Columbia Geometric Topology Seminar

Spring 2019

The GT seminar meets on Fridays at 2:00pm Fridays in room 520. We also have an overflow room 622 from 11 to 1 Fridays for additional talks.
Organizer: Walter Neumann.
Other area seminars. Our e-mail list. Archive of previous semesters

Spring 2019

Date Speaker Title
Jan 25 Khalid Bou-Rabee, CUNY On local residual finiteness of abstract commensurators of Fuchsian groups
Feb 1 David Futer, Temple Effective theorems in hyperbolic Dehn surgery
Feb 8 Aaron Calderon, Yale Mapping class groups and deformations of flat surfaces
Feb 15 Tarik Aougab, Brown Origamis from minimally intersecting filling pairs
Feb 22 Hongbin Sun, Rutgers A characterization of separable subgroups of 3-manifold groups
Mar 1 Rose Morris-Wright, Brandeis Infinite Type Artin Groups and the Clique-Cube Complex
Mar 8 Jenja Sapir, Binghamton Tessellations from long geodesics on surfaces
Mar 15 Tim Susse, Bard College Automorphism groups of graph products
Mar 29, 11:15 am (Double header) Doron Ben-Hadar, NY Lifting a generic surface to a knotted surface is NP-complete.
Mar 29 Ronno Das, Chicago Points and lines on cubic surfaces
Apr 5 Oishee Banerjee, Chicago Cohomology of the space of polynomial morphisms on \mathbb{A}^1 with prescribed ramifications
Apr 12 Caroline Abbott TBA
Apr 19 Laure Flapan, Northeastern TBA
Apr 26 Daryl Cooper, UCSB The Moduli Space of Generalized Cusps in Real Projective Manifolds
May 3 Shea Vela-Vick, LSU TBA

 

Abstracts

Khalid Bou-Rabee, CUNY
January 25
Title: On local residual finiteness of abstract commensurators of Fuchsian groups
Abstract: The abstract commensurator (aka ``virtual automorphisms'') of a group encodes ``hidden symmetries'', and is a natural generalization of the automorphism group. In this talk, I will give an introduction to these mysterious and classical groups and then discuss their residual finiteness. Recall that residual finiteness is a property enjoyed by linear groups (by A. I. Malcev), mapping class groups of closed oriented surfaces (by EK Grossman), and branch groups (by definition!). Moreover, by work of Armand Borel, Gregory Margulis, G. D. Mostow, and Gopal Prasad, the abstract commensurator of any irreducible lattice in any ``nice enough'' semisimple Lie group is locally residually finite (a property is termed ``local'' if it is satisfied by every finitely generated subgroup of the group). ``Nice enough'' is sufficiently broad that the only remaining unknown case is PSL_2(R). Are abstract commensurators of lattices in PSL_2(R) locally residually finite? Lattices here are commensurable with either a free group of rank 2 or the fundamental group of an oriented surface of genus 2. I will present a complete answer to this decades old question with a proof that is computer-assisted. Our answer and methods open up new questions and research directions, so graduate students are especially encouraged to attend. This talk covers joint work with Daniel Studenmund.

David Futer, Temple
February 1
Title: Effective theorems in hyperbolic Dehn surgery
Abstract: I will discuss two effective results about hyperbolic Dehn surgery. The first result is about with cosmetic surgery: namely, distinct long Dehn fillings on a cusped manifold cannot yield the same closed 3-manifold. The second result says that long Dehn fillings yield closed 3-manifolds with large Margulis numbers. These results are effective in the sense that all hypotheses and conclusions (such as ``long'' and ``large'') are explicitly quantified. This is joint work with Jessica Purcell and Saul Schleimer.

Aaron Calderon, Yale
February 8
Title: Mapping class groups and deformations of flat surfaces
Abstract: Flat cone metrics on surfaces (often in the guise of translation surfaces or holomorphic differentials) are a fundamental object of study in Teichmueller theory, billiard dynamics, and complex geometry. Fixing the number and angle of the cone points defines a natural subvariety of the moduli space of flat surfaces called a stratum, the global topology of which is quite enigmatic. In this talk, I will explain which mapping classes are realized by deformations contained in these strata, and how this result can be applied to classify the connected components of Teichmueller spaces of flat cone metrics.

Tarik Aougab, Brown
Feb 15
Title: Origamis from minimally intersecting filling pairs
Abstract: We consider square-tiled surfaces arising from pairs of simple closed curves on a surface of genus g with a single disk in the complement of their union. Our goal is to construct many such surfaces up to the action of the mapping class group. We'll describe two constructions (one topological and one more combinatorial) which can be used to produce factorially many (in genus) of these surfaces, improving dramatically over a previous result of the author and Huang. This represents joint work with Menasco and Nieland.

Hongbin Sun, Rutgers
February 22
Title: A characterization of separable subgroups of 3-manifold groups
Abstract: The subgroup separability is a property in group theory that is closely related to low dimensional topology, especially lifting \pi_1-injective immersed objects in a space to be embedded in some finite cover and the virtual Haken conjecture of 3-manifolds resolved by Agol. We give a complete characterization on which finitely generated subgroups of finitely generated 3-manifold groups are separable. Our characterization generalizes Liu's spirality character on \pi_1-injective immersed surface subgroups of closed 3-manifold groups. A consequence of our characterization is that, for any compact, orientable, irreducible and boundary-irreducible 3-manifold M with nontrivial torus decomposition, \pi_1(M) is LERF if and only if for any two adjacent pieces in the torus decomposition of M, at least one of them has a boundary component with genus at least 2.

Rose Morris-Wright, Brandeis
March 1
Title: Infinite Type Artin Groups and the Clique-Cube Complex
Abstract: Artin groups form a large class of groups including braid groups, free groups, and free abelian groups. Unlike their well understood cousins, Coxeter groups, many basic questions about the properties of Artin groups remain open. In this talk, I will discuss some of these open questions. Then I will introduce the clique-cube complex, a CAT(0) cube complex constructed from a given Artin group. I will discuss some of the properties of this cube complex, as well as how it can be used to show that a large class of Artin groups have trivial center and are acylindrically hyperbolic. This is joint work with Ruth Charney.

Jenja Sapir, Binghamton
March 8
Title: Tessellations from long geodesics on surfaces
Abstract: I will talk about a recent result of Athreya, Lalley, Wroten and myself. Given a hyperbolic surface S, a typical long geodesic arc will divide the surface into many polygons. We give statistics for the geometry of a typical tessellation. Along the way, we look at how very long geodesic arcs behave in very small balls on the surface.

Tim Susse, Bard College
March 15
Title: Automorphism groups of graph products
Abstract: Graph products of abelian groups simultaneously generalize and interpolate between right-angled Coxeter groups and right-angled Artin groups. In this talk we will discuss the structure of their automorphism groups by extending techniques developed in the RAAG setting. In particular, we will show that the automorphism group of a graph product of abelian groups satisfies the Tits' alternative and is residually finite. We will also prove a dichotomy, the outer automorphism group of a graph product of abelian groups is either virtually nilpotent or contains a non-abelian free subgroup, and provide graphical criteria to distinguish between these two cases. This is joint work with Andrew Sale.

Doron Ben-Hadar, NY
March 29, 11:15am (double-header)
Title: Lifting a generic surface to a knotted surface is NP-complete.
Abstract: A knot diagram is made up of a loop in 2-space that intersect itself generically -- at most two segments of the loop intersect at the same point, and they do so transversally -- along with crossing information that tells us which of the two intersecting segment lies above the other in the 3rd dimension. Similarly, knotted surfaces in 4 space are depicted using broken surface diagrams, which are made up of a generic surface in 3 space along with crossing information that tells us, whenever two segments of the surface intersect, which of them lies ``higher'' in the 4th dimension. However, while any generic loop could be ``lifted'' into a knot by giving it arbitrary crossing information, not every generic surface could be lifted into a knot. In this talk, I will explain the obstructions that can stop a surface from being liftable, and prove that the question ``is a given generic surface liftable'' is NP-complete.

Ronno Das, Chicago
March 29
Title: Points and lines on cubic surfaces
Abstract: The Cayley-Salmon theorem states that every smooth cubic surface S in CP^3 has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the moduli space M of smooth cubic surfaces. Similarly, marking a point produces a 'universal family' of cubic surfaces over M. One difficulty in understanding these spaces is that they are complements in affine space of incredibly singular hypersurfaces. In this talk I will explain how to compute the rational cohomology of these spaces. I'll then explain how these purely topological theorems have (via the machinery of the Weil Conjectures) purely arithmetic consequences: the typical smooth cubic cubic surface over a finite field F_q contains 1 line and q^2 + q + 1 points.

Oishee Banerjee, Chicago
April 5
Title: Cohomology of the space of polynomial morphisms on \mathbb{A}^1 with prescribed ramifications.
Abstract: In this talk we will discuss the moduli spaces Simp^m_n of degree n+1 morphisms \mathbb{A}^1_{K} \to \mathbb{A}^1_{K} with "ramification length < m" over an algebraically closed field K. For each m, the moduli space Simp^m_n is a Zariski open subset of the space of degree n+1 polynomials over K up to Aut (\mathbb{A}^1_{K}). It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes -- here we are prescribing, instead, the ramification data. We will also see why and how our results align, in spirit, with the long standing open problem of understanding the topology of the Hurwitz space.

Caroline Abbott,
April 12
Title: TBA
Abstract

Laure Flapan, Northeastern
April 19
Title: TBA
Abstract

Daryl Cooper, UCSB
April 26
Title: The Moduli Space of Generalized Cusps in Real Projective Manifolds
Abstract: In the study of hyperbolic 3-manifolds cusps play an important role. The geometry of a cusp is determined by a similarity structure on the boundary of the cusp. In the finite volume case, the boundary is a torus and the similarity structure is determined by a complex number with positive imaginary part. Properly-convex real-projective manifolds are a generalization of hyperbolic manifolds. In dimension 3 the moduli space of generalized cusps is a bundle over the space of similarity structures on the torus, with fiber a subspace of the space of (real) cubic differentials. Conjecturally a similar statement is true in all dimensions for cusps with compact boundary. There is a 9-dimensional cusp with fundamental group the integer Heisenberg group, and the classification of cusps with non-compact boundary is unknown. Joint: Sam Ballas, Arielle Leitner.

Shea Vela-Vick, LSU
May 3
Title: TBA
Abstract: TBA

Other relevant information.

Previous semesters:

Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, 2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.

Other area seminars.

Our e-mail list: You can subscribe here for announcements for this seminar, as well as occasional related seminars and events.