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

## 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**: Free products and random walks in acylindrically hyperbolic groups

**Abstract**: The properties of a random walk on a group which acts on a
hyperbolic metric space have been well-studied in recent years. In this
talk, I will focus on random walks on acylindrically hyperbolic groups, a
class of groups which includes mapping class groups, Out(F_n), and
right-angled Artin and Coxeter groups, among many others. I will discuss
how a random element of such a group interacts with fixedsubgroups,
especially so-called hyperbolically embedded subgroups. In particular, I
will discuss when the subgroup generated by a random element and a fixed
subgroup is a free product, and I will also describe some of the geometric
properties of that free product. This is joint work with Michael Hull.

**Laure Flapan**, Northeastern

April 19

**Title**: Monodromy of Kodaira fibrations

**Abstract** A long-standing question in studying the topology of complex algebraic varieties is the question of what groups can occur as
the fundamental group of a smooth projective variety. One approximation of this question in the case of fibered varieties is to ask
what groups can occur as the monodromy group of such a fibration. We use Hodge theory to investigate this question in the case of
fibered algebraic surfaces, called Kodaira fibrations, whose fibers are all smooth and draw connections with questions about Shimura
varieties and the moduli space of smooth algebraic curves.

**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**: Transverse knots, cyclic branched cover and grid diagrams

**Abstract**: Branched covers provide a natural setting to better understand
contact manifolds and the transverse knots they contain. In this talk, we
explore how branched covers can generally be used to derivative effective
and computable invariants of transverse knots. We will present specific
example of one such construction using grid diagrams. This is joint work
with C.-M. Mike Wong.

# 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.

- 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.