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

 Thursday January 9 at 34pm in Eck 308
 David Aulicino, University of Chicago
 Higher Rank Orbit Closures in H^{odd}(4)

Abstract: The moduli space of genus 3 translation
surfaces with a single zero has two connected components. We
show that in the odd connected component
H^{odd}(4) the only
GL^{+}(2,R) orbit closures are closed
orbits, the Prym locus
\tilde{Q}(3,1^{3}), and
H^{odd}(4). This is joint work with
DucManh Nguyen and Alex Wright.

 Monday January 13 at 45pm in Eck 308
 Lee Mosher, Rutgers University
 The geometry, topology, and dynamics of free groups (I)

Abstract: Over a long span of time, a geometric
understanding of free groups has emerged from the study of
some very basic questions, such as: How do you tell when a
set of elements of a free group forms a basis up to
conjugacy (Whitehead's algorithm)? What are the possible
values of the cyclically reduced word norm growth for an
automorphism of F_{n}? What are the
possible ranks of fixed subgroups of automorphisms of
F_{n} (the Scott Conjecture, now a
theorem of Bestina and Handel)? Motivated by these questions
and others, in these lectures we shall survey the geometric
methods which over the last 25 years have led to a quite
revolution in the study of free groups. Our survey will
include various deformation spaces of geometric/topological
structures, such as the outer space of the free group
(introduced by Culler and Vogtmann), the sphere complex aka
the free splitting complex (Hatcher), and the free factor
complex (Hatcher and Vogtmann). We also survey the dynamical
theory of Out(F_{n}), the outer
automorphism group, which developed along the lines of
Thurston's dynamical theory for mapping class gorups, using
tools of relative train track maps (Bestvina and Handel),
attracting laminations (Bestvina, Feighn, and Handel), and
Rtrees (Culler, Morgan, Levitt, Lustig, and others).

 Tuesday January 14 at 34pm in Eck 308
 Lee Mosher, Rutgers University
 The geometry, topology, and dynamics of free groups (II)

Abstract: Over a long span of time, a geometric
understanding of free groups has emerged from the study of
some very basic questions, such as: How do you tell when a
set of elements of a free group forms a basis up to
conjugacy (Whitehead's algorithm)? What are the possible
values of the cyclically reduced word norm growth for an
automorphism of F_{n}? What are the
possible ranks of fixed subgroups of automorphisms of
F_{n} (the Scott Conjecture, now a
theorem of Bestina and Handel)? Motivated by these questions
and others, in these lectures we shall survey the geometric
methods which over the last 25 years have led to a quite
revolution in the study of free groups. Our survey will
include various deformation spaces of geometric/topological
structures, such as the outer space of the free group
(introduced by Culler and Vogtmann), the sphere complex aka
the free splitting complex (Hatcher), and the free factor
complex (Hatcher and Vogtmann). We also survey the dynamical
theory of Out(F_{n}), the outer
automorphism group, which developed along the lines of
Thurston's dynamical theory for mapping class gorups, using
tools of relative train track maps (Bestvina and Handel),
attracting laminations (Bestvina, Feighn, and Handel), and
Rtrees (Culler, Morgan, Levitt, Lustig, and others).

 Wednesday January 15 at 2:304pm in Eck 308
 Lee Mosher, Rutgers University
 The geometry, topology, and dynamics of free groups (III)

Abstract: Over a long span of time, a geometric
understanding of free groups has emerged from the study of
some very basic questions, such as: How do you tell when a
set of elements of a free group forms a basis up to
conjugacy (Whitehead's algorithm)? What are the possible
values of the cyclically reduced word norm growth for an
automorphism of F_{n}? What are the
possible ranks of fixed subgroups of automorphisms of
F_{n} (the Scott Conjecture, now a
theorem of Bestina and Handel)? Motivated by these questions
and others, in these lectures we shall survey the geometric
methods which over the last 25 years have led to a quite
revolution in the study of free groups. Our survey will
include various deformation spaces of geometric/topological
structures, such as the outer space of the free group
(introduced by Culler and Vogtmann), the sphere complex aka
the free splitting complex (Hatcher), and the free factor
complex (Hatcher and Vogtmann). We also survey the dynamical
theory of Out(F_{n}), the outer
automorphism group, which developed along the lines of
Thurston's dynamical theory for mapping class gorups, using
tools of relative train track maps (Bestvina and Handel),
attracting laminations (Bestvina, Feighn, and Handel), and
Rtrees (Culler, Morgan, Levitt, Lustig, and others).

 Tuesday January 21 at 34:30pm in Eck 206
 Curt McMullen, Harvard
 Entropy: graphs, surfaces, knots and number fields

Abstract: (joint with dynamics seminar)

 Wednesday January 22 at 2:304pm in Eck 206
 Curt McMullen, Harvard
 Entropy: graphs, surfaces, knots and number fields

Abstract: (joint with dynamics seminar)

 Thursday January 23 at 34:30pm in Ry 277
 Curt McMullen, Harvard
 Entropy: graphs, surfaces, knots and number fields

Abstract: (joint with dynamics seminar)

 Tuesday February 18 at 34pm in Eck 206
 Maia Fraser, University of Toronto
 Biinvariant metrics on contactomorphism groups

Abstract: In contrast to the situation for the group
of Hamiltonian symplectomorphisms of a symplectic manifold
where Hofer's metric has been well studied since the early
90's, nontrivial biinvariant metrics on contactomorphism
groups have only recently been developed: Sandon 2010,
Zapolsky 2012, ColinSandon 2012, AlbersMerry 2013 propose
several. I will describe joint work with Polterovich and
Rosen where we give yet another construction. It applies in
contact manifolds which admit a 1periodic Reeb flow and are
orderable in the sense of Eliashberg and Polterovich. Stable
unboundedness of the norm can be deduced in various settings
including certain prequantization spaces (where the result
follows from basic results on symplectic intersections) and
the standard projective spaces
RP^{2n+1} (using Givental's
nonlinear Maslov index).

 Wednesday February 26 at 2:304pm in Eck 206
 Frank Calegari, Northwestern
 The stable homology of congruence subgroups (Part 1)

Abstract: (joint with Algebraic Topology and with
Number Theory Seminar) Let F be a number field. A
stability result due to Charney and Maazen says that the
homology groups
H_{d}(SL_{N}(O_{F}),Z)
(for d fixed) are independent of N for
N sufficiently large. The resulting stable
cohomology groups are intimately related to the algebraic
Ktheory of
O_{F}. In these talks, we
shall explore the homology of the ppower
congruence subgroups of
SL_{N}(O_{F}) in
fixed degree d as N becomes large. We
show that the resulting homology groups consist of two
parts: an “unstable” part which depends only on
local behavior concerning how the prime p splits
in F, and a “stable” part which
contains global information concerning padic
regulator maps. Our argument consists of two parts. The
first part (which is joint work with Matthew Emerton)
explains how to modify the homology of congruence subgroups
in a suitable way (using completed homology) to obtain
groups which are literally stable for large N. We
prove this by combining recent results in representation
stability with the representation theory of
padic groups. The second part consists of
relating these stable homology groups to Ktheory
using ideas from algebraic topology. We give some
applications of our results. The first concerns the
computation of H_{2} of the
pcongruence subgroup of
SL_{N}(Z) for sufficiently large
N. The second concerns describing the extent to
which the stable classes in characteristic zero constructed
by Borel become more and more divisible by p when
one passes to higher and higher ppower level.

 Thursday February 27 at 34pm in Eck 308
 Michael Farber, Warwick
 Fundamental groups of large random spaces

Abstract: I will discuss several probabilistic
models producing random simplicial complexes and will
describe properties of their fundamental groups (torsion and
cohomological dimension). Besides, I will examine the
probabilistic validity of the Whitehead conjecture for
aspherical 2dimensional subcomplexes. Joint work with A.
Costa.

 Friday February 28 at 2:304pm in Eck 206
 Frank Calegari, Northwestern
 The stable homology of congruence subgroups (Part 2)

Abstract: (joint with Algebraic Topology and with
Number Theory Seminar) Let F be a number field. A
stability result due to Charney and Maazen says that the
homology groups
H_{d}(SL_{N}(O_{F}),Z)
(for d fixed) are independent of N for
N sufficiently large. The resulting stable
cohomology groups are intimately related to the algebraic
Ktheory of
O_{F}. In these talks, we
shall explore the homology of the ppower
congruence subgroups of
SL_{N}(O_{F}) in
fixed degree d as N becomes large. We
show that the resulting homology groups consist of two
parts: an “unstable” part which depends only on
local behavior concerning how the prime p splits
in F, and a “stable” part which
contains global information concerning padic
regulator maps. Our argument consists of two parts. The
first part (which is joint work with Matthew Emerton)
explains how to modify the homology of congruence subgroups
in a suitable way (using completed homology) to obtain
groups which are literally stable for large N. We
prove this by combining recent results in representation
stability with the representation theory of
padic groups. The second part consists of
relating these stable homology groups to Ktheory
using ideas from algebraic topology. We give some
applications of our results. The first concerns the
computation of H_{2} of the
pcongruence subgroup of
SL_{N}(Z) for sufficiently large
N. The second concerns describing the extent to
which the stable classes in characteristic zero constructed
by Borel become more and more divisible by p when
one passes to higher and higher ppower level.

 Thursday March 6 at 34pm in Eck 308
 Anders Sodergern, IAS
 The distribution of angles in hyperbolic lattices

Abstract: We consider certain questions related to
the hyperbolic lattice point problem in general dimension
n≥2. More precisely, we describe the
distribution of angles between geodesic rays associated with
the points in an orbit of a hyperbolic lattice. Our main
interest is in properties of the corresponding pair
correlation density function and the starting point of our
discussion will be the recent results of Kelmer and
Kontorovich in dimension 2. This is joint work with Morten
Risager.

 Tuesday March 18 at 34pm in E308
 Dima Burago, Penn State
 Just so stories (Kipling)

Abstract: This is not a usual type of a seminar
talk, though I have given several talks with almost the same
title lately. Still the talks are different. I made
transparencies for more than 20 topics, two to three slides
per topic. Necessary definitions, formulations of key
results, hints towards proofs, open problems. I chose about
8 topics per talk. The choice depends on the audience, the
weather, what I had for breakfast and such. The only thing
that unites the topics is that they have been of interest to
me in past number of years. They all are related to
geometry, PDEs, dynamics, geometric group theory and such.
If the weather cooperates, the breakfast goes well and there
are no terrible traffic jams, I probably would do something
around there topics:  Uniform approximations of metric
spaces by graphs of bounded geometry;  Graph
approximations of the BeltramiLaplace on Riemannian
manifolds; (including a Marriage Lemma for measures); 
Local generation of metric entropy and surjectivity of the
boundary distance function near simple Finsler metrics; 
Why curvature in Finsler Geometry cannot be responsible for
basic geometric phenomena (no rotten tomatoes, please!); 
On conjugation invariant metrics of unbounded diameter on
groups of geometric origin;  Area minimizers in normed
spaces (including Busemann's Conjecture in D=2 and a formula
for the area of a convex polygon in the Euclidean plane); 
Area spaces: now filling area determines metric; 
Tomography: imaging vs. volume minimality.  Maybe
something completely different.

 Thursday March 20 at 23pm in Ry 352
 Misha Belilopetsky, IMPA
 2systoles and quantum codes

Abstract: I will review a recent paper of Guth and
Lubotzky and discuss some related problems.

 Thursday March 20 at 34pm in Eck 308
 Jing Tao, University of Oklahoma
 Evolving structures on graphs and uniform growth rate of some
families of groups

Abstract: Thurston asserted during his birthday
conference that the growth rate of mapping class group is
independent of genus. In this talk, I will explain what this
means and sketch a proof of this theorem, following
SleatorTarjanThurston.

 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.

 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.

 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.
For questions, contact