Geometry/Topology Seminar
Winter 2012
Thursdays (and sometimes Tuesdays) 3-4pm, in
Eckhart 308
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- Friday January 20 at 3pm in Ry358
- Jim Davis, Indiana University
- Topological Rigidity and H1-negative involutions on tori
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Abstract: An involution on a torus is
H1-negative if it induces
multiplication by -1 on
H1(Tn). We show that all
H1-negative involutions on
Tn 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
Zn \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 E|G-manfiolds are
G-homeomorphic. This is joint work with Frank
Connolly and Qayum Khan.
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- Monday January 23 at 3pm in E308
- Khalid Bou-Rabee, University of Michigan
- Intersection growth of groups
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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.
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- Thursday January 26 at 3pm in E308
- Michael Wolf, Rice University
- Polynomial Pick forms for affine spheres and real
projective polygons
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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.
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- Thursday February 9 at 3pm in E308
- Iddo Samet, University of Illinois Chicago
- Growth of Betti numbers in locally symmetric spaces
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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 LŸck 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.
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- Tuesday February 14 at 1:30pm in E202
- Alex Gamburd, CUNY Graduate Center
- Expander graphs, thin groups and superstrong approximation - I
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Abstract: This is one aspect of exciting recent work on expander graphs, sieve methods, discrete group theory and analytic number theory.
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- Tuesday February 16 at 1:30pm in E202
- Alex Gamburd, CUNY Graduate Center
- Expander graphs, thin groups and superstrong approximation - II
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Abstract: This is one aspect of exciting recent work on expander graphs, sieve methods, discrete group theory and analytic number theory.
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- Thursday February 16 at 3pm in E308
- Qian Yin, University of Chicago
- Lattes maps and combinatorial expansion
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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.
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- Tuesday February 21 at 3pm in E308
- Vadim Kaloshin, University of Maryland
- TBA
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Abstract: TBA
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- Thursday February 23 at 3pm in E308
- Boris Tsygan, Northwestern University
- TBA
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Abstract: TBA
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- Tuesday February 28 at 3pm in E308
- Tatyana Barron, University of Western Ontario
- TBA
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Abstract: TBA
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- Thursday March 1 at 3pm in E308
- Juan Suoto, University of British Columbia
- TBA
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Abstract: TBA
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- Tuesday March 6 at 3pm in E308
- Weiwei Wu, University of Minnesota
- TBA
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Abstract: TBA
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- Tuesday March 8 at 3pm in E308
- Rostyslav Kravchenko, University of Chicago
- TBA
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Abstract: TBA
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- Thursday March 15 at 3pm in E308
- Dan Margalit, Georgia Tech
- Hyperelliptic curves, braid groups, and congruence
subgroups
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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.
For questions, contact