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

 Friday January 20 at 3pm in Ry358
 Jim Davis, Indiana University
 Topological Rigidity and H_{1}negative involutions on tori

Abstract: An involution on a torus is
H_{1}negative if it induces
multiplication by 1 on
H_{1}(T^{n}). We show that all
H_{1}negative involutions on
T^{n} are equivariantly standard if
n is less than 6 or n = 0,1 (mod 4).
Otherwise there are an infinite number of such involutions,
but all are smoothable. Equivalently we show that the group
Z^{n} \rtimes_{1} Z/2 satisfies
equivariant rigidity when n is less than 6 or
n = 0,1 (mod 4) and doesn't otherwise. A discrete
group G satisfies equivariant ridigity if all
cocompact EGmanfiolds are
Ghomeomorphic. This is joint work with Frank
Connolly and Qayum Khan.

 Monday January 23 at 3pm in E308
 Khalid BouRabee, University of Michigan
 Intersection growth of groups

Abstract: Intersection growth concerns the
asymptotic behavior of the index of the intersection of all
subgroups of a group that have index at most n.
We motivate studying this growth and explore some examples
with a focus on nilpotent groups and zeta functions. This
covers joint work with Ian Biringer, Martin Kassabov, and
Francesco Matucci.

 Thursday January 26 at 3pm in E308
 Michael Wolf, Rice University
 Polynomial Pick forms for affine spheres and real
projective polygons

Abstract: (Joint work with David Dumas.) Convex real
projective structures on surfaces, corresponding to discrete
surface group representations into SL(3, R), have
associated to them affine spheres which project to the
convex hull of their universal covers. Such an affine sphere
is determined by its Pick (cubic) differential and an
associated Blaschke metric. As a sequence of convex
projective structures leaves all compacta in its deformation
space, a subclass of the limits is described by polynomial
cubic differentials on affine spheres which are conformally
the complex plane. We show that those particular affine
spheres project to polygons; along the way, a strong
estimate on asymptotics is found. We will carefully describe
the background material.

 Thursday February 9 at 3pm in E308
 Iddo Samet, University of Illinois Chicago
 Growth of Betti numbers in locally symmetric spaces

Abstract: I will describe the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. The main result is a uniform version of the Lueck Approximation Theorem, which is much stronger than the linear upper bounds on Betti numbers given by Gromov. This study leads to the concepts of local convergence of manifolds and invariant random subgroups in Lie groups. I will describe these notions, and explain how they are related to the proof.
Based on a joint work with Abert, Bergeron, Biringer, Gelander, Nikolov, and Raimbault.

 Tuesday February 14 at 1:30pm in E202
 Alex Gamburd, CUNY Graduate Center
 Expander graphs, thin groups and superstrong approximation  I

Abstract: This is one aspect of exciting recent work on expander graphs, sieve methods, discrete group theory and analytic number theory.

 Thursday February 16 at 1:30pm in E202
 Alex Gamburd, CUNY Graduate Center
 Expander graphs, thin groups and superstrong approximation  II

Abstract: This is one aspect of exciting recent work on expander graphs, sieve methods, discrete group theory and analytic number theory.

 Thursday February 16 at 3pm in E308
 Qian Yin, University of Chicago
 Lattes maps and combinatorial expansion

Abstract: Complex dynamics is the study of the
iterates of rational maps from the Riemann sphere to itself.
Lattes maps are a special class of rational maps which play
a fundamental role as exceptional examples in complex
dynamics. Thurston introduced a topological analogue of a
rational map whose critical points have finite orbits,
called a Thurston map. We characterize Lattes maps by their
combinatorial expansion behavior, and deduce a necessary and
sufficient condition for a Thurston map to be topologically
conjugate to a Lattes map. No background is needed, and all
are welcome.

 Tuesday February 21 at 3pm in E308
 Vadim Kaloshin, University of Maryland
 On Conjugacy of Convex Billiards

Abstract: (joint w/ A. Sorrentino) There are indications that in the 80s Guillemin posed a question: If billiard maps are conjugate, can we say that domains are the same up to isometry? We show that if two billiard maps of convex domains are C^2conjugate near the boundary, then the corresponding domains are similar, i.e. they can be ebtained from one another by rescaling and an isometry. As a application, we prove a conditional version of Birkhoff conjecture on the integrability of planar billiards and show that the original conjecture is equivalent to what we call and Extension problem. Quite interestingly, our result and a positive solution to this extension problem would provide an answer to a closely related question in spectral theory: if the marked length spectra of two domains are the same, is it true that they are isometric?

 Thursday February 23 at 3pm in E308
 Boris Tsygan, Northwestern University
 Microlocal methods in symplectic geometry

Abstract: I will outline recent works on applications of microlocal methods in symplectic geometry. I will start with Tamarkin's works on a
sheaftheoretical construction of a category starting from a symplectic
manifold of a certain class. Next, I will sketch a construction of a
category from a symplectic manifold using methods of deformation
quantization. I will state basic properties of both constructions, their
relations to each other and to the Fukaya category.

 Tuesday February 28 at 1pm in E202
 Chen Meiri, IAS
 The group large sieve method  I

Abstract: In this talk we will present the group large sieve method with an emphasis on some of its applications:
1. Most elements of the mapping class group are pseudoAnosov
(Rivin, Kowalski).
2. The Galois group of the characteristic polynomial of most
elements in SL(n,Z) is isomorphic to the symmetric group on n
elements (Rivin, Kowalski).
3. Most elements in a f.g subgroup of GL(n,C) are not powers
(LubotzkyM).

 Tuesday February 28 at 3pm in E308
 Tatyana Barron, University of Western Ontario
 Quantization, complex structures, and automorphic forms

Abstract: After reminding what the BerezinToeplitz quantization is, I will talk about two
questions: (1) how it depends on the choice of the complex structure
(2) how automorphic forms appear in the picture.

 Thursday March 1 at 12pm in the Barn (Ryerson)
 Juan Souto, University of British Columbia
 Special Lunch Talk: Expander knots and metrics

Abstract: We show that every smooth manifold $M$ of dimension $g\ge 3$ admits a sequence of Riemannian metrics with bounded geometry, whose volume tends to infinity and for which the Cheeger constant remains bounded from below by a positive number. Such sequences are known not to exist for surfaces. As an application we show that there are also sequences of hyperbolic knots whose complements have volume tending to infinity but Cheeger constant uniformly bounded. This is joint work with Marc Lackenby.

 Thursday March 1 at 1:30pm in E202
 Chen Meiri, IAS
 The group large sieve method  II

Abstract: In recent years there had been a great progress in the study of propertytau (also known as expanders). We will start the talk by
briefly describing this progress and then focus on proving the large
sieve theorem.

 Thursday March 1 at 3pm in E308
 Juan Souto, University of British Columbia
 Abstract commensurators of the Johnson kernels

Abstract: Let $X$ be a surface of genus at least $4$. We give a condition, which if satisfied by two subgroups $G$ and $G'$ of the Torelli group $Tor(X)$ ensures that every isomorphism $G\to G'$ is just the restriction to $G$ of an inner automorphism of the mapping class group. In particular we obtain that the mapping class group is isomorphic to the abstract commensurator of for instance the Johnson kernels and the members of the derived and lower central series of the Torelli group. This is joint work with Martin Bridson and Alexandra Pettet.

 Tuesday March 6 at 3pm in E308
 Weiwei Wu, University of Minnesota
 Lagrangian isotopies and symplectomorphism groups

Abstract: Given a Lagrangian sphere in a symplectic 4manifold (M, \omega) with b+ = 1, we find embedded symplectic surfaces intersecting it minimally. We describe several applications of this isotopy result. On the uniqueness side, for a symplectic rational manifold with Euler number less than 8, we generalize Hind and Evans' Hamiltonian uniqueness in the monotone case to general symplectic forms. On the existence side, when M is rational or ruled, we give a characterization of classes represented by Lagrangian spheres, which enables us to describe the nonTorelli part of the symplectic mapping class group. If time permits, I will talk about some related applications to ballpacking and nondisplaceable Lagrangians.

 Thursday March 8 at 3pm in E308
 Rostyslav Kravchenko, University of Chicago
 Selfsimilar groups and measures of selfaffine tiles

Abstract: Let $G$ be a group and $\phi:H\rightarrow G$ be a contracting homomorphism from a subgroup $H< G$ of finite index. V. Nekrashevych associated with it a certain topological space, called the limit space of $\phi$. We develop the measure theory of these limit spaces and present an application to the theory of selfaffine tiles. This is joint work with Ievgen Bondarenko.

 Thursday March 15 at 3pm in E207
 Dan Margalit, Georgia Tech
 Hyperelliptic curves, braid groups, and congruence
subgroups

Abstract: The hyperelliptic Torelli group is the
subgroup of the mapping class group of a surface consisting
of elements that act trivially on the homology of the
surface and also commute with some fixed hyperelliptic
involution. This group can also be characterized as the
fundamental group of the branch locus of the period mapping,
as well as the kernel of the (specialized) Burau
representation of the pure braid group. Hain has conjectured
that the hyperelliptic Torelli group is generated by Dehn
twists about separating curves fixed by the hyperelliptic
involution. His conjecture gives a meaningful description of
the topology of the branch locus of the period mapping. We
present some evidence for the conjecture, as well as
progress towards its resolution. Some of the results rely on
some new, intricate relations in the pure braid group. This
is joint work with Tara Brendle.

 Thursday March 29 at 3pm in E308
 Dave Witte Morris, University of Lethbridge, Canada
 Strictly convex norms on amenable groups

Abstract: It is obvious that the usual Euclidean norm is strictly convex, by which we mean that, for all x and all nonzero y, either x + y > x, or x  y > x. We will discuss the existence of such a norm on an abstract (countable) group G. A sufficient condition is the existence of a faithful action of G by orientationpreserving homeomorphisms of the real line. No examples are known to show that this is not a necessary condition, and we will combine some elementary measure theory and dynamics with the theory of orderable groups to show that the condition is indeed necessary if G is amenable. This is joint work with Peter Linnell of Virginia Tech.

 Tuesday April 10 at 3pm in E308
 Mark McLean, MIT
 TBA

Abstract: TBA

 Tuesday May 1 at 3pm in E308
 Frol Zapolsky, Munich
 TBA

Abstract: TBA

 Thursday May 3 at 3pm in E308
 Sobhan Seyfaddini, Berkeley
 TBA

Abstract: TBA

 Tuesday May 8 at 3pm in E308
 Yael Karshon, Toronto
 TBA

Abstract: TBA

 Thursday June 7 at 3pm in E308
 Robert Penner, University of Southern California
 TBA

Abstract: TBA
For questions, contact