Geometry/Topology Seminar
Spring 2014
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday April 3 at 34pm in Eck 308
 Stergios Antonakoudis, Harvard University
 The intrinsic geometry of Teichmuller space and symmetric
domains

Abstract: In this talk we will look at complex
domains in their intrinsic Kobayashi metrics with a goal to
contribute to the following important question: To what
extent does the geometry of Teichmuller space resemble the
geometry of a bounded symmetric domain?

 Monday April 7 at 2:303:30 pm in Ry358
 Satyan Devadoss, Stanford University
 Phylogenetic networks and the real moduli space of curves

Abstract: Infused with visual imagery, this talk is
motivated by the the moduli spaces of punctured Riemann
spheres. In the 1970s, Grothendieck, Deligne, and Mumford
constructed their compactification using Geometric Invariant
Theory. In the 1990s, Gromov and Witten utilized them as
invariants arising from string field theory and quantum
cohomology. We consider real points of these spaces, which
have elegant geometric and combinatorial properties, being
compact hyperbolic manifolds with a beautiful tessellation
by convex polytopes. In recent years, they have gained
importance in their own right, appearing in areas such as
representation theory, geometric group theory, tropical
geometry, and lately reinterpreted as spaces of phylogenetic
networks. In particular, they resolve the singularities of
tree spaces studied by Billera, Holmes, and Vogtmann.

 Tuesday April 8 at 34 pm in E308
 Jonathan Hillman, Sydney
 Poincare duality in low dimensions (1)

Abstract: The goal of this minicourse is to show how
Poincare duality imposes constraints on and interactions
between the two most basic invariants of algebraic topology,
namely, the Euler characteristic \chi(M) and the
fundamental group \pi_{1}(M), for
M a manifold (or a Poincare duality complex) of
dimension 3 or 4. (In lower dimensions these invariants
determine each other, while in higher dimensions they are
decoupled.) We shall concentrate on recent work and open
questions. In dimension 3 there is a simple system of
homotopy invariants, and every orientable
PD_{3}complex is a connected sum of
indecomposables. These are either aspherical or have
virtually free fundamental group. The latter class is
largely known, and the main question here is whether every
aspherical PD_{3}complex is homotopy
equivalent to a closed 3manifold. In dimension 4 the
situation is less clear, but aspherical
PD_{4}complexes may be characterised in
terms of \pi and \chi, while the
fundamental group, StiefelWhitney classes and the homotopy
intersection pairing determine the homotopy type in many
cases with c.d.\pi≤2.

 Thursday April 10 at 34 pm in E308
 Jonathan Hillman, Sydney
 Poincare duality in low dimensions (2)

Abstract: The goal of this minicourse is to show how
Poincare duality imposes constraints on and interactions
between the two most basic invariants of algebraic topology,
namely, the Euler characteristic \chi(M) and the
fundamental group \pi_{1}(M), for
M a manifold (or a Poincare duality complex) of
dimension 3 or 4. (In lower dimensions these invariants
determine each other, while in higher dimensions they are
decoupled.) We shall concentrate on recent work and open
questions. In dimension 3 there is a simple system of
homotopy invariants, and every orientable
PD_{3}complex is a connected sum of
indecomposables. These are either aspherical or have
virtually free fundamental group. The latter class is
largely known, and the main question here is whether every
aspherical PD_{3}complex is homotopy
equivalent to a closed 3manifold. In dimension 4 the
situation is less clear, but aspherical
PD_{4}complexes may be characterised in
terms of \pi and \chi, while the
fundamental group, StiefelWhitney classes and the homotopy
intersection pairing determine the homotopy type in many
cases with c.d.\pi≤2.

 Tuesday April 15 at 34 pm in E203
 Jonathan Hillman, Sydney
 Poincare duality in low dimensions (3)

Abstract: The goal of this minicourse is to show how
Poincare duality imposes constraints on and interactions
between the two most basic invariants of algebraic topology,
namely, the Euler characteristic \chi(M) and the
fundamental group \pi_{1}(M), for
M a manifold (or a Poincare duality complex) of
dimension 3 or 4. (In lower dimensions these invariants
determine each other, while in higher dimensions they are
decoupled.) We shall concentrate on recent work and open
questions. In dimension 3 there is a simple system of
homotopy invariants, and every orientable
PD_{3}complex is a connected sum of
indecomposables. These are either aspherical or have
virtually free fundamental group. The latter class is
largely known, and the main question here is whether every
aspherical PD_{3}complex is homotopy
equivalent to a closed 3manifold. In dimension 4 the
situation is less clear, but aspherical
PD_{4}complexes may be characterised in
terms of \pi and \chi, while the
fundamental group, StiefelWhitney classes and the homotopy
intersection pairing determine the homotopy type in many
cases with c.d.\pi≤2.

 Thursday April 17 at 34 pm in E308
 Jonathan Hillman, Sydney
 Poincare duality in low dimensions (4)

Abstract: The goal of this minicourse is to show how
Poincare duality imposes constraints on and interactions
between the two most basic invariants of algebraic topology,
namely, the Euler characteristic \chi(M) and the
fundamental group \pi_{1}(M), for
M a manifold (or a Poincare duality complex) of
dimension 3 or 4. (In lower dimensions these invariants
determine each other, while in higher dimensions they are
decoupled.) We shall concentrate on recent work and open
questions. In dimension 3 there is a simple system of
homotopy invariants, and every orientable
PD_{3}complex is a connected sum of
indecomposables. These are either aspherical or have
virtually free fundamental group. The latter class is
largely known, and the main question here is whether every
aspherical PD_{3}complex is homotopy
equivalent to a closed 3manifold. In dimension 4 the
situation is less clear, but aspherical
PD_{4}complexes may be characterised in
terms of \pi and \chi, while the
fundamental group, StiefelWhitney classes and the homotopy
intersection pairing determine the homotopy type in many
cases with c.d.\pi≤2.

 Tuesday April 22 at 34pm in Eck 308
 Weiwei Wu, Michigan State University
 Symplectic mapping class groups in dimension four, and their
relations with Lagrangians

Abstract: This talk will survey some recent progress
and open problems on understanding symplectic mapping class
groups in dimension four. In particular, we explain their
interactions with Lagrangians in both directions: on the one
hand, we show some evidence that, in nice situations,
Lagrangian spheres creates Dehn twists that generates the
symplectic mapping class groups; on the other hand, the
understandings on symplectic mapping class groups can be
used to resolve problems on isotopy classes of Lagrangian or
symplectic objects, not limited to spheres. The bridge
between the two problems is provided by the socalled
ballswapping construction.

 Thursday May 1 at 34pm in Eck 308
 Burglind Joricke, Indiana
 Braids, Conformal Module and Entropy

Abstract: We will discuss two invariants of
conjugacy classes of braids. The first invariant is the
conformal module which occurred in connection with the
interest in the 13. Hilbert problem. The second is a popular
dynamical invariant, the entropy. It occurred in connection
with Thurston's theory of surface homeomorphisms. It turns
out that these invariants are related: They are inverse
proportional. This allows to use known results on entropy
for applications to the concept of conformal module, in
particular to give a conceptional proof of a previous
theorem. We will also give an application of the concept of
conformal module to the problem of isotopy of continuous
objects involving braids to the respective holomorphic
objects.

 Thursday May 8 at 34pm in Eck 308
 Rich Schwartz, Brown
 The projective heat map on pentagons

Abstract: I'll describe several geometrically
defined and projectively natural iterations on polygons. One
of them, the pentagram map, is now known to be a discrete
completely integrable system. The other one, which I call
the projective heat map, is sort of a marriage of the
pentagram map and heat flow. I'll sketch a computerassisted
(but rigorous) analysis of how the projective heat map acts
on the moduli space of projective equivalence classes of
pentagons.

 Friday May 9 at 45pm in Ry 251
 Ian Agol, University of California at Berkeley
 The virtual Haken conjecture (Namboodiri lecture 1)

Abstract: Waldhausen conjectured in 1968 that every
aspherical 3manifold has a finitesheeted cover which is
Haken (contains an embedded essential surface). Thurston
conjectured that hyperbolic 3manifolds have a
finitesheeted cover which fibers over the circle. The first
lecture will be an overview of 3manifold topology in order
to explain the meaning of Waldhausen's virtual Haken
conjecture and Thurston's virtual fibering conjecture, and
how they relate to other problems in 3manifold theory. The
second lecture will give some background on geometric group
theory, including the topics of hyperbolic groups and CAT(0)
cube complexes after Gromov, and explain how the above
conjectures may be reduced to a conjecture of Dani Wise in
geometric group theory. The third lecture will discuss the
proof of Wise's conjecture, that cubulated hyperbolic groups
are virtually special, and some discussion of a new proof of
Wise's Malnormal Special Quotient Theorem. Part of these
results are joint work with Daniel Groves and Jason Manning.

 Monday May 12 at 45pm in Ry 251
 Ian Agol, University of California at Berkeley
 The virtual Haken conjecture (Namboodiri lecture 2)

Abstract: Waldhausen conjectured in 1968 that every
aspherical 3manifold has a finitesheeted cover which is
Haken (contains an embedded essential surface). Thurston
conjectured that hyperbolic 3manifolds have a
finitesheeted cover which fibers over the circle. The first
lecture will be an overview of 3manifold topology in order
to explain the meaning of Waldhausen's virtual Haken
conjecture and Thurston's virtual fibering conjecture, and
how they relate to other problems in 3manifold theory. The
second lecture will give some background on geometric group
theory, including the topics of hyperbolic groups and CAT(0)
cube complexes after Gromov, and explain how the above
conjectures may be reduced to a conjecture of Dani Wise in
geometric group theory. The third lecture will discuss the
proof of Wise's conjecture, that cubulated hyperbolic groups
are virtually special, and some discussion of a new proof of
Wise's Malnormal Special Quotient Theorem. Part of these
results are joint work with Daniel Groves and Jason Manning.

 Tuesday May 13 at 34pm in Eck 308
 Jorgen Ellegaard Andersen, QGM Aarhus
 The WittenReshetikhinTuraev quantum representations of
mapping class groups

Abstract: As part of the WittenReshetikhinTuraev
topological quantum field theory one has the corresponding
quantum representations of the mapping class groups. They
can be constructed both by purely combinatorial means and by
applying geometric quantization to the moduli spaces of flat
connections on a two dimensional surface. We shall review
this geometric construction of these representations and
turn to a number of applications of these representations to
the study of mapping class groups.

 Tuesday May 13 at 4:305:30pm in Eck 206
 Ian Agol, University of California at Berkeley
 The virtual Haken conjecture (Namboodiri lecture 3)

Abstract: Waldhausen conjectured in 1968 that every
aspherical 3manifold has a finitesheeted cover which is
Haken (contains an embedded essential surface). Thurston
conjectured that hyperbolic 3manifolds have a
finitesheeted cover which fibers over the circle. The first
lecture will be an overview of 3manifold topology in order
to explain the meaning of Waldhausen's virtual Haken
conjecture and Thurston's virtual fibering conjecture, and
how they relate to other problems in 3manifold theory. The
second lecture will give some background on geometric group
theory, including the topics of hyperbolic groups and CAT(0)
cube complexes after Gromov, and explain how the above
conjectures may be reduced to a conjecture of Dani Wise in
geometric group theory. The third lecture will discuss the
proof of Wise's conjecture, that cubulated hyperbolic groups
are virtually special, and some discussion of a new proof of
Wise's Malnormal Special Quotient Theorem. Part of these
results are joint work with Daniel Groves and Jason Manning.

 Tuesday May 27 at 34pm in Eck 308
 Jon Chaika, University of Utah
 There exists a mixing locally hamiltonian flow on a surface with only nondegenerate saddles

Abstract: We construct a locally hamiltonian flow
with simple saddles on a genus 2 surface that is mixing with
respect to the volume. Kocergin showed that this impossible
for the torus. C. Ulcigrai showed this was exceptional for
any genus. This is joint work with A. Wright.

 Wednesday May 28 at 1:302:20pm in Eck 202
 Dave Witte Morris, University of Chicago
 Vanishing of first cohomology for arithmetic groups of higher real rank (after G. A. Margulis)

Abstract: Let Gamma be an arithmetic subgroup of G =
SL(n,R), with n > 2. (More generally, Gamma could be any
irreducible arithmetic subgroup of any semisimple Lie group
of higher real rank.) We will describe a proof (due to
Margulis) that the first cohomology of Gamma vanishes (with
coefficients in any finitedimensional real representation).
The result is a consequence of the much stronger statement,
known as the Margulis Superrigidity Theorem, that, roughly
speaking, every finitedimensional representation of Gamma
extends to a finitedimensional representation of G.

 Wednesday May 28 at 2:304:30pm in Eck 308
 Steve Ferry, Rutgers
 Torsion (Lecture Series)

Abstract: Whitehead torsion is an invariant of
homotopy equivalences between finite complexes. It is strong
enough to distinguish when lens spaces, for instance, are
homotopy equivalent but not homeomorphic to each other. In
these talks I will explain the basic theory as well as the
result, due to Chapman, that torsion is a homeomorphism
invariant.

 Thursday May 29 at 34pm in Eck 308
 Barry Minemyer, Alfred University
 Isometric Embeddings of Polyhedra

Abstract: In 195456 John Nash solved the isometric
embedding problem for Riemannian manifolds. This, along with
a result due to Zalgaller, prompted Gromov to ask whether or
not Euclidean polyhedra admit isometric embeddings into
Euclidean space. Here, a Euclidean polyhedra is just a
metric space which admits a locally finite triangulation
such that every kdimensional simplex is affinely
isometric to a simplex in Euclidean space
E^{k}. In this talk we will
discuss recent results toward answering Gromov's question
and, if time permits, we will discuss indefinite metric
analogues to the above results.

 Wednesday June 4 at 2:304:30pm in Eck 308
 Steve Ferry, Rutgers
 Torsion (Lecture Series)

Abstract: Whitehead torsion is an invariant of
homotopy equivalences between finite complexes. It is strong
enough to distinguish when lens spaces, for instance, are
homotopy equivalent but not homeomorphic to each other. In
these talks I will explain the basic theory as well as the
result, due to Chapman, that torsion is a homeomorphism
invariant.
For questions, contact