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

## Fall 2018

## Abstracts

**Nick Salter,** Columbia

Sept 7,

**Title**: Continuous sections of families of complex algebraic varieties

**Abstract**: Families of algebraic varieties exhibit a wide range of fascinating topological phenomena. Even families of zero-dimensional varieties
(configurations of points on the Riemann sphere) and one-dimensional varieties (Riemann surfaces) have a rich theory closely related to the theory of
braid groups and mapping class groups. In this talk, I will survey some recent work aimed at understanding one aspect of the topology of such
families: the problem of (non)existence of continuous sections of "universal" families. Informally, these results give answers to the following sorts
of questions: is it possible to choose a distinguished point on every Riemann surface of genus g in a continuous way? What if some extra data (e.g. a
level structure) is specified? Can one instead specify a collection of n distinct points for some larger n? Or, in a different direction, if one is
given a collection of n distinct points on CP^1, is there a rule to continuously assign an additional m distinct points? In this last case there is a
remarkable relationship between n and m. For instance, we will see that there is a rule which produces 6 new points given 4 distinct points on CP^1,
but there is no rule that produces 5 or 7, and when n is at least 6, m must be divisible by n(n-1)(n-2). These results are joint work with Lei Chen.

**Thang Nguyen,** NYU

Sept 14,

**Title**: Rigidity of QI-embeddings of non-uniform lattices

**Abstract**: Non-uniform lattices in Lie groups have a strong rigidity phenomenon. That is
we can describe completely all quasi-isometries by commensurators of the lattices, due
to the works or Schwartz, Eskin, and Drutu. When we study QI-embedding of a non-uniform
lattice into another non-uniform lattice, a similar rigidity also happens in some
situations. In this talk, we will state precisely the rigidity theorem and will focus
on difficulties and resolving ideas. This is from a joint work with David Fisher.

**Hung C Tran,** UGA

Sept 21,

**Title**:On the coarse geometry of certain right-angled Coxeter groups

**Abstract**: I will briefly talk about progress on the problem of quasi-isometry classification of right-angled Coxeter groups. Then I will focus on my joint work with Hoang Thanh Nguyen on the coarse geometry of
right-angled Coxeter groups which are virtually 3-manifold groups. More precisely, we study the relatively hyperbolic structure of these such groups and give a complete quasi-isometric classification
of peripheral subgroups of this structure. We give a necessary and sufficient conditions for our groups to be quasi-isometric to a right-angled Artin group.

**Tullia Dymarz,** Wisconsin

Sept 28,

**Title**: Quasi-isometries of the Baumslag-Gersten group

**Abstract**: The Baumslag-Gersten group is a commonly used example/counterexample in geometric and combinatorial group theory. For example its Dehn
function grows faster than any iterated tower of exponentials. It is a one relator group that can be viewed as the HNN extension of a solvable
Baumslag-Solitar group that identifies two different cyclic subgroups. We study quasi-isometries of this group as well as quasi-isometries of related
graphs of groups with solvable Baumslag-Solitar vertex groups and cyclic edge groups. This is joint work with Jen Taback and Kevin Whyte.

**Balazs Strenner,** Gatech

Oct 5,

**Title**: Fibrations of 3-manifolds and nowhere continuous functions

**Abstract**: Given a 3-manifold fibering over the circle, we investigate how the
pseudo-Anosov monodromies change as we vary the fibration. Fried proved that
the stretch factor of the monodromies (normalized with the Thurston norm)
varies continuously. We study how another numerical invariant, the asymptotic
translation length in the arc complex (also normalized with the Thurston norm)
varies. We show that the functions that describe how this quantity varies are
rather strange: they are nowhere continuous, but the set of accumulation points
of the graphs of these functions are, in certain cases, graphs of very simple
continuous functions such as $1/(1-x^2)$. Most of the talk will explain the
background and results using lots of pictures. I will mention a few interesting
ingredients of the proof as well — for example, the Frobenius Coin Problem.

**Javier Aramayona,** Universidad Autónoma de Madrid, Yale

Oct 12,

**Title**: On the abelianization of (pure) big mapping class groups.

**Abstract**: A classical theorem of Powell (building up on work of Birman and Mumford) asserts that the mapping class group of an orientable surface of finite topological type and genus at least
three has trivial abelianization. The first part of the talk will be devoted to explaining a proof of this result, as well as discussing the remaining low-genus cases.
We will then show that, in stark contrast, mapping class groups of infinite-type surfaces can have infinite abelianization. More concretely, we will explain how to construct non-trivial
integer-valued homomorphisms from mapping class groups of infinite-genus surfaces. Further, we will give a description the first integral cohomology group of pure mapping class groups in terms
of the first homology of the underlying surface. This is joint work with Priyam Patel and Nick Vlamis.

**Bena Tshishiku,** Harvard

Oct 19,

**Title**: Symmetries of exotic negatively curved manifolds

**Abstract**: Let N be a smooth manifold that his homeomorphic but not
diffeomorphic to a hyperbolic manifold M. How much symmetry does N have? In
particular, does Isom(M) act on N by diffeomorphisms? Farrell-Jones showed
that in general the answer is "No"; however, for their examples, it is still
possible for the orientation-preserving subgroup to act on N. In this talk,
we will discuss this problem and its relation to Nielsen realization. We
give examples of N such that Isom(M) does act on N, as well as examples
where the largest subgroup of Isom(M) that acts on N has arbitrarily large
index. This is ongoing joint work with Mauricio Bustamante.

**Daniel Woodhouse,** Technion

Oct 26,

**Title**: Revisiting Leighton's graph covering theorem

**Abstract**: Leighton's graph covering theorem states that two finite graphs with
isomorphic universal covers have isomorphic finite covers. I will discuss a
new proof that involves using the Haar measure to solve a set of gluing
equations. I will discuss generalizations to graphs with fins, and
applications to quasi-isometric rigidity.

**Ken Baker,** Miami U

Nov 2,

**Title**: Asymmetric L-space Knots

**Abstract**: Based on the known examples, it had been conjectured that all L-space knots in S3 are strongly invertible. We show this conjecture is false by constructing
large families of asymmetric hyperbolic knots in S3 that admit a non-trivial surgery to the double branched cover of an alternating link. The construction easily adapts
to produce such knots in any lens space, including S1xS2. This is joint work with John Luecke.

**Tian Yang,** Texas A&M

Nov 9,

**Title**: Some recent progress on the volume conjecture for the Turaev-Viro invariants

**Abstract**:
In 2015, Qingtao Chen and I conjectured that at the root of unity
$\exp(2\pi i/r)$ instead of the usually considered root $\exp(\pi i/r)$, the
Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic
3-manifold grow exponentially with growth rates respectively the
hyperbolic and the complex volume of the manifold. In this talk, I
will present a recent joint work with Giulio Belletti, Renaud
Detcherry and Effie Kalfagianni on an infinite family of cusped
hyperbolic $3$-manifolds, the fundamental shadow links complement, for
which the conjecture is true.

**Kathryn Mann,** Brown

Nov 16,

**Title**: Punctured mapping class group actions on the circle

**Abstract**: The mapping class group of a surface S with a marked point can be identified with the group Aut(pi_1(S)) of automorphisms of the fundamental group of the surface. I will explain a new
rigidity theorem, joint with M. Wolff, that shows that any nontrivial action of Aut(pi_1(S)) on the circle is semi-conjugate to its natural action on the Gromov boundary of pi_1(S); solving a problem
posed by Farb. As a consequence, we can also quickly recover and extend some older results on the regularity (non-smoothability) of these group actions.

**Kevin Kordek, ** Gatech

Nov 30,

**Title**: The rational cohomology of the level 4 braid group

**Abstract**: In this talk I will describe recent joint work with Dan Margalit on the rational cohomology of the level 4 braid group, a finite-index subgroup of the
braid group, which is the kernel of the mod 4 reduction of the integral Burau representation. The main result of our work is an explicit formula for the
rational first Betti number. I will also discuss some applications of this formula to the structure of the rational cohomology ring and to the structure of some
closely related groups.

**Qingtao Chen**, NYU in Abu Dhabi

Dec 7,

**Title**:

**Abstract**: In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants.
Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic length of the
corresponding closed geodesic on the pair of pants.
Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem.
I will discuss a proof of this conjecture obtained in joint work with I. Gekhtman and S. Taylor, and its generalizations
to other hyperbolic groups.

**Giulio Tiozzo**, Toronto

Dec 7,

**Title**: A central limit theorem for random closed geodesics

**Abstract**: In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants.
Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic length of the
corresponding closed geodesic on the pair of pants.
Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem.
I will discuss a proof of this conjecture obtained in joint work with I. Gekhtman and S. Taylor, and its generalizations
to other hyperbolic groups.

# Other relevant information.

## Previous semesters:

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.