The GT seminar meets on Fridays
in Math. 520,
at 1:15PM.

Organizer:
Walter Neumann.

Other
area seminars. Our e-mail
list. Archive of previous semesters

### FALL 2011

## Abstracts.

#### September 16. Note room change: Math 622

Kristen Hendricks, “A rank inequality for knot Floer homology of branched double covers”

**Abstract**: Given a knot K in the three sphere, we compare
the knot Floer homology of (S^3, K) with the knot Floer homology of
(Sigma(K), K), where Sigma(K) is the double branched cover of the
three-sphere over K. By studying an involution on the symmetric
product of a Heegaard surface for (Sigma(K), K) whose fixed set is a
symmetric product of a Heegaard surface for (S^3, K), and applying
recent work of Seidel and Smith, we produce an analog of the classical
Smith inequality for cohomology for knot Floer homology. To wit, we
show that the rank of the knot Floer homology of (S^3,K) is less than
or equal to the rank of the knot Floer homology of (Sigma(K), K).

#### September 23

Dan Margalit, (Georgia Tech), “Mapping class groups, symplectic groups, and the Burau representation”

**Abstract:** There is a natural map from the braid group on
2g strands to the mapping class group of a surface of genus g. By
considering the action on the homology of the surface, we obtain a
representation of the braid group. This representation is nothing
other than the Burau representation evaluated at t = -1. We are
interested in the kernel of this representation, as this group is
isomorphic to the fundamental group of the branch locus of the period
mapping from Torelli space to the Siegel upper half-plane. Hain has
conjectured that this kernel is generated by Dehn twists. After
explaining the necessary background, we will provide evidence for and
progress towards this conjecture. We will also discuss other
cohomological properties of the kernel. This is joint work with Tara
Brendle, Leah Childers, and Allen Hatcher.

#### September 30

Anastasiia Tsvietkova (UTK), “An alternative approach to hyperbolic structures on link complements”

**Abstract:**
Thurston demonstrated that every link in $S^3$ is a torus link, a
satellite link or a hyperbolic link and these three categories are
mutually exclusive. It also follows from work of Menasco that an
alternating link represented by a prime diagram is either hyperbolic or a
$(2,n)$--torus link.
A new method for computing the hyperbolic structure of the complement of a
hyperbolic link, based on ideal polygons bounding the regions of a diagram
of the link rather than decomposition of the complement into ideal
tetrahedra, was suggested by M. Thistlethwaite. Although the method is
applicable to all diagrams of hyperbolic links under a few mild
restrictions, it works particularly well for alternating (non-torus)
links. The talk will introduce the basics of the method. Some applications
will be discussed, including a surprising rigidity property of certain
tangles, a new numerical invariant for tangles, and formulas that allow
one to calculate the volume of 2--bridged links directly from the diagram.

#### October 7

Jenny Wilson (Chicago) “The cohomology groups of the pure string motion group are uniformly representation stable”

**Abstract:**
The string motion group Sn, the group of motions of n
disjoint, unlinked, unknotted circles in 3-space, is a generalization
of the braid group. It can be identified with the symmetric
automorphism group of the free group on n letters. The pure string
motion group PSn, which is the analogue of the pure braid group,
admits an action by the hyperoctahedral group Wn. The rational
cohomology of PSn is not stable in the classical sense -- the
dimension of the degree k cohomology group tends to infinity as n
grows -- however, Church and Farb have recently developed a notion of
stability for a sequence of vector spaces with a group action, which
they call representation stability. Inspired by their recent work on
the cohomology of the pure braid group, they conjectured that for each
k>0, the degree k rational cohomology of PSn is uniformly
representation stable with respect to the induced action of Wn, that
is, the description of the decomposition of the cohomology group into
irreducible Wn-representations stabilizes for n>>k. In this talk, I
will give an overview of the theory of representation stability, and
briefly outline a proof verifying this conjecture. This result has
implications for the cohomology of the string motion group, and the
permutation-braid group.

#### October 14

Vaibhav Gadre (Harvard), “Curve complex translation lengths”

**Abstract:** The curve complex C(S) of a closed orientable
surface S of genus g is an infinite graph with vertices isotopy
classes of essential simple closed curves on S with two vertices
adjacent by an edge if the curves can be isotoped to be disjoint. By a
celebrated theorem of Masur-Minsky, the curve complex is Gromov
hyperbolic. Moreover, a pseudo-Anosov map f of S acts on C(S) as a
hyperbolic isometry with an invariant quasi-axis. This allows one to
define an asymptotic translation length for f on C(S). In joint work
with Chia-yen Tsai, we prove that the minimal pseudo-Anosov asymptotic
translation lengths on C(S) are of the order 1/g^2. We shall also
outline related interesting results and questions.

#### October 20, 11-12am

Matthias Kreck (Bonn) “From cutting and pasting to topological quantum field theories”

**Abstract: **
A cut and paste invariant (SK-invariant) is an invariant on closed smooth
manifolds M (oriented, Spin or with other structure) which is unchanged if
one cuts M along a codimension 1 submanifold into two pieces and glues
them differently by a structure preserving diffeomorphism. A weaker
condition is that the difference of the invariants only depends on the
gluing diffeomorphism, not on the pieces. These are called cut and paste
controlled invariants (SKK-invariants). Both the cut and paste invariants
and the cut and paste controlled invariants were determined long ago in
joint work with Karras, Neumann and Ossa. The partition function of an
invertible topological quantum filed theory is a cut and paste controlled
invariant. In joint work with Stolz and Teichner we investigate the
question, which cut and paste controlled invariants are such partition
functions. Using deep work by Galatius, Madsen, Tillmann and Weiss one can
give a complete answer. We also try to construct such field theories
explicitely. First examples in dimension 3 are explained.

#### October 21

Feng Luo (Rutgers) “Solving Thurston's equation in the real numbers”

**Abstract: ** Thurston's equation defined on triangulated
3-manifolds tends to find hyperbolic structures. It is usually solved
in the complex numbers. We are interested in solving Thurston's
equation in the real numbers and we establish a variational principle
associated to such solutions.

#### October 28

Danny Ruberman (Brandeis) “Applications of Heegaard-Floer theory to link concordance”

**Abstract: **
Invariants derived from Heegaard-Floer theory have proved to be very powerful in investigating smooth knot concordance. I will
discuss some applications of these invariants, especially the so-called correction term (or d-invariant) to problems of link
concordance. For example, answering a question of Jim Davis, I will describe links with trivial (two-variable) Alexander polynomial
and unknotted components that are not concordant to the Hopf link. I will also discuss concordance aspects of a simple cabling
construction of links.

#### November 4 9:30am

Tali Pinsky, “Knotted geodesics on Hecke triangles”

**Abstract: ** Ghys discovered in 2006 that the periodic orbits of the
modular flow are identical to the set of orbits of the Lorenz flow,
studied by Birman and Williams. Thus, for example, these orbits are
all prime knots, fibered, and have positive signature. This is the
first result regarding the topological properties of closed geodesics
on Hecke triangles, which are otherwise a well studied class. The
results were obtained using a "template" for the flow, which reduces
the original three dimensional flow to a much simpler two dimensional
system. In the talk I will present a method for computing templates
for geodesic flows on an infinite class of Hecke triangle groups. We
will then use the arising templates to show that as knots, all closed
geodesics corresponding to any of these groups are prime.

#### November 4

Joan Licata, “Legendrian knot invariants in Seifert fibered spaces”

**Abstract: ** Knot theory beyond the three-sphere has seen
increased attention in recent years, and in this talk I will focus on
knot theory in Seifert fibered spaces. In particular, we'll consider
Legendrian knots in Seifert fibered spaces equipped with a special
contact form. This setting gives rise to both topological and contact
geometric questions, and I'll describe some of the ingredients used to
prove the Legendrian non-simplicity of an infinite family of knot
types representing torsion homology classes. This is joint work with
J. Sabloff.

#### November 11

Anne Pichon (Marseille), “Bilipschitz classification of singularities of complex surfaces”

**Abstract:** This is a joint work with Lev Birbrair and
Walter Neumann.

We study the geometry of a normal complex surface \(X\) in a
neighbourhood of a singular point \(p \in X\). It is well known that
for all sufficiently small \(\epsilon > 0\) the intersection of \(X\)
with the sphere \(S^{2n-1}_\epsilon\) of radius \(\epsilon\) about
\(p\) is transverse, and \(X\) is therefore locally "topologically
conical," i.e., homeomorphic to the cone on its link \(X\cap
S^{2n-1}_\epsilon\). However, as shown by Birbrair and Fernandez,
\((X,p)\) need not be "metrically conical", i.e., bilipschitz
equivalent to a standard metric cone when \(X\) is equipped with the
Riemanian metric induced by the ambient space. In fact, it was shown
by Birbrair, Fernandez and Neumann that it rather rarely is.

I will present a complete classification of the bilipschitz geometry
of \((X,p)\). It starts with a decomposition of a normal complex
surface singularity into its "thick" and "thin" parts. The
former is essentially metrically conical, while the latter shrinks
rapidly in thickness as it approaches the origin. The thin part is
empty if and only if the singularity is metrically conical. Then the
complete classification consists of a refinement of the thin part
into geometric pieces. I will describe it on an example, and I will
present a list of open problems related with this new point of view
on classifying complex singularities.

#### November 18

Greg Kuperberg (Davis), “Denseness and Zariski denseness of Jones braid representations”

**Abstract: **For several applications in computer science, it is useful to
know that various braid group representations in quantum algebra are dense.
The first and still most important result of this type is the theorem
of Freedman, Larsen, and Wang, that the Jones braid representations at
principal roots of unity are dense as unitary representations. However,
the original proof was in the setting of compact groups, and depended on the
classification of finite simple groups. I will discuss another approach
which is successful in both compact and non-compact cases. One idea is
to work at the level of Lie algebras and their representations whenever
possible. A second idea is to change the question to the easier case of
the Zariski topology, and postpone the analytic topology until the end.
A third idea is to rely on a powerful surjectivity lemma of Phillip Hall,
adapted to the setting of Lie groups or algebraic groups.

If time permits, I can also discuss an important negative consequence of
the denseness results, that any fair approximation to a dense value of
the Jones polynomial is #P-hard.

#### December 2

Jeff Brock (Brown), “The Weil-Petersson metric and the geometry of hyperbolic 3-manifolds”

**Abstract: **
Many coarse relationships support the existence of a deep connection
between the synthetic geometry of the Weil-Petersson metric and the
geometry of hyperbolic 3-manifolds. In this talk, I'll elaborate on
some new, finer examples of such connections, as well as some
contrary evidence to a fundamental link. This talk will present joint
work with Yair Minsky and Juan Souto.

#### December 9

Harold Sultan (Columbia), “Large Scale Geometry of Teichmuller Space; Thickness and Divergence”

**Abstract: **
I will talk about the asymptotic geometry of Teichmuller
space equipped with the Weil-Petersson metric. In particular, I will
give a criterion for determining when two points in the asymptotic
cone of Teichmuller space can be separated by a point; motivated by a
similar characterization in mapping class groups by
Behrstock-Kleiner-Minsky-Mosher and in right angled Artin groups by
Behrstock-Charney. I will also explain two new ways to uniquely
characterize the Teichmuller space of the genus two once punctured
surface amongst all Teichmuller spaces: one is that it is thick of
order two and the other is that it has a divergence function which is
superquadratic yet subexponential.

#### December 16

Martin Bridson (Oxford), “Profinite completions of groups, geometry, and decision problems”

**Abstract: **
I'll discuss the geometric background to the questions: what
can one say about groups with the same profinite completion, and which
properties of a finitely presented group are visible in its profinite
completion? I shall describe recent results in this direction.

I'll then sketch a proof of a recent theorem with Henry Wilton that
implies, among other things, that the following problems for finitely
presented discrete groups G (even CAT(0) groups) are undecidable:

1. Does G have a non-trivial finite quotient?

2. Is every subgroup of finite index in G perfect?

Moreover, the isomorphism problem for finitely presented
residually-finite groups with a fixed profinite completion is unsolvable.

# Other relevant information.

## Previous semesters:

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.

## Our e-mail list.

Announcements for this seminar, as well as for related seminars and events, are sent to the GT seminar mailing list. You can subscribe directly or by contacting Walter Neumann.