### 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.

1. Michelle Bucher: Bounded cohomology and volumes of representations
2. Jeff Danciger: Geometric structures on manifolds
3. Charles Frohman: Quantum topology
4. Andres Sambarino: Anosov representations
5. Laura Schaposnik: An introduction to Higgs bundles
6. Alex Wright: Dynamics on moduli spaces

The second part of the program will consist of 30 minute talks given by graduate students. If you are graduate student and interested in giving a talk you should submit an abstract when you register for the retreat.

### 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.

### Minicourse Abstracts

1. Michelle Bucher: Bounded cohomology and volumes of representations
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
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: Dynamics on moduli spaces

### Schedule

A detailed schedule will be posted here closer to the event.