Speaker: Mikhail Belolipetsky (IMPA)
Title: Computing covolumes of arithmetic lattices
Abstract: Volume is a natural measure of complexity of Riemannian manifolds and orbifolds. In these lectures, I will discuss computation of volumes of arithmetic quotient orbifolds based on G. Prasad's volume formula.

Speaker: Antonin Guilloux (IMJ-PRG)
Title: Character variety of 3-manifolds: restriction to the boundary, dimension and volume
Abstract: Let M be an oriented, compact with boundary 3-manifold - e.g. a knot complement. We will study the character variety of its fundamental group with values in SL(n,C).
I will first present a lower bound for the dimension of a (reasonably generic) component of this variety, generalizing a bound of Thurston and Culler-Shalen for SL(2,C). This is a joint work with Elisha Falbel. Next I will present the volume function on the character variety and its relevance to understand its geometry. An important tool in several situations will be the restriction of representations to fundamental groups of the boundary.

Speaker: Jeff Meyer (University of Oklahoma)
Title: Quantitative Results in Spectral Geometry of Arithmetic Hyperbolic Manifolds
Abstract: The goal of these lectures is to discuss recent quantitative results in the spectral geometry of arithmetic hyperbolic 3-manifolds. I will begin by giving a quick introduction to arithmetic 3-manifolds, connecting manifolds, their geodesics, and their totally geodesic surfaces with quaternion algebras, their maximal subfields, and their subalgebras. Next I will discuss the types of arithmetic tools that have recently been leveraged to prove quantitative results in spectral geometry. Lastly I will give an application of these techniques by looking at a quantitative answer to the following question: Do there exist standard arithmetic hyperbolic 3-orbifolds whose “short” geodesics do not lie on any totally geodesic surfaces? This is joint work with Benjamin Linowitz and Paul Pollack.

Speaker: Joan Porti (Universitat Autònoma de Barcelona)
Title: Representations and character varieties
Abstract: The set of homomorphisms of the fundamental group of a three manifold in SL(2,C) is called the variety of representations. The variety of characters encodes the conjugacy classes of the representations. Both varieties are key tools to study the geometry and topology of three manifolds.
I will start introducing the basic definitions and properties, and I will compute some examples. Then I will overview two applications. Firstly Culler-Shalen theory about essential surfaces. Secondly I will discuss an application to distinguish symmetries of knots that are free or that have fixed points (also called periods).

Speaker: Stephan Tillmann (Sydney)
Title: Multisections of manifolds
Abstract: Dave Gay and Rob Kirby recently introduced trisections of smooth 4-manifolds arising from their study of broken Lefschetz fibrations and Morse 2-functions. Dave asked us if this could be established using triangulations. We have done this and extended the theory to all dimensions. The idea is to split a 2k- or (2k+1)-manifold into k
handlebodies, such that intersections of the handlebodies have special properties. The splitting can be viewed as mapping the manifold into a k-simplex and pulling back a decomposition into dual cubes. I'll outline the construction, give some applications and conclude with open questions. This is joint work with Hyam Rubinstein.

Speaker: Genkai Zhang (Chalmers University of Technology / KIAS)
Title: Representations of rank one Lie groups and their branching under symmetric subgroups
Abstract: We study the branching rule of complementary series of rank one Lie groups U(n+1, 1; K) under the symmetric subgroup U(n, 1; K).