|January 24||Ryan Budney||Isotopy in dimension 4|
|January 31||Nick Salter||Framed mapping class groups and strata of abelian differentials|
|February 7||Jacob Russell||Relative hyperbolicity in hierarchically hyperbolic spaces|
|February 14||Colin Adams||Hyperbolicity and Turaev Hyperbolicity of Classical and Virtual Knots|
|February 21||Daniele Alessandrini||Non commutative cluster coordinates for Higher Teichmüller Spaces|
|February 28||Abhijit Champanerkar||Right-angled polyhedra and alternating links|
|March 6||Peter Shalen||Quantitative Mostow Rigidity|
|March 13||Lei Chen||Actions of Homeo and Diffeo on manifolds|
|March 27||Keiko Kawamuro|
|April 3||Talia Fernos|
|April 10||Harrison Bray|
|April 24||Rylee Lyman|
|May 1||Hans Boden|
|May 15||Johanna Mangahas|
Title: Actions of Homeo and Diffeo on manifolds
Abstract: In this talk, I discuss the general question of how to obstruct and construct group actions on manifolds. I will focus on large groups like Homeo(M) and Diff(M) about how they can act on another manifold N. The main result is an orbit classification theorem, which fully classifies possible orbits. I will also talk about some low dimensional applications and open questions. This is a joint work with Kathryn Mann.
Abstract: To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume of the alternating link. We give an explicit procedure to compute the right-angled volume from any alternating link diagram, and prove that it is a new lower bound for the hyperbolic volume of the link. This is joing work with Ilya Kofman and Jessica Purcell.
Abstract: In higher Teichmuller theory we study subsets of the character varietiesof surface groups that are higher rank analogs of Teichmuller spaces,
e.g. the Hitchin components and the spaces of maximal representations.
Fock-Goncharov generalized Thurston's shear coordinates and Penner's
Lambda-lengths to the Hitchin components, showing that they have a
beautiful structure of cluster variety.
Here we apply similar ideas to Maximal Representations and we find new
coordinates on these spaces that give them a structure of non-commutative
cluster varieties, in the sense defined by Berenstein-Rethak.
This is joint work with Guichard, Rogozinnikov and Wienhard.
Abstract: We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume. We will discuss what is known about this invariant. We further employ a construction of Turaev to associate a family of hyperbolic 3-manifolds of finite volume to any classical or virtual link, even if non-hyperbolic. These are in turn used to define the Turaev volume of a link, which is the minimal volume among all the hyperbolic 3-manifolds associated via this Turaev construction. We will talk about what is known.
Title: Relative hyperbolicity in hierarchically hyperbolic spaces
Abstract: Relative hyperbolicity and thickness describe incompatible ways that the non-negatively curved parts of a metric space can be organized. In several classes of spaces (Teichmuller space, Coxeter groups, 3-manifold groups) there exists a strict dichotomy between relative hyperbolicity and thickness that produces strong geometric consequences. Behrstock, Drutu, and Mosher have thus asked for which additional classes of spaces can such a dichotomy be established. We investigate this question in the class of hierarchically hyperbolic spaces and produce a combinatorial criteria for detecting relative hyperbolicity. We apply this criteria to prove the separating curve graph of a surface has the relatively hyperbolic versus thick dichotomy.
Title: Framed mapping class groups and strata of abelian differentials
Abstract: A holomorphic 1-form on a Riemann surface admits a geometric incarnation as a so-called translation surface. The moduli spaces of translation surfaces are known as strata. The dynamics of translation surfaces is an intense area of active study, but the topological properties of strata are almost entirely unknown. I will outline some work, joint with Aaron Calderon, aimed at obtaining information about the fundamental groups of strata by means of a monodromy representation into the mapping class group. The core of our approach is a study of the ``framed mapping class group'', a natural infinite-index subgroup that, very surprisingly, turns out to admit a very simple finite set of generators.
Title: Isotopy in dimension 4
Abstract: I will describe why the trivial knot S2-->S4 has non-unique spanning discs up to isotopy. This comes from a chain of deductions that include a description of the low-dimensional homotopy-groups of embeddings of S1 in S1xSn (for n>2), a group structure on the isotopy-classes of reducing discs of S1xDn, and the action of the diffeomorphism group Diff(S1xSn) on the embedding space Emb(S1, S1xSn). Roughly speaking, these results say there is no direct translation from dimension 3 to 4, for the Hatcher-Ivanov theorems on spaces of incompressible surfaces.
- Columbia Symplectic Geometry/Gauge Theory Seminar
- All Columbia Math Dept Seminars
- CUNY Geometry and Topology Seminar
- CUNY Complex Analysis & Dynamics Seminar
- CUNY Magnus Seminar
- Princeton Topology Seminar.