Geometry/Topology Seminar

Winter 2012

Thursdays (and sometimes Tuesdays) 3-4pm, in Eckhart 308

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• 1/20/12
• 1/23/12
• 1/26/12
• 2/16/12
• 2/21/12
• 2/23/12
• 2/28/12
• 3/1/12
• 3/6/12
• 3/8/12

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• contacts

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• previous quarters
• F2011
• S2011
• W2011
• F2010
• S2010
• W2010
• F2009
• S2009
• W2009
• F2008
• S2008
• W2008
• F2007
• S2007
• W2007
• F2006
• S2006
• W2006

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• • Friday January 20 at 3pm in Ry358
  • Jim Davis, Indiana University
  • Topological Rigidity and H₁-negative involutions on tori
  • 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.
• • Monday January 23 at 3pm in E308
  • Khalid Bou-Rabee, 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^2-conjugate 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 sheaf-theoretical 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 pseudo-Anosov (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 (Lubotzky-M).
• • Tuesday February 28 at 3pm in E308
  • Tatyana Barron, University of Western Ontario
  • Quantization, complex structures, and automorphic forms
  • Abstract: After reminding what the Berezin-Toeplitz 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 property-tau (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 4-manifold (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 non-Torelli part of the symplectic mapping class group. If time permits, I will talk about some related applications to ball-packing and non-displaceable Lagrangians.
• • Thursday March 8 at 3pm in E308
  • Rostyslav Kravchenko, University of Chicago
  • Self-similar groups and measures of self-affine 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 self-affine 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 orientation-preserving 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

• Benson Farb, farb at math.uchicago.edu
• Alex Eskin, eskin at math.uchicago.edu
• Shmuel Weinberger, shmuel at math.uchicago.edu
• Howard Masur, masur at math.uchicago.edu
• Leonid Polterovich, polterov at math.uchicago.edu
• David Constantine, dc at math.uchicago.edu
• William Lopes, lopes at math.uchicago.edu
• Jonathan Chaika, jonchaika at math.uchicago.edu