal bundles that detect the number of group reductions associated to characteristic classes that a principal bundle admits.

Multiple solutions for a higher order variational problem in conformal geometry

When: Mon, February 12, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Paolo Piccione (Universidade de São Paulo) -
Abstract: I will consider the total Q-curvature functional in the conformal class of a compact Riemannian manifold of dimension n greater than or equal to 5. This is the quadratic functional associated to the Paneitz operator, a fourth order differential operator given by the bi-Laplacian plus lower order terms. I will present two results of existence of multiple critical points of the total Q-curvature, that corresponds to metrics with constant Q-curvature in a fixed conformal class. This is a joint work with Renato Bettiol (UPenn) and Yannick Sire (Johns Hopkins Univ.).

Hyperbolic surfaces as singular flat surfaces

When: Mon, February 19, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Aaron Feynes (University of Toronto) -
Abstract: It's long been known that a hyperbolic surface with a maximal measured geodesic lamination is the same thing, loosely speaking, as a half-translation surface: a singular flat surface with a geodesic foliation. I say "loosely" to mean that corresponding hyperbolic and half-translation surfaces are only identified up to isotopy. I'll present a tighter version of this correspondence, due to Gupta, which maps each hyperbolic surface to its corresponding half-translation surface in a geometrically rigid way. This mapping turns the nonabelian flat bundle encoding the hyperbolic structure into the abelian flat bundle encoding the half-translation structure, carrying out a concrete instance of Gaiotto, Hollands, Moore, and Neitzke's abelianization process.

Progress in the Riemann-Hilbert correspondence (note the unusual time)

When: Tue, February 20, 2018 - 10:00am
Where: Kirwan Hall 3206
Speaker: Andy Neitzke (University of Texas) -

Abstract: Given a linear ordinary differential equation in one complex variable z, e.g. a "Schrodinger equation" (d2 / dz2 + P(z)) f(z) = 0, one would like to understand the solutions as well as possible. One concrete question is: what is the monodromy of the solutions when z goes around a loop? I will describe a conjectural scheme for solving this problem, which gives more precise information than was previously available, and which connects the problem to various other areas such as the combinatorics of cluster algebras, the theory of enumerative invariants (generalized Donaldson-Thomas invariants of 3-Calabi-Yau categories), and the geometry of trajectories of quadratic differentials (and higher analogues).

Geometry of weighted Riemannian manifolds

When: Mon, February 26, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: William Wylie (Syracuse University) -
Abstract: A weighted Riemannian manifold is simply a Riemannian manifold equipped with a (variable) density function. For example, a surface with a positive function that describes the density of the material that makes up the surface. In this talk we'll discuss a new geometric approach to weighted Riemannian manifolds that takes a natural torsion free connection as the fundamental object of study. This approach gives new comparison results that are valid under weaker Ricci curvature assumptions than have previously been considered in the literature, and also leads to novel rigidity phenomena. Time permitting, we'll also discuss how the connection leads to a theory of sectional curvature bounds for weighted Riemannian manifolds.

Symplectic geometry of the moduli space of projective structures and the monodromy map.

When: Mon, March 12, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Chaya Norton (Concordia University) -
Abstract: The moduli space of projective connections/structures on Riemann surfaces of genus g can be identified with the moduli space of quadratic differentials, and hence the total space of $T^*\mathcal M_g$, by choosing a base projective connection which varies holomorphically in moduli. The monodromy map from the moduli space of projective connections to the character variety of $PSL(2,\mathbb C)$ representations of the fundamental group of the Riemann surface maps a holomorphic projective connection $u(z)$ to the monodromy group associated to the second order equation $\psi''-u(z)\psi=0$.
In joint work with Bertola and Korotkin, we study the symplectic geometry induced via these maps and highlight the role played by the base projective connection. We introduce the homological symplectic structure and use it to characterize base projective connections which induce equivalent symplectic structure on the moduli space of projective connections. We prove the equivalence of Bergman, Schottky, and Wringer projective connections. By an explicit computation we show that the monodromy map with with the base Bergman projective connection is a symplectomorphism from the moduli space of quadratic differentials with the homological symplectic structure to the character variety with the Goldman bracket. We compare our results with those of Kawai 96.

3-manifolds not obtained by surgery on a knot

When: Mon, April 2, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Thomas Mark (University of Virginia) -
Abstract: A well-known theorem of Lickorish and Wallace states that any closed orientable 3-manifold can be obtained by surgery on a link in the 3-sphere. For a given 3-manifold one can ask how ``simple’’ a link can be used to obtain it, e.g., whether a manifold satisfying certain obvious necessary conditions on its fundamental group always arises by surgery on a knot. This question turns out to be rather subtle, and progress has been limited, but in general the answer is known to be ``no.’’ Here I’ll summarize some recent results including joint work with Matt Hedden, Min Hoon Kim, and Kyungbae Park that give the first examples of 3-manifolds with the homology of S^1 x S^2 and having fundamental group of weight 1 that do not arise by surgery on a knot in the 3-sphere.

Domains of discontinuity for (Quasi-)Hitchin representationsTBA

When: Mon, April 9, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Daniele Alessandrini (University of Heidelberg) -
Abstract: Among representations of surface groups into Lie groups, the Anosov representations are the ones with the nicest dynamical properties. Guichard-Wienhard and Kapovich-Leeb-Porti have shown that their actions on generalized flag manifolds often admit co-compact domains of discontinuity, whose quotients are closed manifolds carrying interesting geometric structures. Dumas and Sanders studied the topology and the geometry of the quotient in the case of Quasi-Hitchin representations (Anosov representations which are deformations of Hitchin representations). In a conjecture they ask whether these manifolds are homeomorphic to fiber bundles over the surface.In a joint work with Qiongling Li, we can prove that the conjecture is true for (Quasi-)Hitchin representations in SL(n,\R) and SL(n,C), acting on projective spaces and partial flag manifolds parametrizing points and hyperplanes.

From point-picking to sections of surface bundles

When: Mon, April 16, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Lei Chen (University of Chicago) -
Abstract: Given any n points on a manifold, how can we systematically and continuously find a new point? What if we ask them to be distinct? In this talk, I will try to answer this question in surfaces. Then I will connect this question to sections of surface bundles. The slogan is "there is no center of mass on closed hyperbolic surfaces".