Geometry/Topology Seminar
Winter 2015
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday February 5 at 34pm in Eck 308
 Nathan Dunfield, University of Illinois at UrbanaChampaign
 Certifying the Thurston norm via twisted homology

Abstract: From the very beginning of 3manifold
topology, a fundamental task has been to find the simplest
surface in a given 2dimensional homology class, e.g. the
Seifert genus of a knot in the 3sphere. The behavior of the
minimal topological complexity as the homology class varies
is encapsulated in the Thurston norm. In this talk, I will
discuss tools for proving that a particular surface has
minimal genus. These are generalizations of the classical
Alexander polynomial, but are defined using homology with
coefficients twisted by some finitedimensional
representation of the fundamental group of the manifold. I
will discuss recent work with Ian Agol on situations where
using representations coming from hyperbolic geometry
suffices to provide such certificates. If time permits, I
will sketch how all of this relates to fundamental questions
about the computational complexity of finding the Thurston
norm. Only basic facts about manifolds and homology will be
assumed.

 Thursday February 12 at 34pm in Eck 308
 Juliette Bavard, Institut de Mathematiques de Jussieu
 About a big mapping class group

Abstract: The mapping class group of the complement
of a Cantor set in the plane arises naturally in dynamics.
To get informations about this "big mapping class group", we
can look at its action on a hyperbolic space: the ray graph.
In this talk, I will explain why this ray graph has infinite
diameter and is hyperbolic. I will then exhibit an element
of the big MCG which has a loxodromic action on the ray
graph, and explain why this element is useful to construct
non trivial quasimorphisms on the group.

 Monday February 23 at 34 pm in Eck 202
 Jordan Ellenberg, University of Wisconsin
 Topology and counting

Abstract: Weil wrote about the theory of function
fields of curves over finite fields as a Godgiven "bridge"
between number theory and topology. In modern terms, that
bridge is provided by the GrothendieckLefschetz trace
formula, which allows us to say something about the number
of points on a variety X over a finite field (resp. the
average value of an interesting function on those points) in
terms of the etale cohomology of X (resp. etale cohomology
with coefficients in an interesting sheaf.) In particular,
this bridge allows us to connect conjectures and theorems in
topology (concerning stable cohomology of moduli spaces)
with conjectures and theorems in number theory (concerning
distribution and asymptotics of arithmetic functions.) I'll
give a survey of results and ideas in this direction,
including some subset of: the CohenLenstra conjectures on
class groups of random number fields, the PoonenRains
conjectures on Selmer groups of random elliptic curves,
distribution of numbers of primes in short intervals, the
BatyrevManin conjectures, etc. The overall theme is that
ideas from topology provide a consistent, geometrically
principled machine for generating conjectures in number
theory, and sometimes for proving these conjectures in the
global function field case.

 Thursday February 26 at 34 pm in Eck 308
 Jonathan Bowden, University of Munich
 Tight contact structures and Reebless Foliations

Abstract: As an important step in Mrowka and
Kronheimer's proof of Property P, Eliashberg and Thurston
showed in the late 90's that any codimension1 foliation on
a closed oriented 3manifold can be approximated by contact
structures. This raised many interesting questions about how
foliation theory and contact topology relate and allowed for
quite a fruitful interplay between the two theories. In this
talk I will discuss the relationship between various notions
of tightness/inflexibility for foliations and contact
structures and how they correspond in light of Eliashberg
and Thurston's theory. This yields information about the
structure of the space of contact structures and its closure
in the space of all plane fields.

 Tuesday March 3 at 1:302:30pm in Eck 202
 Holger Reich, Freie Universitat Berlin
 Algebraic Ktheory of group algebras and topological cyclic homology

Abstract: The talk will report on joint work with
Wolfgang Lueck (Bonn), John Rognes (Oslo) and Marco Varisco
(Albany).The Whitehead group Wh(G) and its higher
analogues defined using algebraic Ktheory play an important
role in geometric topology. There are vanishing conjectures
in the case where G is torsionfree. For groups
containing torsion the FarrellJones conjectures give a
conjectural description in terms of group homology. After an
introduction to this circle of ideas, I will report on the
following new result, which for example detects a large
direct summand inside the rationalized Whitehead group of a
group like Thompson's group T: The FarrellJones assembly
map for connective algebraic Ktheory is rationally
injective, under mild homological finiteness conditions on
the group and assuming that a weak version of the
LeopoldtSchneider conjecture holds for cyclotomic fields.
This generalizes a result of Boekstedt, Hsiang and Madsen,
and leads to a concrete description of a large direct
summand of K_{n} (ZG) \otimes_{Z}
Q in terms of group homology. Since the number
theoretic assumption holds in low dimensions, this also
computes a large direct summand of Wh(G)
\otimes_{Z} Q. The proof uses the cyclotomic
trace to topological cyclic homology,
BoekstedtHsiangMadsen's functor C, and new general
injectivity results about the assembly maps for THH and C.

 Tuesday March 3 at 34pm in Eck 308
 Lev Buhovsky, Tel Aviv University
 C^{0} properties of the group of Hamiltonian diffeomorphisms

Abstract: I will talk about Hamiltonian and
symplectic diffeomorphisms of a closed symplectic manifold.
The main subject of this talk is the flux group of a closed
symplectic manifold, and related conjectures ("flux
conjectures")

 Tuesday March 10 at 34pm in Eck 308
 Andrzej Szczepanski, Gdansk
 Outer automorphism group of crystallographic groups with
trivial center

Abstract: Let \Gamma be a
crystallographic group of dimension n, i.e. a
discrete, cocompact subgroup of
Isom(R^{n}) = O(n) \ltimes
R^{n}. For any n≥ 2, we
shall construct a crystallographic group with trivial center
and a trivial outer automorphism group. Moreover, we shall
present properties of an example (constructed by R.
Waldmuler in 2003) of the torsion free crystallographic
group of dimension 141 with a trivial center and a trivial
outer automorphism group.

 Thursday March 12 at 34pm in Eck 308
 David Dumas, UIC
 The moduli space of convex real projective structures

Abstract: I will describe several ways to understand
the simplest example of a "higher Teichmueller space",
namely, the moduli space of convex real projective
structures on a surface. This space can be seen as a
connected component of the character variety, as a bundle
over Teichmueller space, and most concretely as a set of
parameters that describe how to build a surface out of
polygons in RP^{2}. For the most part
this talk will be expository, though if time allows I will
describe a connection to my recent work with Michael Wolf on
moduli spaces of convex polygons and polynomial cubic
differentials.
For questions, contact