Geometry/Topology Seminar
Fall 2012
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 joint with dynamics seminar
 Tuesday October 16 at 3pm in Eck 308
 Sarah Koch, Harvard University
 Eigenvalues of the Thurston Pullback Map

Abstract: Given a critically finite rational map,
one can define a holomorphic endomorphism of a Teichmueller
space associated to it; this endomorphism is called the
Thurston pullback map. With the exception of one class of
examples, this endomorphism has a unique fixed point, and
the eigenvalues of the derivative at this fixed point are
all *algebraic*. What do these eigenvalues mean? What
algebraic numbers arise this way? We establish some facts
about these eigenvalues if the rational map is a quadratic
polynomial (for example, we prove in this case that there
are no "small eigenvalues"), but the situation is still
mysterious.

 Thursday October 25 at 3pm in Eck 308
 Eric Zaslow, Northwestern University
 Skeleta for Affine Hypersurfaces

Abstract: Any smooth affine hypersurface
Z of complex dimension n deformation
retracts to a cell complex of real dimension n.
Starting from the Newton polytope of the defining function,
I will give an explicit combinatorial construction of a
compact space S comprised, of
ndimensional components, which embeds into
Z as a deformation retract. In particular,
Z is homotopy equivalent to S. If time
allows, I will discuss the relation of this work to the
homological mirror symmetry program. This work is joint with
Helge Ruddat, Nicolò Sibilla, and David Treumann.

 Thursday November 1 at 3pm in Eck 308
 Frank Calegari, Northwestern University
 Congruence subgroups and stability.

Abstract: The cohomology groups of
SL_{n}(Z) in fixed degree are known to
stabilize for large n. For trivial reasons, this
is not always true when one replaces
SL_{n}(Z) by a congruence subgroup. We
present various approaches to understanding the cohomology
of congruence subgroups in the stable range, and discuss
their (conjectural) relationship with number theory.

 Tuesday November 6 at 3pm in Eck 308
 Matt Bainbridge, Indiana University
 Effective Veech dichotomy

Abstract: The Veech dichotomy tells us that the
geodesic flow on certain very symmetric flat surfaces has
very wellbehaved dynamics: every orbit is either periodic
or uniformly distributed. In this talk, I will discuss a
more effective strengthening of this dichotomy in some
cases. The proof involves geometry of numbers and diagonal
flows on homogeneous spaces. This is joint work with Martin
Moller.

 Algebraic Toplogy Seminar
 Tuesday November 6 at 4:30pm in Eck 203
 Ilya Grigoriev, Stanford University
 Relations in the cohomology of classifying spaces of manifold bundles.

Abstract: The nth cohomology of the
classifying space of surface bundles with fiber of genus g,
denoted H^{n}(BDiffΣ_{g}),
is known in the "stable range" n ≤ (2g2)/3)
by theorems of MadsenWeiss, Harer, and others. In this
range, the map from a free algebra generated by the
socalled "kappaclasses" to
H^{*}(BDiffΣ_{g}) is an
isomorphism. Recently, Soren Galatius, Oscar
RandalWilliams, Ib Madsen, and Alexander Berglund have
obtained similar results for a highdimensional
generalization of this space, with the surface
Σ replaced by the connect sum of
g copies of the cross product S^{k}
× S^{k}. Outside the stable range, the
kernel of the abovementioned map for surfaces has been
studied by Morita, Faber, Looijenga, Pandharipande and many
others. In this talk, I will describe a method for producing
a vast family of elements in the kernel that also works in
the the highdimensional case (for odd k ≥ 1
). This method is based on a refinement of results in
the surface case due to Oscar RandalWilliams. The kernel is
large enough to imply that the image of this map ("the
tautological subring") is finitelygenerated for all odd
k, rationally, even though there are infinitely
many kappa classes. It also implies upper bounds on the
stable range of cohomology for fixed g and k.

 Special time and date
 Wednesday November 7 at 5:10pm in Eck 308
 Alejandro Adem, University of British Columbia
 Topology and spaces of representations for abelian groups

Abstract: In this talk we describe basic topological
properties for spaces of commuting elements in a compact Lie
group, as well as their equivariant structure under
conjugation. From this we derive information about their
stable homotopy type and equivariant Ktheory. This is joint
work with José Gómez.

 joint with number theory seminar
 Tuesday November 13 at 3pm in Eck 308
 Aaron Silberstein, University of Pennsylvania
 The Birational Anabelian Theorem for Surfaces over Q

Abstract: The "yoga" of Alexandre Grothendieck's
program of anabelian geometry dictates that if the
étale fundamental group of an algebraic variety X is
rich enough, then it should encode much of the information
about X as a variety; such varieties X are called anabelian
in the sense of Grothendieck, and different anabelian
varieties have nonisomorphic étale fundamental
groups. An anabelian theorem is a theorem which asserts that
a class of varieties are anabelian. Grothendieck took
inspiration in part from differentialgeometric rigidity
theorems such as the celebrated MostowPrasad rigidity
theorem for finitevolume hyperbolic manifolds, but until
recently they were thought to be disjoint phenomena;
Grothendieck stated in his letter to Faltings that "the
reason for [anabelian phenomena] seems...[to spring] from
the fac that the (outer) action of the 'arithmetic' part of
[the étale fundamental group of a variety over a
finitelygenerated field] is extraordinarily strong." Fedor
Bogomolov had the surprising insight that as long as the
dimension of a variety is greater than or equal to 2,
anabelian phenomena can be exhibited even in the complete
absence of the "arithmetic" part of the group Grothendieck
referenced. We outline the proof of such an anabelian
theorem using techniques from complex geometry and Hodge
theory, and give applications to fundamental groups of open
algebraic surfaces and to absolute Galois groups of number
fields.

 Thursday November 15 at 3pm in Eck 308
 Steven Lalley, University of Chicago
 Random Walk on a Hyperbolic Group

Abstract: According to a fundamental theorem of
Harry Kesten, the probability that a nondegenerate random
walk on a nonamenable group returns to its starting point
after 2n steps must decay exponentially with
n. The precise asymptotic behavior, however,
depends on finer details of the geometry of the group, and
for discrete groups has been established in only a few
special cases. I will discuss the case of random walks on
hyperbolic groups, for which Sebastien Gouezel and I have
recently proved the following general result: For any
nondegenerate random walk X_{n} whose
step distribution is finitely supported, the probability of
return satisfies P{X_{2n} = 1} ~
Cn^{3/2}R^{2n}, where R >
1 and C > 0 are constants that depend on
the step distribution. The proof centers on showing that the
Martin boundary of the random walk (more precisely, the
Martin boundary for Rpotentials) coincides with
the Gromov boundary of the group. This in turn entails
showing that the Green's function of the random walk has
sharp exponential decay along geodesics.

 joint with the logic seminar: part 1
 Tuesday November 27 at 3pm in Eck 308
 Justin Moore, Cornell University
 An analysis of the amenability problem for Thompson's group

Abstract: Thompson's group F has served as an
important example in group theory since it was introduced by
Richard Thompson in the 1960s. One of the major open
problems concerning this group is whether it is amenable.
The first part of this talk will give a proof that the
growth of the Folner function for F is faster than any
finite iteration of the exponential function. The proof also
reveals qualitative features of invariant means on F  if
they exist  which suggest an approach to proving the
amenability of F. I will then discuss the relationship
between the amenability of a group (and F in particular) and
Ramsey theory.

 joint with the logic seminar: part 2 (special time and room)
 Wednesday November 28 at 1011:30am in The Barn
 Justin Moore, Cornell University
 An analysis of the amenability problem for Thompson's group

Abstract: Thompson's group F has served as an
important example in group theory since it was introduced by
Richard Thompson in the 1960s. One of the major open
problems concerning this group is whether it is amenable.
The first part of this talk will give a proof that the
growth of the Folner function for F is faster than any
finite iteration of the exponential function. The proof also
reveals qualitative features of invariant means on F  if
they exist  which suggest an approach to proving the
amenability of F. I will then discuss the relationship
between the amenability of a group (and F in particular) and
Ramsey theory.

 Tuesday December 4 at 3pm in Eck 308
 T.N. Venkataramana, Tata Inst. for Fund. Res.
 Monodromy and Arithmetic Groups

Abstract: Monodromy groups arise naturally in
algebraic geometry and in differential equations, and often
preserve an integral lattice. It is of interest to know
whether the monodromy groups are arithmetic or thin. In this
talk we review the DeligneMostow theory and show that for
cyclic coverings of degree d of the projective
line, with a prescribed number m of branch points
and prescribed ramification indices, the monodromy is an
arithmetic group provided m≥ 2d2. This is
closely related to the arithmeticity of the image of the
Burau/Gassner representation at dth roots of
unity. We also show that the monodromy asociated to certain
hypergeometric differential equations of type
_{n} F_{n}1 is arithmetic in a
number of cases, providing a counterpart to results of
FuchsMeiriSarnak and of BravThomas.

 Wednesday December 5 at 4pm in Eck 308
 Damien Gaboriau, UMPA
 Measured Group Theory, Percolation and NonAmenability

Abstract: Amenability of groups is a concept
introduced by J.~von~Neumann in his seminal article (1929)
to explain the socalled BanachTarski paradox. It is easily
shown that the free groups F on two generators are
nonamenable. It follows that the countable discrete groups
containing F are nonamenable. von Neumann's problem asked
whether the converse holds true. In the 80's Ol'shanskii
showed that his Tarski monsters are counterexamples.
However, in order to extend certain results from groups
containing F to any nonamenable countable group Gamma, it
may be enough to know that Gamma “contains” F in
a weaker sense. Namely, to know that Gamma admits an ergodic
probability measure preserving action on some standard space
for which the orbits can be partitioned into orbits of some
ergodic free action of F. The solution to this measurable
von Neumann's problem involves percolation theory on Cayley
graphs and measured laminations by subgraphs. I shall
present an introduction to this subject and some instances
where notions from group theory admit an interesting
analogue in the context of measure preserving actions.

 Thursday December 6 at 3pm in Eck 308
 Spencer Dowdall, UIUC
 Kleinian convex cocompact subgroups of mapping class groups

Abstract: Convex cocompact subgroups of mapping
class groups, as introduced by Farb and Mosher, are
subgroups whose action on Teichmuller space is analogous to
that of convex cocompact Kleinian groups acting on
hyperbolic space. In this talk I will describe a setting in
which there is a concrete connection between these two
notions of convex cocompactness: Because of the Birman exact
sequence, fibered hyperbolic 3manifold groups give many
ways to realize Kleinian groups as subgroups of mapping
class groups. Expanding on earlier work of
KentLeiningerSchleimer, I will explain a result that
certain subgroups of these 3manifold groups which are
convex cocompact in the Kleinian sense are also convex
cocompact as subgroups of the mapping class group. This is
joint work with Richard Kent and Christopher Leininger.

 Special time
 Tuesday December 11 at 4pm in Eck 308
 Miklos Abert, Alfred Renyi Institute of Mathematics
 On the growth of L^{2}invariants for sequences of lattices in Lie groups

Abstract: We study the asymptotic behavior of Betti
numbers, twisted torsion and other spectral invariants of
sequences of locally symmetric spaces. Our main results are
uniform versions of the DeGeorgeWallach Theorem, of a
theorem of Delorme and various other limit multiplicity
theorems. The idea is to adapt BenjaminiSchramm convergence
(BSconvergence), originally introduced for sequences of
finite graphs of bounded degree, to sequences of Riemannian
manifolds. This is joint work with Bergeron, Biringer,
Gelander, Nikolov, Raimbault and Samet.
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