### Schedule

The retreat will begin at approximately 9:00 am on Monday, August 7th and end at approximately 5:00 pm on Friday, August 11th. A detailed schedule can be found here.

### Speakers and Abstracts

• Sam Ballas (Florida State) Generalized cusps in convex projective manifolds
Abstract: A generalized cusp is a generalization of a cusp of a finite volume hyperbolic manifold. These types of cusps occur naturally as the ends of properly convex projective manifolds. In this talk I will discuss the geometry of these cusps, how they can be classified, and interesting transitional properties. This is joint work with Daryl Cooper and Arielle Leitner.

• Olivier Biquard (ENS-Paris) Hitchin component for $SL(\infty,R)$
Abstract: I will describe a construction of a Hitchin component for the limit group $SL(\infty,R)$, by considering Hitchin’s selfduality equations for the group of Hamiltonian diffeomorphisms of the 2-sphere. I will also describe an approximation procedure by finite dimensional representations.

• Kenneth Bromberg (Utah) The gradient flow of renormalized volume
Abstract: Renormalized volume is a way of assigning a finite volume to an convex co-compact hyperbolic 3-manifold whose usual volume is infinite. On the space of all convex co-compact structures the renormalized volume is a smooth function whose derivative has been calculated by Krasnov-Schlenker. We will discuss the gradient flow of renormalized volume and show how studying this flow leads to implications on the geometry of the manifolds. This is joint work with M. Bridgeman and J. Brock.

• Marc Burger (ETH) Compactifying the space of maximal representations
Abstract: Let $S$ be a surface of finite genus with possibly finitely many punctures and $G=Sp(2n,R)$. We define a compactification $C(S,n)$ of the topological space of $G$- conjugacy classes of maximal representations of the fundamental group $F(S)$ of $S$ into $G$ on which the mapping class group $Map(S)$ of $S$ acts and which is functorial with respect to restriction to incompressible subsurfaces. To every point in the boundary $B(S,n)$ of $C(S,n)$ we associate canonically an action of $F(S)$ on an affine Bruhat -Tits building , well defined up to homothety, which leads via the associated length function to an equivariant projection of $C(S,n)$ onto the Parreau compactification of the $G$-equivalence classes of maximal representations. The subset $U(S)$ of $B(S,n)$ consisting of actions for which $0$ is not a limit point of the length spectrum is open and we show that Map(S) acts properly discontinously on it. The complement of $U(S)$ consist of actions which can be explained in terms of $\mathbb{R}$-tree actions of $F(S)$ with small stabilizers, and pieces belonging to $U(T)$, for subsurfaces $T$ of $S$. We also show that the the stabilizer in $Map(S)$ of any point in $B(S,n)$ is virtually abelian. This is joint work with A.Iozzi, A.Parreau and B. Pozzetti.

• Brian Collier (Maryland) The geometry of maximal $SO(2,n)$ representations
Abstract: Like Hitchin representations, the set of maximal representations of a surface group into a Lie group of Hermitian type define connected components of the character variety which consist entirely of Anosov representations. For such representations, Guichard and Wienhard have constructed geometric structures modeled on certain flag varieties. In this talk we will use Higgs bundle techniques to construct a unique maximal space like surface in the pseudo-Riemannian hyperbolic space $H^{2,n-1}$? associated to any maximal $SO(2,n)$ representation. Using the geometry of this surface, we construct geometric structures on certain homogeneous bundles on the surface and prove that they agree with those of Guichard-Wienhard. As a corollary, we also prove (a generalization of) a conjecture of Labourie for all maximal representations into rank 2 Hermitian Lie groups. Time permitting we will explore how some the these features of the maximal $SO(2,n)$ representations can be generalized to $SO(p,q)$ surface group representations. This is based mainly on joint work with Nicholas Tholozan and Jeremy Toulisse.

• Simion Filip (Harvard) Integral-affine structures, with singularities
Abstract: An integral-affine structure on a manifold is a collection of charts with transition maps in $SL(n,\mathbb{Z})$ plus translations. Such structures appear naturally in symplectic geometry on the base of a Lagrangian fibration, via the Arnold-Liouville theorem. They also appear in algebraic geometry, describing maximal degenerations of Calabi-Yau manifolds. In both cases, the integral-affine structure has singularities and this leads to a rich geometry. After providing an introduction to the above concepts, I will discuss some of the geometric properties of integral-affine structures, including their moduli spaces, and some associated dynamical systems. Based on joint work in progress with Phil Engel.

• Jeremy Kahn (Brown) Surface subgroups for non-uniform lattices

• Fanny Kassel (IHES) Convex projective structures and Anosov representations
Abstract: In this talk we will discuss recent joint work with Jeff Danciger and Franois Guritaud, investigating a notion of convex cocompactness in real projective space, and relating it to Anosov representations.

• Autumn Kent (Wisconsin) On word hyperbolic surface bundles
Abstract: There is a characterization of hyperbolicity of the fundamental group of a surface bundle due to Farb-Mosher-Hamenstaedt, namely that the bundle has hyperbolic fundamental group if and only if the fundamental group is “convex cocompact,” a notion analogous to the synonymous notion in Kleinian groups. I will discuss joint work with Bestvina, Bromberg, and Leininger that gives a new characterization of convex cocompactness, namely that the group is purely pseudo-Anosov and undistorted in the mapping class group.

• Francois Labourie (Nice) Surface groups in uniform lattices of simple complex group
Abstract: This talk will discuss joint work with J.Kahn and S.Mozes that gives the existence of many surface subgroups in certain lattices. I will also explain that a notion of quasi-symmetric mapping arises from this work.

• Christopher Leininger (UIUC) Surface bundles over Teichmüuller curves
Abstract: I will discuss joint work-in-progress with Dowdall, Durham, and Sisto on the coarse geometry of the canonical surface bundle over a Teichmüller curve with the goal of developing a notion of geometric finiteness in the mapping class group.

• Qiongling Li (Caltech) Cyclic Higgs bundles on noncompact surfaces
Abstract: In this talk, I will discuss the solutions to Hitchin equations for cyclic Higgs bundles on noncompact surfaces. In particular, I mainly focus on the uniqueness of Toda-like solutions.

• Sara Maloni (Virginia) The geometry of quasi-Hitchin symplectic Anosov representations
Abstract: In this talk we will focus on our joint work in progress with Daniele Alessandrini and Anna Wienhard about quasi-Hitchin representations in $Sp(4,C)$, which are deformation of Fuchsian representations. In particular, we will show that the quotient of the domain of discontinuity for the action of these representations on the space of complex lagrangians $Lag(C^4)$ is a fiber bundle over the surface and we describe the fiber. In particular, we will describe how the projection comes from an interesting parametrization of $Lag(C^4)$ as the space of regular ideal hyperbolic tetrahedra and their degenerations.

• Kathryn Mann (Berkeley) Characterizing Fuchsian representations by topological rigidity
Abstract: A remarkable rigidity theorem, due to Matsumoto, states that actions of surface groups on $S^1$ with maximal Euler number satisfy a strong topological rigidity property: all are semi-conjugate to each other, and semi-conjugate to discrete, faithful representations into $PSL(2,\mathbb{R})$. In this talk I will describe the proof of a converse to this theorem, characterizing Fuchsian representations by strong rigidity up-to-semiconjugacy. This is joint work with Maxime Wolff.

• Martin Möller (Frankfurt) Smooth compactifications of strata of abelian differentials
Abstract: We present a smooth complex orbifold that compactifies strata of abelian differentials with a normal crossing boundary divisor and that has a modular interpretation.

• Andrew Neitzke (Texas) Precision studies of nonabelian Hodge for irregular Higgs bundles on $CP^1$
Abstract: The nonabelian Hodge correspondence is a diffeomorphism between two moduli spaces associated to a compact complex curve: the space of Higgs bundles and the space of flat connections. In general it is difficult to determine what this diffeomorphism is concretely, say, to find the flat connection corresponding to some specific given Higgs bundle.
There is a simple “model” case for this problem, namely Higgs bundles on $CP^1$ with one irregular singularity (wild ramification). In this case I will report two recent developments:
1) the nonabelian Hodge correspondence can be solved exactly at some special points of the moduli space, for any $G = \operatorname{SU}(K)$ [joint work with Laura Fredrickson and Fei Yan],
2) the nonabelian Hodge correspondence can be solved approximately, but to high precision, for Higgs bundles lying on the Hitchin section, when $G = \operatorname{SU}(2)$ or $G = \operatorname{SU}(3)$ [joint work with David Dumas]; the results match well with conjectured asymptotic formulas coming from the technology of spectral networks.

• Beatrice Pozzetti (Warwick) The geometry of maximally framed representations
Abstract: For limits of maximal representations we show that there is a clear distinction between phenomena already present in the boundary of the Teichmüller space and new flat features. Whenever a limit representation belongs to the first case of this dichotomy, we reconstruct a canonical tree isometrically embedded in the asymptotic cone of the symmetric space, provided the latter is endowed with a suitable Finsler metric. We also describe several examples of new degenerations. Joint work with Marc Burger, Alessandra Iozzi and Anne Parreau.

• Carlos Simpson (Nice) Betti Hitchin fibration
Abstract: We’ll look at how some aspects of the Hitchin fibration show up in the structure of the neighborhood at infinity for a normal-crossings compactification of the character variety.

• Nicolas Tholozan (ENS) A Highest Teichmüller space?
Abstract: A number of authors have studied refined dynamical properties of Anosov representations by associating to them certain Hölder reparametrizations of the geodesic flow on the unit tangent bundle of a hyperbolic surface. In this talk, I will explain that the space of such reparametrizations identifies with the infinite dimensional Teichmüller space of complex structures along the weakly stable foliation of the geodesic flow. I will also discuss the geometry of this space. This is joint work with Bertrand Deroin.

• Tian Yang (Stanford) Volume Conjectures for Reshetikhin-Turaev and Turaev-Viro invariants
Abstract: In a joint work with Qingtao Chen, we conjecture that at the root of unity ${\rm exp} \, \left(2 \pi \sqrt{\frac{-1}{r}} \,\right)$, instead of the usually considered root ${\rm exp} \, \left(\pi \sqrt{\frac{-1}{r}} \,\right)$, the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially with growth rates, respectively, the hyperbolic and the complex volume of the manifold. This reveals a different asymptotic behavior of the relevant quantum invariants than that of Witten’s invariants (that grow polynomially by the Asymptotic Expansion Conjecture), which may indicate a different geometric interpretation of the Reshetikhin-Turaev invariants than the $SU(2)$ Chern-Simons gauge theory. Recent progress toward these conjectures will be summarized, including a joint work with Renaud Detcherry and Effie Kalfagianni.

• Tengren Zhang (Caltech) The Goldman symplectic form and the Hitchin component
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 $\operatorname{PSL}(V)$ 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 $\operatorname{PSL}(V)$ 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.