Geometry, Topology and Dynamics of Moduli Spaces
(1 - 19 Aug 2016)



~ Abstracts ~

 

Remarks on parahoric bundles on curves
Vikraman Balaji, Chennai Mathematical Institute, India


In this talk I will review the work on parahoric bundles and its moduli spaces and take a re-look at the notion of stability of such torsors especially when the weights are real. The dimensions of the moduli spaces will also be reviewed as well as a closer look at their geometry.

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Integrability of moduli spaces of parahoric Higgs bundles
David Baraglia, University of Adelaide, Australia


Hitchin showed that the moduli space of Higgs bundles on a curve is an integrable system, where the Poisson commuting functions are obtained by applying invariant polynomials to the Higgs field. The number of such functions, to quote Hitchin, "somewhat miraculously" equals exactly half the dimension of the moduli space. In joint work with Masoud Kamgarpour and Rohith Varma we have shown that surprisingly this miracle persists in the setting of parahoric Higgs bundles, at least for certain classes of parahoric subgroups. Our results hold in particular for parabolic Higgs bundles of types A,B,C and G2.

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Homogeneous principal bundles
Indranil Biswas, Tata Institute of Fundamental Research, India


The holomorphic Hermitian principal bundles on a Hermitian symmetric space are classified. This is a joint work with Harald Upmeier.

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Renormalized volume of 3-manifolds and the Weil-Petersson geometry of moduli space
Jeffrey F. Brock, Brown University, USA


A unifying theme in the study of hyperbolic 3-manifolds has been the role of combinatorial study of the mapping class group in understanding the fine structure of hyperbolic metrics. In turn, combinatorial models have led to a better grasp of the geometry of metrics on moduli space, leading to relationships between geometry of 3-manifolds and geometry of moduli spaces that factor through this combinatorial perspective. In the case of hyperbolic volume, the Weil-Petersson translation distance of surface automorphisms, and the translation distance in a "pants graph" the volume of a fibered 3-manifold are coarsely related, leading one to speculate on more direct connections. In this talk I will discuss how the notion of renormalized volume provides a more explicit and effective relationship between Weil-Petersson geometry of moduli spaces and volumes of fibered hyperbolic 3-manifolds. Paradoxically, these results lead to some of the first effective geometric estimates on the systoles and diameter lower bounds for the Weil-Petersson moduli space. This talk will survey the history of these connections, frame the current state of affairs, and pose future directions for study. This is joint work with Ken Bromberg.

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Hilbert schemes of points and quiver varieties
Ugo Bruzzo, International School for Advanced Studies (SISSA), Trieste, Italy


Hilbert schemes of points on the total spaces of the line bundles O(-n) on P1 (desingularizations of singularities of type (1/n)(1,1)) can be given an ADHM description, and as a result, they can be realized as varieties of quiver representations.

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Maximal representations of surface groups over real closed fields
Marc Burger, ETH Zurich, Switzerland


We give an outline of a structure theory for maximal representations of surface groups into symplectic vector spaces over non-archimedean real closed fields; we will in particular illustrate the use of the real spectrum compactification of representation varieties. This is joint work with A.Iozzi, A.Parreau, and B.Pozzetti.

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Simple length rigidity
Richard Canary, University of Michigan, USA


It is well known that a discrete, faithful representation of a closed surface group into PSL(2,R) is determined, up to conjugacy in PGL(2,R), by the translation lengths of finitely many elements represented by simple closed curves. We will discuss generalization of this result to the setting of quasifuchsian representations into PSL(2,C) and Hitchin representations into PSL(n,R). We will also discuss applications of our simple length rigidity results to the study of the pressure metric. (This is joint work with Martin Bridgeman, Francois Labourie and Andres Sambarino.)

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Tori in the Einstein universe of dimension three
Virginie Charette, Université de Sherbrooke, Canada


We will discuss how tori intersect in three-dimensional Einstein Universe, using the correspondence between this and the space of Lagrangian planes in a four-dimensional symplectic vector space. As an application, we will look at disjointness of so-called << crooked surfaces >>, in the Einstein Universe as well as three-dimensional anti-de Sitter space. This is joint work with Jean-Philippe Burelle, Dominik Francoeur and Bill Goldman.

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Group actions on CAT(0) cube complexes
Indira Lara Chatterji, Université Nice Sophia Antipolis, France


I will give an introduction of CAT(0) cube complexes and spaces with walls. As an illustration I will sketch a proof of Bergeron-Wise result that fundamental groups of closed hyperbolic 3-manifolds act cocompactly on a CAT(0) cube complex (this uses Kahn-Markovic's result that every hyperbolic 3-manifold contains a closed quasi-fuchsian immersed surface).

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Parameterizing connected components of SO(p,p+1)-Higgs bundles
Brian Collier, University of Maryland, USA


In this talk I will discuss a parameterization of n(2g-2) connected components of the SO0(n,n+1)-Higgs bundle moduli space. We will see how this parameterization generalizes both Hitchin's parameterization of the Hitchin component as a vector space of holomorphic differentials of degree 2,4,...,2n and Hitchin's parameterization of the nonzero Toledo invariant components of the PSL(2,R)=SO0(1,2)-Higgs bundle moduli space as vector bundles over certain symmetric products of the Riemann surface.

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Complex deformations of n-Fuchsian representations
David Dumas, University of Illinois at Chicago, USA


We study the topology and complex geometry associated with representations of surface groups in the complex Lie group SL(n,C). We focus on representations in a small neighborhood of an n-Fuchsian representation, i.e. one that factors as the inclusion of a uniform lattice in SL(2,R) composed with the irreducible representation SL(2,R) -> SL(n,R). This is joint work with Andrew Sanders.

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Surface group representations and parabolic Higgs bundles
Oscar García-Prada, Instituto de Ciencias Matemáticas, Spain


In this talk I will introduce parabolic G-Higgs bundles over a compact Riemann surface X with marked points D, when G is a reductive real Lie group. I will describe the correspondence of these objects with representations of the fundamental group of X\D with fixed conjugacy classes around the punctures (joint work with O. Biquard and I. Mundet). I will also mention some applications, including the study of the action of a finite subgroup of the mapping class group on the moduli space of representations of the fundamental group of X in G (on going joint work with G. Wilkin).

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Stability and moduli spaces of twisted parahoric bundles
Jochen Heinloth, Universität Duisburg-Essen, Germany


In artihmetic applications it is often useful to replace moduli spaces of G-bundles by torsors under group schemes over curves that are not constant. Such objects also arise in geometry when studying fibers of the Hitchin system.

In this talk, we would like to give some examples motivating the notion and then explain how one can construct moduli spaces for this type of bundles without going through a GIT construction.

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Some results on the Yang-Mills flow and its application
Min-Chun Hong, The University of Queensland, Australia


At first, I will outline some old and new results on the Yang-Mills flow. Then we establish a parabolic gauge fixing theorem and used it to present a new proof of the existence of the weak solution of the Yang-Mills flow over a four dimensional manifold with initial value in $H^1$. Finally, we improve a key lemma of Uhlenbeck and prove the uniqueness of the weak solution of the Yang-Mills flow on a four dimensional manifold.

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The Voevodsky motive of the moduli stack of vector bundles
Victoria Hoskins, Freie Universität Berlin, Germany


We start by introducing Voevodsky motives of varieties (and stacks); the guiding principle is that the motive of a variety should encode various cohomology theories as well as algebraic cycles. We give a formula for the (compactly supported) motive of the moduli stack of vector bundles on a curve, which is reminiscent of the Atiyah-Bott description of the cohomology. We sketch the proof which uses a forgetful map from an ind-variety of matrix divisors to the stack of vector bundles and Bialynicki-Birula decompositions of varieties of matrix divisors. This is joint work with Simon Pepin Lehalleur.

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Categories and filtrations
Ludmil Katzarkov, Universität Wien, Austria


In this talk we will introduce some new categorical phenomena coming from a Kaehler approach to category theory.

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Obstruction for a virtual C2 action on the circle
Sang-hyun Kim, Seoul National University, Korea


When does a group virtually admit a faithful C2 action on the circle? We provide such an obstruction using a RAAG. Applications include all (non-virtually-free) mapping class groups, Out(Fn) and Torelli groups. (Joint work with Hyungryul Baik and Thomas Koberda)

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Primitive stable representations in semisimple Lie groups
Inkang Kim, Korea Institute for Advanced Study, Korea


I will present some recent results on primitive stable representations into general semisimple Lie groups covering from hyperbolic 3-manifolds to higher rank examples. Some generalization of the notion will be given.

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Currents and Hitchin representations
François Labourie, Université Nice Sophia Antipolis, France


In this course, I will start by recalling the notion of currents for the geodesic flow of hyperbolic surfaces. I will explain some of the basic constructions of currents and what we can do with them: Bonahon intersection, pairing with Hölder parametrisation, as well as constructing them as << Gibbs equilibrium states >> using the thermodynamic formalism. Then, I will explain the interplay of currents with Teichmüller theory and Hitchin representations: how representations give rise to currents (and in particular the "Liouvile current"), what intersection tells us for Hitchin representations and how mixing currents with complex structures produces interesting holomorphic differentials.

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Exotic compact quotients of SO(d,1)
Christopher J. Leininger, University of Illinois at Urbana-Champaign, USA


Given a discrete, torsion free subgroup $\Gamma < SO(d,1)$, the quotient $\Gamma \setminus SO(d,1)$ is an $SO(d)$--bundle over the associated hyperbolic d--manifold $\Gamma \setminus H^d$. For $d=3$, this is also an example of a complete holomorphic Riemannian manifold of constant curvature. Ghys studied deformations of these structures to a more general class of $SO(3,1)$ quotients, and this theory was expanded and further studied by Kobayashi and Gueritaud-Kassel. By a result of Tholozan, the volume of compact quotients remains constant under these deformations, and he asked whether there were "exotic" compact quotients, i.e. bundles over compact hyperbolic manifolds with different volume than the standard $SO(d)$--bundle. Building on a construction of Agol and the work of Gueritaud--Kassel, we describe an infinite family of such exotic compact quotients for $d \leq 4$, and calculate their volumes using Tholozan's volume formula. This is joint work with Grant Lakeland.

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Minimal surfaces for Hitchin representations
Qiongling Li, California Institute of Technology, USA


Given a reductive representation from surface group into a Lie group $G$, there exists a equivariant harmonic map from the universal cover of a fixed Riemann surface to the symmetric space $G/K$ associated to $G$. If the Hopf differential of the harmonic map vanishes, the harmonic map is then minimal. In this talk, we investigate the properties of immersed minimal surfaces inside symmetric space associated to a subloci of Hitchin component: $q_n$ and $q_{n-1}$ case. This is joint work with Song Dai.

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Compactness of symplectic critical surfaces
Jiayu Li, University of Science and Technology, China


In this talk, we introduce new functionals to study the existence of holomorphic curves in Kähler surfaces. We study the properties of the critical surfaces of the functionals. We study the compactness of the critical surfaces.

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Convex polytopes in hyperbolic 3-space and discrete conformal geometry of surfaces
Feng Luo, Rutgers University, USA


Our recent work on discrete conformal geometry on compact polyhedral surfaces shows that a discrete version of the uniformization theorem holds. The relationship between the discrete uniformization theorem and convex ideal convex polytopes in hyperbolic 3-space the convergence of discrete conformal geometry to the smooth conformal geometry will be addressed. This is based on a joint work with Jian Sun and Tianqi Wu.

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Asymptotic properties of the Hitchin moduli spaces
Rafe Mazzeo, Stanford University, USA


In these lectures I will describe recent results concerning large-scale features of the Hitchin moduli space. The first main theme reviews the construction of 'large' solutions of Hitchin's equations, and the consequences of this construction for the asymptotics of the natural hyperKaehler metric on this moduli space. The goal is to understand a string-theoretic conjecture by Gaiotto, Moore and Neitzke. Key issues involve the analysis of families of model solutions on P^1, which in the simplest case reduces to the study of solutions of a classical Painleve equation. This is joint work with Swoboda, Weiss and Witt, and I will also report on recent work by Fredrickson. I will also describe recent work, with Dumitrescu, Fredrickson, Kydonakis, Mulase and Neitzke, about a different asymptotic limit of this moduli space along the Hitchin section, which is phrased in the language of opers.

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Generalized Theta functions, strange duality and odd orthogonal bundles on curves
Swarnava Mukhopadhyay, University of Maryland, USA


Generalized theta functions are a non-abelian generalization for the classical theta functions. In this talk, we study the space of generalized theta functions for odd orthogonal bundles with nontrivial Stiefel-Whitney class and the associated space of twisted spin bundles. We will present a Verlinde type formula that was conjectured by Oxbury-Wilson and address the issue of strange duality for odd orthogonal bundles. We will show that the naive conjecture fails in general. A consequence of this is the reducibility of projective representations of mapping class groups arising from the Hitchin connection for these moduli spaces. This is a joint work with Richard Wentworth.

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Schematic Harder-Narasimhan stratification for families of principal bundles in higher dimensions
Nitin Nitsure, Tata Institute of Fundamental Research, India


Let $G$ be a connected split reductive group over a field $k$. Let $X\to S$ be a smooth projective morphism of $k$-schemes, with geometrically connected fibers. We formulate a natural definition of a relative canonical reduction, under which principal $G$-bundles of any given Harder-Narasimhan type $\tau$ on fibers of $X/S$ form an Artin algebraic stack $Bun_{X/S}^{\tau}(G)$ over $S$, and prove that in characteristic zero, as $\tau$ varies, these stacks define a stratification of the stack $Bun_{X/S}(G)$ by locally closed substacks. This result extends to principal bundles in higher dimensions the earlier such result for principal bundles on families of curves. The result is new even for vector bundles, that is, for $G = GL_{n,k}$. (Joint work with Sudarshan Gurjar).

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Geometric realisation for degenerations of hyperbolic structures
Ken'ichi Ohshika, Osaka University, Japan


In dimension 2, by work of Bestvina, Paulin and Skora, degenerations of hyperbolic structures can be described using projective laminations via limit R-tree actions. We consider the same problem in dimension 3. It turns out that in general there is a divergent sequence of hyperbolic structures on a 3-manifold M whose limit R-tree action is not dual to any projective lamination in M. We show even in such a case, there is a 3-manifold homotopy equivalent to M in which the limit R-action is dual to a projective lamination. We shall also touch upon its relevance to Thurston's "Broken windows only" theorem.

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Reidemeister torsion and complex volume of hyperbolic 3-manifolds
Jinsung Park, Korea Institute for Advanced Study, Korea


In this talk, I will explain a formula relating the Reidemeister torsion attached to the geometric representation with the complex volume, which is a complex valued invariant defined by the hyperbolic volume and the Chern-Simons invariant, for hyperbolic 3-manifolds.

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A complex hyperbolic Riley slice
John Parker, Durham University, UK
Tokyo Institute of Technology, Japan



(Joint work with Pierre Will) In the late 1970s Robert Riley investigated subgroups of SL(2,C) generated by two parabolic transformations. The conjugacy classes of such groups may be parametrised by one complex number. Riley investigated for which values of this complex number the group is discrete. In our work we investigate subgroups of SU(2,1) generated by two unipotent parabolic maps whose product is also unipotent. The conjugacy classes of such groups may be parametrised by two real numbers and we investigate values of these parameters where the group is discrete. We give an explicit open set where the group is discrete and free. Moreover, two consequences of our work are, first, that we prove a conjecture of Schwartz for complex hyperbolic (3,3,infinity) triangle groups and, secondly, that we give a spherical CR uniformisation of the Whitehead link complement with unipotent holonomy.

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Constructing thin surface groups in lattices in SL(3,R)
Alan Reid, University of Texas, USA


In this talk we show how to construct thin surface groups in all non-uniform lattices in SL(3,R).

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An explicit geometry of moduli spaces of Higgs bundles and Singular connections on a smooth curve and differential equations of Painlev'e types
Masahiko Saito, Kobe University, Japan


We will start by reviewing algebraic constructions of moduli spaces of stable parabolic Higgs bundles and parabolic connections on a smooth curve via GIT. Due to the works of Maruyama and Yokogawa, and Inaba, Iwasaki and Saito, one can show that the moduli spaces are smooth quasiprojective algebraic schemes with natural holomorphic symplectic structures.

We will also explain about Riemann-Hilbert correspondence from moduli spaces of parabolic connections to moduli spaces of monodromy representations.

In the later part of the talk, we will give an explicit description of moduli spaces of parabolic Higgs bundles and parabolic connections by apparent singularities and their duals (a joint work of S. Szabo). We will explain the relation between the geometry of moduli spaces and spectral curves.

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A higher rank Morse Lemma
Andrés Sambarino, Institut de Mathématiques de Jussieu-Paris Rive Gauche, France


The classical Morse Lemma in negative curvature stablishes that a quasi- geodesic is at bounded distance from a geodesic. In a recent work, Kapovich-Leeb-Porti have found an analog statement that holds for higher rank symmetric spaces. The purpose of the talk is to give a dynamical proof of this result. This is joint work with Jairo Bochi and Rafael Potrie.

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On the geometry and topology of branes through Higgs bundles
Laura Schaposnik, University of Illinois at Chicago, USA


The moduli space of Higgs bundles provides an ideal setting to study branes (Lagrangian or complex submanifolds). Along the talk we shall stroll through different paths that allow us to recover old and new results by studying spectral data associated to different spaces. We will begin by defining some interesting branes through both involutions and finite group actions. Then we will focus on real G-Higgs bundles to see how K-theory and Langlands duality can be used to describe the topological invariants. We will look at the case of SO(n,n+1)-Higgs bundles, and more generally, how one can recover the Toledo invariant, Milnor-Wood type inequalities and Cayley partners by considering auxiliary spectral curves. We will finish with some comments about products of branes and their relation to geometric structures. Some of the results presented here are in joint (and ongoing) work with D. Baraglia, S. Bradlow, S. Heller and R. Rubio.

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Anti-de Sitter geometry and polyhedra inscribed in quadrics
Jean-Marc Schlenker, University of Luxembourg, Luxembourg


Anti-de Sitter geometry is a Lorentzian analog of hyperbolic geometry. In the last 25 years a number of connections have emerged between 3-dimensional anti-de Sitter geometry and the geometry of hyperbolic sufaces. We will explain how the study of ideal polyhedra in anti-de Sitter space leads to an answer to a question of Steiner (1832) on the combinatorics of polyhedra that can be inscribed in a quadric.
Joint work with Jeff Danciger and Sara Maloni.

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Gauge-theoretic moduli spaces as spaces of stability conditions
Tom Sutherland, University of Pavia, Italy


Following the work of Seiberg and Witten, it has been understood that certain N=2 supersymmetric gauge theories in four dimensions can by "geometrically engineered" by studying string compactfications on Calabi-Yau 3-folds. In particular the BPS spectrum at a point in the vacuum moduli space of the gauge theory can be computed through (weighted) Euler characteristics of semistable objects in a Calabi-Yau-3 triangulated category with respect to a corresponding stability condition. The jumps in the piecewise-continuous BPS spectrum across walls in the moduli space can be effectively computed in the formalism of stability conditions on triangulated categories through the wall-crossing formula of Kontsevich and Soibelman.

In this talk I will introduce these ideas through a class of examples encompassing those in Seiberg and Witten's original paper where the BPS spectrum and its wall-crossing can be understood combinatorially through triangulations of marked oriented surfaces with boundary. I will discuss how the special geometry on the gauge theory moduli space (which can be viewed as the base of Hitchin's integrable system on a moduli space of meromorphic Higgs bundles on an algebraic curve) might be studied through the BPS invariants and their wall-crossing.

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Yang-Mills flow in dimension four
Alex Waldron, Stony Brook University, USA


I will describe my thesis results on YM flow in the critical dimension, and some generalizations in progress. These include long-time existence near the minimum energy, and smooth convergence to the Uhlenbeck limit in the unobstructed case. I'll then discuss the problem of uniqueness of limits. As for any geometric flow, this is very difficult to establish when singularities are present.

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SL(2,C) Higgs bundles with smooth spectral data
Michael Wolf, Rice University, USA


We show that for every nonelementary representation of a surface group into SL(2,C) there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration. Along the way in the argument, we encounter a number of constructions in the geometry of surfaces: complex projective structures, pleated surfaces, harmonic maps to R-trees and the Thurston compactification. We then turn more speculative, interpreting the 'limit configurations' of a principal stratum of the space of SL(2,C) surface group representations in terms of pleated surfaces. (Joint with Richard Wentworth and Andreas Ott, Jan Swoboda and Richard Wentworth.)

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On Atiyah's independence conjecture for four points in a hyperbolic plane
Ying Zhang, Soochow University, China


Atiyah proposed his independence conjecture in about 2000 aiming at a solution to a problem in physics. Given $n$ distinct points in a Euclidean space, a set of $n-1$ unit vectors is naturally associated to each of the given points, and, regarding each unit vector as a complex number via the stereographic projection, one obtains a monic polynomial of degree $n-1$ having as roots the $n-1$ complex numbers corresponding to the unit vectors. The conjecture asserts that the set of $n$ polynomials so obtained is linearly independent. There is a similar conjecture for points in a hyperbolic space. The conjecture has been proved for the case of four points in a Euclidean space, and the case of four points in a hyperbolic space which do not lie in a hyperbolic plane. In joint work with Jiming Ma, we have confirmed Atiyah's independence conjecture for the case of four points in a hyperbolic plane.

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Positively ratioed representations
Tengren Zhang, California Institute of Technology, USA


Given a cocycle (satisfying some conditions) on the fundamental group of a closed surface, one can construct a cross ratio whose periods agree with the periods of the cocycle. In this talk, I will explain that under some additional conditions, this cross ratio gives rise to a geodesic current m with the property that the Bonahon intersection number of m with closed curves agree with the periods of the cocycle. Applying this to maximal and Hitchin representations allows one to obtain some systolic inequalities. This is joint work with Giuseppe Martone.

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