August 29th: Subhojoy Gupta (Indian Institute of Technology)
Please note this meeting is in room 6495.
Title: Meromorphic quadratic differentials and geometric structures on surfaces
Abstract: Let Q(X) be the vector space of holomorphic quadratic differentials on a Riemann surface X of genus greater than one. This parametrizes the following two spaces of geometric objects on the underlying topological surface S:
First, by a theorem of Wolf, and independently Hitchin, Q(X) provides a parametrization of marked hyperbolic structures, namely the Teichmüller space of S.
Second, by a theorem of Hubbard and Masur, there is a bijective correspondence between Q(X) and measured foliations on S.
In this talk I shall describe generalisations of these results to the case when Q(X) is replaced by the space of meromorphic quadratic differentials with poles of higher order. The proofs involve harmonic maps of infinite energy.
Part of this is joint work with Michael Wolf.
September 5th: Ivan Levcovitz (CUNY Graduate Center)
Title: Coarse geometry and right-angles Coxeter groups
Abstract: A main goal of Geometric Group theory is to understand finitely generated groups up to the coarse equivalence (quasi-isometry) of their Cayley graphs. Right-angled Coxeter groups, in particular, are important classical objects that have been unexpectedly linked to the theory of hyperbolic 3-manifolds through recent results, including those of Agol and Wise. After giving the appropriate background, I will discuss what is currently known regarding the quasi-isometric classification of right-angled Coxeter groups. I will give special focus to a new computable quasi-isometry invariant and its relation to other known invariants.
September 12th: Ivan Levcovitz (CUNY Graduate Center)
Title: Coarse geometry and right-angles Coxeter groups (part 2)
September 19th: No meeting (No classes)
September 26th: No meeting (No classes)
October 3rd: Ivan Levcovitz (CUNY Graduate Center)
Title: Coarse geometry and right-angles Coxeter groups (part 3)
October 10th: Wenbo Li (CUNY Graduate Center)
Title: Quasisymmetric embeddings of Slit Sierpiński Carpets
Abstract: The study of quasiconformal geometry of fractal metric spaces has received much attention recently, In particular, the spaces homeomorphic to the standard Sierpiński carpet.
I will present a result of Hrant Hakobyan and Wenbo Li which partially answered a special situation of Kapovich-Klein conjecture. We were able to define a class of spaces (which we called Dyadic Slit Sierpiński Carpets) where we obtained a complete characterization of spaces admitting a quasisymmetric embedding into the plane. The main tools are classical and transboundary modulus of families of curves.
October 17th: Yan Mary He (University of Chicago)
Title: Free Fuchsian groups and quantititative geometry of hyperbolic surfaces
Abstract: In this talk, I will discuss a displacement constraint theorem in the spirit of Margulis Lemma for Fuchsian groups. Our main theorem gives an inequality that must be satisfied by the displacement of generators of a free Fuchsian group, which can be applied to study quantitative geometry of hyperbolic surfaces such as the two-dimensional Margulis constant, lengths of loops on hyperbolic surfaces and generalized collar lemma. Along the way, I will introduce Culler-Shalen's paradoxical decomposition of the Patterson-Sullivan measure which gives good estimates for the displacement.
October 24th: No meeting (Scheduling conflict)
October 31st: Wenbo Li (CUNY Graduate Center)
Title: Quasisymmetric embeddings of Slit Sierpiński Carpets (continued)
November 7th: Meeting cancelled
November 14th: Francesco Preta (New York University)
Title: On Bers's embedding of deformation spaces
Abstract: One of Bers's most celebrated achievements is the simultaneous uniformization of Riemann surfaces. As an immediate consequence of his construction, one obtains a canonical (up to a choice of base point) embedding of the Teichmüller space of Riemann surfaces of genus g and n punctures, as a domain of holomorphy in a complex linear space of dimension 3g-3+n. In his paper ``Finite dimensional Teichmüller spaces and generalizations'' he also announces a similar theorem for deformation spaces, analogues of Teichmüller spaces in which homotopy classes of homeomorphisms are replaced by deformations. Those are maps between nodal Riemann surfaces, which are generically one-to-one, except for possibly retracting some loops onto nodes. In my talk I will present a proof of this theorem, based on a construction with Kleinian groups. As the outline was sketched in the aforementioned paper, I will present a more detailed and complete account. This is a joint work with F. Buonerba.
November 21st: No meeting (Scheduling conflict)
November 28th: No meeting (Scheduling conflict)
December 5th: Francesco Preta (New York University)
Title: On Bers's embedding of deformation spaces
Diversity and Inclusion at the GC