Abstracts/Lecture Notes

Schedule

Abstracts/Lecture Notes

Title: Properly convex manifolds and generalized cusps.

Speaker: Daryl Cooper

Abstract: We will discuss properly convex manifolds with ends that are generalized cusps. This involves the thick-thin decomposition, the classification and moduli space of generalized cusps, and the deformation theory of manifolds with these ends. This is a summary of work over many years of the author with Long, Tillmann, Leitner and Ballas. 

Title: Flags, snakes and stingrays: Configurations of flags and positivity 

Speaker: Francis Bonahon

Abstract:

The minicourse will be an exposition of the Fock-Goncharov classification of positive configurations of flags in R^n. There will be an emphasis on the nitty-gritty aspects of proofs, and the goal is to justify why positivity is a natural condition.  

Title: Slice-ribbon conjecture and cables of figure-eight knots

Speaker: Sungkyung Kang

Abstract:

Even cables of the figure-eight knot are known to be potential counterexamples of the slice-ribbon conjecture. More precisely, if the (2n,1)-cable of the figure-eight knot is (smoothly) slice, then the slice-ribbon conjecture is false. We eliminate this possibility for n=1, and then for all odd n. This consists of joint works with various authors, including Dai, Mallick, Park, Stoffregen, and Taniguchi.

Title : Structural stability of meandering hyperbolic group actions

Speaker: Sungwoon Kim

Abstract :

Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. We relax Sullivan’s axioms and introduce a notion of meandering hyperbolicity for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain nonhyperbolic groups, such as actions of uniform lattices in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust, and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions.

Title: Earthquakes on the once-punctured torus

Speaker: Grace Garden

Abstract:

We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in linear recurrence relations and uses the character variety, the second in hyperbolic geometry and uses maximal collar neighbourhoods. The two methods align, providing both an algebraic and geometric interpretation of the earthquake deformations. We convert the expressions to other coordinate systems for Teichmüller space. Examining the limiting behaviour gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.

Title: Knot Concordance and Satellite Operation 

Speaker: JungHwan Park

Abstract:

This is a survey talk on problems in knot concordance related to satellite operations. 

Title: Bifoliations on the plane and flows in 3-manifolds  

Speaker: Hyungryul Harry Baik

Abstract: The orbit space of a (pseudo-)Anosov flows in 3-manifold is an example of bifoliated plane. Barthelmé-Bonatti-Mann recently gave a necessary and sufficient condition for a pair of circle prelaminations to be induced from a so-called pA-bifoliated plane (the class of pA-bifoliated planes includes all bifoliated planes from (pseudo-)Anosov flows in 3-manifolds). We first discuss this correspondence and then discuss how to reconstruct the 3-manifold and its flow from a group action on the bifoliated plane. This talk is based on the joint work with Chenxi Wu and Bojun Zhao.    

Title: Entropy Rigidity for cusped Hitchin representations

Speaker: Tengren Zhang

Abstract: 

Let G be a geometrically finite subgroup of PSL(2,R). We say that a representation r from G to PGL(d,R) is a Hitchin representation if there is an r-equivariant positive map from the real projective line to the space of complete flags in R^d. We then prove a rigidity result for the entropy of Hitchin representations, generalizing previous work of Potrie-Sambarino. This is joint work with Richard Canary and Andrew Zimmer.

Title: Hamiltonian flows on Hitchin components and their applications

Speaker: Hongtaek Jung

Abstract:

In the first talk we review Hitchin components. We explore their smooth, and symplectic structures. We also construct special Hamiltonian flows generalizing twisting flows on Teichmuller spaces. In the second talk, we give two applications of these Hamiltonian flows. We show that the mapping class group actions on G-Hitchin components have infinite Atiyah-Bott-Goldman covolume if G=PSL(n+1,R), PSO(n+1,n), PSp(2n,R) with n>1. We also show that, for any split real forms G of a simple complex Lie group, the set of strongly dense G-Hitchin representations is generic in the space of G-Hitchin representations.

Title: Holonomies of complex projective structures on surfaces

Speaker: Thomas Le Fils

Abstract:

A complex projective structure on a surface is the datum of an atlas of charts in the Riemann sphere, with changes of coordinates that are Möbius transformations. Such a structure gives rise to a representation of the surface group in PSL(2, C): its holonomy. I will discuss a theorem of Gallo, Kapovich and Marden characterising the representations that arise in this way. I then will present a generalisation of this theorem that applies to structures with branch points.

Title: Convex real projective structures on reflection orbifolds

Speaker: Gye-Seon Lee

Abstract: 

Let O be a compact reflection n-orbifold whose underlying space is homeomorphic to a truncation n-polytope, i.e. a polytope obtained from an n-simplex by successively truncating vertices. In this talk, I will give a complete description of the deformation space of convex real projective structures on the orbifold O of dimension at least 4.  Joint work with Suhyoung Choi and Ludovic Marquis.