Geometry/Topology Seminar

Spring 2014

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

- Thursday April 3 at 3-4pm in Eck 308
- Stergios Antonakoudis, Harvard University
- The intrinsic geometry of Teichmuller space and symmetric domains
- Abstract: In this talk we will look at complex domains in their intrinsic Kobayashi metrics with a goal to contribute to the following important question: To what extent does the geometry of Teichmuller space resemble the geometry of a bounded symmetric domain?
- Monday April 7 at 2:30-3:30 pm in Ry358
- Satyan Devadoss, Stanford University
- Phylogenetic networks and the real moduli space of curves
- Abstract: Infused with visual imagery, this talk is motivated by the the moduli spaces of punctured Riemann spheres. In the 1970s, Grothendieck, Deligne, and Mumford constructed their compactification using Geometric Invariant Theory. In the 1990s, Gromov and Witten utilized them as invariants arising from string field theory and quantum cohomology. We consider real points of these spaces, which have elegant geometric and combinatorial properties, being compact hyperbolic manifolds with a beautiful tessellation by convex polytopes. In recent years, they have gained importance in their own right, appearing in areas such as representation theory, geometric group theory, tropical geometry, and lately reinterpreted as spaces of phylogenetic networks. In particular, they resolve the singularities of tree spaces studied by Billera, Holmes, and Vogtmann.
- Tuesday April 8 at 3-4 pm in E308
- Jonathan Hillman, Sydney
- Poincare duality in low dimensions (1)
- Abstract: The goal of this minicourse is to show how Poincare duality imposes constraints on and interactions between the two most basic invariants of algebraic topology, namely, the Euler characteristic \chi(M) and the fundamental group \pi₁(M), for M a manifold (or a Poincare duality complex) of dimension 3 or 4. (In lower dimensions these invariants determine each other, while in higher dimensions they are decoupled.) We shall concentrate on recent work and open questions. In dimension 3 there is a simple system of homotopy invariants, and every orientable PD₃-complex is a connected sum of indecomposables. These are either aspherical or have virtually free fundamental group. The latter class is largely known, and the main question here is whether every aspherical PD₃-complex is homotopy equivalent to a closed 3-manifold. In dimension 4 the situation is less clear, but aspherical PD₄-complexes may be characterised in terms of \pi and \chi, while the fundamental group, Stiefel-Whitney classes and the homotopy intersection pairing determine the homotopy type in many cases with c.d.\pi≤2.
- Thursday April 10 at 3-4 pm in E308
- Jonathan Hillman, Sydney
- Poincare duality in low dimensions (2)
- Abstract: The goal of this minicourse is to show how Poincare duality imposes constraints on and interactions between the two most basic invariants of algebraic topology, namely, the Euler characteristic \chi(M) and the fundamental group \pi₁(M), for M a manifold (or a Poincare duality complex) of dimension 3 or 4. (In lower dimensions these invariants determine each other, while in higher dimensions they are decoupled.) We shall concentrate on recent work and open questions. In dimension 3 there is a simple system of homotopy invariants, and every orientable PD₃-complex is a connected sum of indecomposables. These are either aspherical or have virtually free fundamental group. The latter class is largely known, and the main question here is whether every aspherical PD₃-complex is homotopy equivalent to a closed 3-manifold. In dimension 4 the situation is less clear, but aspherical PD₄-complexes may be characterised in terms of \pi and \chi, while the fundamental group, Stiefel-Whitney classes and the homotopy intersection pairing determine the homotopy type in many cases with c.d.\pi≤2.
- Tuesday April 15 at 3-4 pm in E203
- Jonathan Hillman, Sydney
- Poincare duality in low dimensions (3)
- Abstract: The goal of this minicourse is to show how Poincare duality imposes constraints on and interactions between the two most basic invariants of algebraic topology, namely, the Euler characteristic \chi(M) and the fundamental group \pi₁(M), for M a manifold (or a Poincare duality complex) of dimension 3 or 4. (In lower dimensions these invariants determine each other, while in higher dimensions they are decoupled.) We shall concentrate on recent work and open questions. In dimension 3 there is a simple system of homotopy invariants, and every orientable PD₃-complex is a connected sum of indecomposables. These are either aspherical or have virtually free fundamental group. The latter class is largely known, and the main question here is whether every aspherical PD₃-complex is homotopy equivalent to a closed 3-manifold. In dimension 4 the situation is less clear, but aspherical PD₄-complexes may be characterised in terms of \pi and \chi, while the fundamental group, Stiefel-Whitney classes and the homotopy intersection pairing determine the homotopy type in many cases with c.d.\pi≤2.
- Thursday April 17 at 3-4 pm in E308
- Jonathan Hillman, Sydney
- Poincare duality in low dimensions (4)
- Abstract: The goal of this minicourse is to show how Poincare duality imposes constraints on and interactions between the two most basic invariants of algebraic topology, namely, the Euler characteristic \chi(M) and the fundamental group \pi₁(M), for M a manifold (or a Poincare duality complex) of dimension 3 or 4. (In lower dimensions these invariants determine each other, while in higher dimensions they are decoupled.) We shall concentrate on recent work and open questions. In dimension 3 there is a simple system of homotopy invariants, and every orientable PD₃-complex is a connected sum of indecomposables. These are either aspherical or have virtually free fundamental group. The latter class is largely known, and the main question here is whether every aspherical PD₃-complex is homotopy equivalent to a closed 3-manifold. In dimension 4 the situation is less clear, but aspherical PD₄-complexes may be characterised in terms of \pi and \chi, while the fundamental group, Stiefel-Whitney classes and the homotopy intersection pairing determine the homotopy type in many cases with c.d.\pi≤2.
- Tuesday April 22 at 3-4pm in Eck 308
- Weiwei Wu, Michigan State University
- Symplectic mapping class groups in dimension four, and their relations with Lagrangians
- Abstract: This talk will survey some recent progress and open problems on understanding symplectic mapping class groups in dimension four. In particular, we explain their interactions with Lagrangians in both directions: on the one hand, we show some evidence that, in nice situations, Lagrangian spheres creates Dehn twists that generates the symplectic mapping class groups; on the other hand, the understandings on symplectic mapping class groups can be used to resolve problems on isotopy classes of Lagrangian or symplectic objects, not limited to spheres. The bridge between the two problems is provided by the so-called ball-swapping construction.
- Thursday May 1 at 3-4pm in Eck 308
- Burglind Joricke, Indiana
- Braids, Conformal Module and Entropy
- Abstract: We will discuss two invariants of conjugacy classes of braids. The first invariant is the conformal module which occurred in connection with the interest in the 13. Hilbert problem. The second is a popular dynamical invariant, the entropy. It occurred in connection with Thurston's theory of surface homeomorphisms. It turns out that these invariants are related: They are inverse proportional. This allows to use known results on entropy for applications to the concept of conformal module, in particular to give a conceptional proof of a previous theorem. We will also give an application of the concept of conformal module to the problem of isotopy of continuous objects involving braids to the respective holomorphic objects.
- Thursday May 8 at 3-4pm in Eck 308
- Rich Schwartz, Brown
- The projective heat map on pentagons
- Abstract: I'll describe several geometrically defined and projectively natural iterations on polygons. One of them, the pentagram map, is now known to be a discrete completely integrable system. The other one, which I call the projective heat map, is sort of a marriage of the pentagram map and heat flow. I'll sketch a computer-assisted (but rigorous) analysis of how the projective heat map acts on the moduli space of projective equivalence classes of pentagons.
- Friday May 9 at 4-5pm in Ry 251
- Ian Agol, University of California at Berkeley
- The virtual Haken conjecture (Namboodiri lecture 1)
- Abstract: Waldhausen conjectured in 1968 that every aspherical 3-manifold has a finite-sheeted cover which is Haken (contains an embedded essential surface). Thurston conjectured that hyperbolic 3-manifolds have a finite-sheeted cover which fibers over the circle. The first lecture will be an overview of 3-manifold topology in order to explain the meaning of Waldhausen's virtual Haken conjecture and Thurston's virtual fibering conjecture, and how they relate to other problems in 3-manifold theory. The second lecture will give some background on geometric group theory, including the topics of hyperbolic groups and CAT(0) cube complexes after Gromov, and explain how the above conjectures may be reduced to a conjecture of Dani Wise in geometric group theory. The third lecture will discuss the proof of Wise's conjecture, that cubulated hyperbolic groups are virtually special, and some discussion of a new proof of Wise's Malnormal Special Quotient Theorem. Part of these results are joint work with Daniel Groves and Jason Manning.
- Monday May 12 at 4-5pm in Ry 251
- Ian Agol, University of California at Berkeley
- The virtual Haken conjecture (Namboodiri lecture 2)
- Abstract: Waldhausen conjectured in 1968 that every aspherical 3-manifold has a finite-sheeted cover which is Haken (contains an embedded essential surface). Thurston conjectured that hyperbolic 3-manifolds have a finite-sheeted cover which fibers over the circle. The first lecture will be an overview of 3-manifold topology in order to explain the meaning of Waldhausen's virtual Haken conjecture and Thurston's virtual fibering conjecture, and how they relate to other problems in 3-manifold theory. The second lecture will give some background on geometric group theory, including the topics of hyperbolic groups and CAT(0) cube complexes after Gromov, and explain how the above conjectures may be reduced to a conjecture of Dani Wise in geometric group theory. The third lecture will discuss the proof of Wise's conjecture, that cubulated hyperbolic groups are virtually special, and some discussion of a new proof of Wise's Malnormal Special Quotient Theorem. Part of these results are joint work with Daniel Groves and Jason Manning.
- Tuesday May 13 at 3-4pm in Eck 308
- Jorgen Ellegaard Andersen, QGM Aarhus
- The Witten-Reshetikhin-Turaev quantum representations of mapping class groups
- Abstract: As part of the Witten-Reshetikhin-Turaev topological quantum field theory one has the corresponding quantum representations of the mapping class groups. They can be constructed both by purely combinatorial means and by applying geometric quantization to the moduli spaces of flat connections on a two dimensional surface. We shall review this geometric construction of these representations and turn to a number of applications of these representations to the study of mapping class groups.
- Tuesday May 13 at 4:30-5:30pm in Eck 206
- Ian Agol, University of California at Berkeley
- The virtual Haken conjecture (Namboodiri lecture 3)
- Abstract: Waldhausen conjectured in 1968 that every aspherical 3-manifold has a finite-sheeted cover which is Haken (contains an embedded essential surface). Thurston conjectured that hyperbolic 3-manifolds have a finite-sheeted cover which fibers over the circle. The first lecture will be an overview of 3-manifold topology in order to explain the meaning of Waldhausen's virtual Haken conjecture and Thurston's virtual fibering conjecture, and how they relate to other problems in 3-manifold theory. The second lecture will give some background on geometric group theory, including the topics of hyperbolic groups and CAT(0) cube complexes after Gromov, and explain how the above conjectures may be reduced to a conjecture of Dani Wise in geometric group theory. The third lecture will discuss the proof of Wise's conjecture, that cubulated hyperbolic groups are virtually special, and some discussion of a new proof of Wise's Malnormal Special Quotient Theorem. Part of these results are joint work with Daniel Groves and Jason Manning.
- Tuesday May 27 at 3-4pm in Eck 308
- Jon Chaika, University of Utah
- There exists a mixing locally hamiltonian flow on a surface with only non-degenerate saddles
- Abstract: We construct a locally hamiltonian flow with simple saddles on a genus 2 surface that is mixing with respect to the volume. Kocergin showed that this impossible for the torus. C. Ulcigrai showed this was exceptional for any genus. This is joint work with A. Wright.
- Wednesday May 28 at 1:30-2:20pm in Eck 202
- Dave Witte Morris, University of Chicago
- Vanishing of first cohomology for arithmetic groups of higher real rank (after G. A. Margulis)
- Abstract: Let Gamma be an arithmetic subgroup of G = SL(n,R), with n > 2. (More generally, Gamma could be any irreducible arithmetic subgroup of any semisimple Lie group of higher real rank.) We will describe a proof (due to Margulis) that the first cohomology of Gamma vanishes (with coefficients in any finite-dimensional real representation). The result is a consequence of the much stronger statement, known as the Margulis Superrigidity Theorem, that, roughly speaking, every finite-dimensional representation of Gamma extends to a finite-dimensional representation of G.
- Wednesday May 28 at 2:30-4:30pm in Eck 308
- Steve Ferry, Rutgers
- Torsion (Lecture Series)
- Abstract: Whitehead torsion is an invariant of homotopy equivalences between finite complexes. It is strong enough to distinguish when lens spaces, for instance, are homotopy equivalent but not homeomorphic to each other. In these talks I will explain the basic theory as well as the result, due to Chapman, that torsion is a homeomorphism invariant.
- Thursday May 29 at 3-4pm in Eck 308
- Barry Minemyer, Alfred University
- Isometric Embeddings of Polyhedra
- Abstract: In 1954-56 John Nash solved the isometric embedding problem for Riemannian manifolds. This, along with a result due to Zalgaller, prompted Gromov to ask whether or not Euclidean polyhedra admit isometric embeddings into Euclidean space. Here, a Euclidean polyhedra is just a metric space which admits a locally finite triangulation such that every k-dimensional simplex is affinely isometric to a simplex in Euclidean space E^k. In this talk we will discuss recent results toward answering Gromov's question and, if time permits, we will discuss indefinite metric analogues to the above results.
- Wednesday June 4 at 2:30-4:30pm in Eck 308
- Steve Ferry, Rutgers
- Torsion (Lecture Series)
- Abstract: Whitehead torsion is an invariant of homotopy equivalences between finite complexes. It is strong enough to distinguish when lens spaces, for instance, are homotopy equivalent but not homeomorphic to each other. In these talks I will explain the basic theory as well as the result, due to Chapman, that torsion is a homeomorphism invariant.

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
Danny Calegari, dannyc at math.uchicago.edu
Qian Yin, qyin at math.uchicago.edu
Sebastian Hensel, hensel at math.uchicago.edu
Jonathan Chaika, jonchaika at math.uchicago.edu