### Minicourses

The first component of the program will consist of 6 minicourses covering a broad range of topics. Each minicouse will consist of two 90 minute lectures by the speaker. These lectures will be supplemented by problems sessions led by the speakers and graduate student assistants.

1. Michelle Bucher: Bounded cohomology and rigidity
Assisted by: Elia Fioravanti and Alessio Savini
2. Jeff Danciger: Geometric structures on manifolds
Assisted by: Martin Bobb and Jean-Philippe Burelle
3. Charles Frohman: Quantum topology
Assisted by: Daniel Douglas and Thomas Kindred
4. Andres Sambarino: Anosov representations: some general aspects
Assisted by: Giuseppe Martone and Nicolaus Treib
5. Laura Schaposnik: An introduction to Higgs bundles
Assisted by: Johannes Horn and Georgios Kydonakis
6. Alex Wright: The $\operatorname{GL}(2,\mathbb{R})$ action on the Hodge bundle
Assisted by: Florent Ygouf and Benjamin Dozier

The second part of the program will consist of 30 minute talks given by graduate students.

### Minicourse Abstracts

1. Michelle Bucher: Bounded cohomology and rigidity

Abstract: Bounded cohomology was introduced by Gromov in the beginning of the 80’s. One of its first striking application was an elegant proof by Gromov and Thurston of Mostow Rigidity for hyperbolic manifolds based on Haagerup and Munkholm’s result that hyperbolic simplices of maximal volume are the ideal regular simplices. In this mini course we will see which properties of bounded cohomology make it a natural tool for the study of rigidity of representations. The aim will be to give a detailed proof of the rigidity of representations of lattices in $\operatorname{SL}(2,\mathbb{C})$ into $\operatorname{SL}(m,\mathbb{C})$ (the case $m=2$ leading back to Mostow Rigidity), which is a joint result with M. Burger and A. Iozzi.

Lecture 1: Overview on characteristic numbers of representations and rigidity

Lecture 2: Background on bounded cohomology

Lecture 3: A cocycle representing the degree 3 Borel class for $\operatorname{SL}(m,\mathbb{C})$

Lecture 4: Detailed proof of the main rigidity result

2. Jeff Danciger: Geometric structures on manifolds

Abstract: Let $X$ be a homogeneous space of the Lie group $G$. We will introduce the notion of a $(G,X)$ structure on a manifold $M$. To each such structure is associated a representation, called the holonomy representation, of the fundamental group of $M$ in $G$. We discuss the ways in which the properties of the $(G,X)$ structure may be reflected in the properties of the holonomy representation and vice versa. We will briefly survey several well-studied examples and then discuss the ongoing effort to find natural $(G,X)$ structures associated to various special representations.

3. Charles Frohman: Quantum topology

Abstract: A central question of quantum topology is how the hyperbolic volume of a three manifold with a hyperbolic structure is reflected in its quantum invariants. The quantum invariants of a manifold can be understood as an example of a rigorously defined path integral (Feynman integral) over the states of a quantum system. The quantum invariants of a three manifold correspond to the computation of a partition function in a quantum field theory- a theory that models physical situations where quantum and relativistic effects are both significant.

Quantum hyperbolic geometry corresponds to a simplified physical setting where only quantum mechanical phenomena are in play. Let (M, $[\rho]$) consist of an oriented three manifold, and $[\rho]$ the conjugacy class of $\rho : \pi_1(M, \ast) \to \operatorname{SL}(2, \mathbb{C})$ a homomorphism from the fundamental group of $M$ into $\operatorname{SL}(2, \mathbb{C})$. Quantum hyperbolic geometry assigns to the pair $(M,[\rho])$ a numerical invariant. If $\rho$ is a lifting of holonomy of a hyperbolic structure on $M$, then the invariant assigned to $(M, [\rho])$ can be thought of as a quantization of the complex Chern-Simons invariant of the structure. More simply, you can think of quantum hyperbolic geometry as assigning a holomorphic function for each odd n ∈ N to a Zariski open subset of the $\operatorname{SL}(2, \mathbb{C})$-character variety of the fundamental group of $M$.

These lectures will be an introduction to quantum hyperbolic geometry. The approach we will take is based on the work of Bonahon, Wong and Liu. In the first lecture we will focus on the representation theory of algebras that are of the type produced by geometric quantization. After an overview of the representation theory of algebras, we show that irreducible representations of nice algebras are generically in finite to one correspondence with the points of the underlying space. Hence it makes sense to associate a quantum object to a classical object. In the second lecture we will introduce the Kauffman bracket skein algebra of a finite type surface, and explain how it is a quantization of the coordinate ring of the $\operatorname{SL}(2, \mathbb{C})$ character variety of $1(F)$. In the third lecture we will show that when the parameter in the Kauffman bracket skein algebra is set equal to a root of unity, it acts like a geometric quantization. In the last lecture we will use this to define quantum hyperbolic invariants of $(M, [\rho])$ where $M$ is a three manifold that fibers over the circle with fiber a finite type surface. In the activity we will compute quantum hyperbolic invariants for the figure eight knot equipped with a representation of its fundamental group into $\operatorname{SL}(2, \mathbb{C})$.

4. Andres Sambarino: Anosov representations: some general aspects

Abstract: The course is intended as a broad-audience introduction to the subject. Anosov representations were introduced by Labourie around 2006 and further developed by Guichard-Wienhard. They are representations of Gromov-hyperbolic groups into semi-simple Lie groups verifying a dynamical condition. (As we will see) This condition is analogous, in non-positive curvature, of what is known in (pinched) negative curvature as ‘convex-cocompact’.

We will prove a recent characterization of Anosov representations, due to Kapovich-Leeb-Porti, as quasi-isometrically embedded groups ‘whose eigenvalues miss some wall of the Weyl chamber’. We will then construct a natural flow associated to a given such representation. This flow provides several ways to distinguish two Anosov representations which in turn give canonical shapes to the deformation space.

If time permits, we will further study rigidity results for the topological entropy of this flow.

The main references for the topics covered are:

Bochi-Potrie-S: Anosov representations and dominated splittings

Bridgeman-Canary-Labourie-S: The pressure metric for Anosov representations

Guéritaud-Guichard-Kassel-Wienhard: Anosov representations and proper actions

Guichard-Wienhard: Anosov representations: domains of discontinuity and applications

Kapovich-Leeb-Porti: A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings

5. Laura Schaposnik: An introduction to Higgs bundles

Abstract: During the first two lectures we shall introduce classical Higgs bundles and the Hitchin fibration, and describe the associated spectral data in the case of principal Higgs bundles for classical complex Lie groups. Whilst bibliography is provided in the text, the main references followed are Hitchin’s papers [Hit87, Hit87a, Hit92, Hit07]. During the last two lectures, we shall construct Higgs bundles for real forms of classical complex Lie groups as fixed points of involutions, and describe their appearance inside the Hitchin fibration. Along the way, we shall mention different applications and open problems related to the methods introduced in all lectures, including the study of other interesting subspaces of the moduli space of Higgs bundles. For the first three lectures, a useful reference will be https://arxiv.org/abs/1408.0333, and for the last one https://arxiv.org/abs/1603.06691.

6. Alex Wright: The $\operatorname{GL}(2,\mathbb{R})$ action on the Hodge bundle

Abstract: Abstract: We will begin with the geometry of translation surfaces (a.k.a. Abelian differentials), and build up to topics surrounding the theorem of Eskin-Mirzakhani-Mohammadi on $\operatorname{GL}(2,\mathbb{R})$ orbit closures in moduli spaces of translation surfaces.

Some introductory materials can be found below:

### Student Talk Abstracts

• Dmitri Gekhtman (Caltech) Holomorphic retractions onto Teichmuller disks
By recent work of Markovic, most Teichmuller disks are not holomorphic retracts of Teichmuller space. On the other hand, Kra earlier proved that a Teichmuller disk is a holomorphic retract if it is generated by a quadratic differential with no odd-order zeros. We conjecture the converse: if a quadratic differential has an odd-order zero, then the disk it generates is not a retract. Using dynamical results of Minsky, Smillie, and Weiss, we show that it suffices to consider disks generated by Jenkins-Strebel differentials. We then prove a complex-analytic criterion characterizing Jenkins-Strebel differentials which generate retracts. Finally, we use this criterion to prove the conjecture for the five-times punctured sphere and twice-punctured torus.

• Andrea Tamburelli (University of Luxembourg) The volume of GHMC Anti-de Sitter manifolds
In this talk I will introduce GHMC Anti-de Sitter manifolds, trying to underline the many similarities with hyperbolic quasi-Fuchsian manifolds. I will then focus on the study of their volume, which turns out to be related to $L^1$ energy between hyperbolic surfaces.

• Qiyu Chen (Sun Yat-sen University) Constant Gauss curvature foliations of AdS spacetimes with particles
Motivated by the work of Barbot, Béguin and Zeghib about the K-foliations (constant Gauss curvature foliations) of 3-dimensional globally hyperbolic maximal compact (GHMC) spacetimes of constant curvature, we study the analogous question for convex GHM AdS manifolds with particles (cone singularities of angles less than $\pi$ along time-like curves).
We will show that the complement of the convex core in a convex GHM AdS manifold with particles admits a unique K-foliation. This extends, and provides a new proof of, a result of Barbot, B'{e}guin and Zeghib. As an application of this result, we generalize to hyperbolic surfaces with cone singularities (of angles less than $\pi$) a number of results concerning landslides, which are smoother analogs of earthquakes sharing some of their key properties. This is a joint work with Jean-Marc Schlenker.

• Hongtaek Jung (KAIST) Expression of symplectic forms of $\operatorname{PSL}(4,\mathbb{R})$ Hitchin components
We will first recall the construction of the symplectic forms on the $\operatorname{PSL}(4,\mathbb{R})$-character varieties of surface groups due to Atiyah, Bott and Goldman. We know that the Fenchel-Nielsen coordinates of the $\operatorname{PSL}(2,\mathbb{R})$-Hitchin components are symplectic. After brief review of these results, we will propose, using some Mathematica computations, parameterizations of the $\operatorname{PSL}(4,\mathbb{R})$-Hitchin components where the symplectic forms can be explicitly written. We will use machineries from the Bonahon-Dreyer parameterizations and the previous work of Goldman, Zocca, and Hongchan Kim.

• Stefano Riolo (University of Pisa) Hyperbolic Dehn filling in dimension four
By gluing copies of a deforming polytope, we describe some deformations of complete, finite-volume hyperbolic cone 4-manifolds. Despite the fact that hyperbolic lattices are locally rigid in dimension greater than three (Garland-Raghunathan), we see a four-dimensional instance of Thurston’s hyperbolic Dehn filling: a path of cone-manifolds Mt interpolating between two cusped hyperbolic 4-manifolds $M_0$ and $M_1$. This is a joint work with Bruno Martelli.

• Michelle Chu (University of Texas at Austin) Essential surfaces and intersections in the character variety
I will describe the $\operatorname{SL}(2,\mathbb{C})$ character variety for a family of hyperbolic knots. These character varieties have multiple components which intersect at points corresponding to non-integral irreducible representations. These carry lots of interesting topological information. In particular, they are associated to splittings of the fundamental group along essential surfaces.

• Brian Freidin (Brown) Harmonic maps and Rigidity
Harmonic maps have found rich applications in the theory of geometric rigidity. Following the works of Gromov-Schoen and Korevaar-Schoen, we describe the theory of harmonic maps into singular spaces and how the geometry of spaces of non-positive curvature can be used to approach rigidity problems.

• Florestan Martin-Baillon (ENS Paris) Domination of CAT(-1) surface group representations by Fuchsian ones
We explain how for every representation of a surface group into a negatively curved metric space, we can find a Fuchsian representation which is uniformly bigger, in an appropriate sense. This is a generalization of works of Deroin-Tholozan and Guéritaud-Kassel-Wolff.

• Léo Brunswic (Université d’Avignon et des Pays de Vaucluse) Alexandrov Theorem for singular spacetimes
Alexandrov theorem states that any singular locally euclidean metric on the sphere with cone angles lesser than $2\pi$ can be realized as the boundary of a convex polyhedra in $\mathbb R^3$. This theorem has been extended in many ways. We show that modern technics developped by Volkov, Izmetiev and Fillastre give rise to a version of Alexandrov Theorem for singular spacetimes. Let $\Sigma$ be a closed singular locally Euclidean surface of genus $g\geq 2$ with $s>0$ cone angles $(\theta_i)_{1\leq i \leq s}$ all bigger than $2\pi$ and let $(\kappa_i)_{1\leq i \leq s}$ lesser than $2\pi$. There a couple $(M,\iota)$ with $M$ is a singular radiant locally minkowski maximal 3-manifold with $s$ massive particles of angles $(\kappa_i)_{1\leq i \leq s}$ and where $\iota$ is an isometric embedding of $\Sigma$ into $M$ such that $\iota(\Sigma)$ is a Cauchy-surface of $M$ and the boundary of a convex polyhedra in $M$. Furthermore, this couple is essentially unique.

• Jean-Philippe Burelle (University of Maryland) Free groups acting on cyclically ordered spaces
Schottky groups are a classical combinatorial way of constructing free group actions on $CP^1$ or $RP^1$. I will define a generalization of this construction to sets admitting a type of ternary relation called a partial cyclic order. Then, I will discuss examples and an application to maximal representations of surfaces with boundary into Lie groups of Hermitian type.

• Marissa Loving (University of Illinois at Urbana-Champaign ) Bounding the Least Dilatation of Pure Surface Braids
Pure surface braid groups are most often defined as the fundamental group of the configuration space of points on a surface. However, pure surface braids can also be viewed as elements of the pure mapping class group of a punctured surface, which makes it natural to consider their dilatation. Currently there are no known bounds on the dilatation of pure surface braids that are independent of both genus and the number of punctures. In fact, the current bounds tend to zero as the number of punctures increases. We give a constant lower bound and a constant upper bound in the case of a sufficient number of punctures. We also provide a way to interpolate between the constant upper bound in the case of a sufficient number of punctures and the log(g) upper bound given by Aougab and Taylor in the case of a single puncture.

• Nicolaus Treib (University of Heidelberg) Schottky groups in $\mathrm{SL}(n,\mathbb{R})$
Partial cyclic orders allow for a natural generalization of classical Schottky groups in $\mathrm{PSL}(n,\mathbb{R})$. After defining the required notions, I will focus on one example: The space of complete oriented flags in $\mathbb{R}^n$ admits a partial cyclic order with strong connections to total positivity. I will explain how this gives rise to Schottky-type representations of free groups into $\mathrm{SL}(n,\mathbb{R})$ if $n$ is odd. These representations admit continuous boundary maps into the space of oriented flags. Furthermore, there exists a domain of discontinuity in the sphere $S^{n-1}$, obtained by removing a set of half-hyperplanes determined by the limit set.

• Florian Stecker (Heidelberg University) Domains of discontinuity of Anosov representations
A representation of an infinite discrete group Gamma into a semisimple Lie group G gives an action of Gamma on any flag manifold of G. This action is never properly discontinuous, but sometimes one can find open subsets on which it is. For Anosov representations and certain flag manifolds, Guichard-Wienhard and Kapovich-Leeb-Porti have constructed cocompact examples of such domains of discontinuity. But are these all flag manifolds where such domains exist? And how many different domains are there? I want to answer this at least for Hitchin representations, and discuss what is different in the general case.

• Lorenzo Ruffoni (University of Bologna) Bubbling complex projective structures with quasi-Fuchsian holonomy
A branched complex projective structure is a geometric structure on a surface which is locally modelled on $(PSL(2,C),CP^1)$, possibly with integral conical singularities; motivating examples come from the study of constant curvature metrics and linear ODEs on Riemann surfaces. We investigate the interactions between some geometric surgeries which can be performed on a given structure without changing its holonomy; we show in particular that bubbling (i.e. taking a connected sum with a copy of the Riemann sphere) is essentially enough to describe structures with quasi-Fuchsian holonomy and at most two branch points.

• Thomas Kindred (University of Iowa) Plumbing is a natural operation
The operation of plumbing, also called Murasugi sum, involves gluing two spanning surfaces (unoriented, embedded surfaces with boundary) $F,F'\subset S^3$ along disks so as to produce a new spanning surface that some sphere cuts into $F,F'$. For example, any geometrically $\partial$-compressible spanning surface decomposes as a connect sum—the simplest type of plumbing—in which one factor is a mobius band spanning the unknot. Following work of Stallings, Gabai, and Ozawa, this talk will survey some basic properties of spanning surfaces and plumbing, while allowing lots of time with pictures. We will show how checkerboard surfaces, if they de-plumb, do so in an “obvious” way, and we will examine plumbing structures on state surfaces, with an eye toward Gordon-Litherland linking pairings and Khovanov homology.

• Gianluca Faraco (University of Parma) Complex projective structures with maximal number of M"obius transformation
Abstract: In this talk we consider the group of projective automorphisms for a complex projective structure on a Riemann surface. Projective automorphisms are in particular biholomorphisms for the underlying conformal structure, therefore their number subject to the classical Hurwitz bound. We show the existence of structures achieving the maximal possible number of projective automorphisms, i.e. the Hurwitz bound can not be improved. This leads to adopt a relative point view and look for which projective structures the group of projective automorphisms coincides with the group of biholomorphisms. We show that Galois Bely\u{\i} curves are precisely those Riemann surfaces admitting a unique complex projective structure invariant under the full group of biholomorphisms. This is a joint work with Lorenzo Ruffoni.

• Leona Sparaco (Florida State University) The Eigenvalue Variety and the Geometry of Hyperbolic Link Complements
Let $M$ be an orientable hyperbolic manifold of finite volume. The $SL_{2}(\mathbb{C})$ character variety of $M$ is essentially the set of all representations $\rho : \pi_{1}(M) \rightarrow SL_{2}(\mathbb{C})$ up to trace equivalence. This algebraic set is connected to many geometric properties of the manifold $M$. In this talk we will examine some properties of the character variety of $M$ when $M$ is a link complement with a non-trivial symmetry. We will do this using properties of the quotient orbifold as well as the eigenvalue variety, which is a generalization of the $A$-$polynomial for an$n-component link.

• Benjamin Dozier(Stanford) Equidistribution of saddle connections on translation surfaces
I will show that on any translation surface, the collection of saddle connections (straight segments connecting singular points) of length at most R becomes more and more evenly distributed on the surface as R tends to infinity. This implies that on a rational-angled billiard table, if a player takes the shortest shot that starts and ends at a corner, then the next shortest, and so on, the table will get worn evenly. The proof will use a new result about angles of saddle connections, together with the classic result of Kerckhoff-Masur-Smillie that the directional flow on a surface is uniquely ergodic in almost every direction.

• Jonas Beyrer (University of Zürich) Cross ratios on higher rank symmetric spaces
Rank one symmetric spaces of non-compact type carry a natural cross ratio on the ideal boundary which has many different applications. We want to generalize this to higher rank symmetric spaces. A similar geometric construction as for the rank one case yields a collection of generalized cross ratios on the Furstenberg boundaries. We will collect some facts about those cross ratios. In particular we motivate that those are suitable generalizations by the fact that they characterize the isometry group, as in the rank one case.

• Tommaso Cremaschi (Boston College) Hyperbolization on infinite type 3-manifold
We will answer the following question by Ian Agol. Is there a 3-dimensional manifold $M$ with no divisible subgroups in $\pi_1(M)$ that is locally hyperbolic but not hyperbolic? Specifically we construct an example of such a 3-manifold. By looking at a particular class of 3-manifolds, that contains our example, we will be able to characterize the hyperbolizable ones by looking at their topology.

### Schedule

The schedule for the Junior Retreat can be found here.

### Funding

Funding will preferentially go to GEAR members and graduate students/postdocs at GEAR nodes; excess resources may be available for other participants. Applications received by April 1 are guaranteed full consideration, although applications will be accepted after this target date.

If you receive GEAR funding you must buy a ticket with a US air carrier or provide documentation of why this is unreasonable.

We will update information on the reimbursement procedure very soon. Please, keep all your receipts safe in the meanwhile.

Please note that GEAR will not be able to reimburse participants for lodging on the evening of Saturday July 29.