Geometry/Topology Seminar

Spring 2013

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

- Special Time and Special Room
- Wednesday April 3 at 4pm in Eck 202
- Rob Kirby, University of California, Berkeley
- Trisections of 4-manifolds
- Abstract: Any smooth, orientable 4-manifold X has a trisection (analogous with Heegaard splittings in dimension 3) where each sector is diffeomorphic to a connected sum of k copies of S¹ × B³. These trisections are isomorphic up to stabilization, which is a connected sum with a simple trisection of S⁴. Similar results hold when X has a boundary.
- Thursday April 4 at 3pm in Eck 308
- Izzet Coskun, UIC
- The birational geometry of the Hilbert scheme of points on P²
- Abstract: I will discuss the basic properties of the Hilbert scheme of points on smooth surfaces. I will then describe joint work with Daniele Arcara, Aaron Bertram and Jack Huizenga on the birational geometry of the Hilbert scheme of points on P². I will explain how to calculate the cones of ample and effective divisors and how to run the minimal model program.
- Thursday April 11 at 3pm in Eck 308
- Craig Westerland, University of Melbourne
- Homology of Hurwitz spaces and the Cohen-Lenstra heuristics
- Abstract: We will examine Hurwitz moduli spaces of branched covers of Riemann surfaces, and study their stable homology (as the number of branch points tends to infinity). The results may be used to prove a function field analogue of the Cohen-Lenstra heuristics on the distribution of class groups of imaginary quadratic number fields. The main tools are a homological stability theorem for these spaces, as well as an identification of their limit as a space of functions into a target which supports a certain "universal" branched cover.
- Tuesday April 16 at 3pm in Eck 308
- Avner Ash, Boston College
- Resolutions of the Steinberg module for GL(n)
- Abstract: I will discuss several different resolutions for the Steinberg module for GL(n,A) where A is a PID, including the sharbly resolution, which extends the presentation by modular symbols. Then I will give some applications to the homology of GL(n.Z). Some of this is joint work with Paul Gunnells and Mark McConnell.
- Thursday April 25 at 3pm in Eck 308
- Jim Conant, University of Tennessee
- Topological interpretations of generalized Morita classes in the homology of Aut(F_n)
- Abstract: The definition of Morita classes in the homology of Aut(F_n) originally relied on an identification of the homology of Aut(F_n) with the stable cohomology of a certain complicated Lie algebra, and in particular an analysis of the abelianization of this Lie algebra. We show how these Morita classes can be reinterpreted in terms of homological gluing maps arising from gluing punctured 3-dimensional handlebodies together. In particular, statements about the abelianization of the Lie algebra can be translated into calculations of the homology of the mapping class groups (modulo torsion), Gamma_n,s, of punctured 3-dimensional handlebodies. In particular, a result of Conant-Kassabov-Vogtmann translates into a calculation of H_*(Gamma_2,s) in terms of modular forms and symmetric group representations. Known calculations in dihedral homology lead to a classification of H_*(Gamma_1,s). Not only does this point of view open up a whole new world of questions to explore, it gives a direct interpretation of how classes of small degree assemble to create classes of large degree. (Joint with Martin Kassabov and Karen Vogtmann)
- Thursday May 2 at 3pm in Eck 308
- Ronen Mukamel, Stanford University
- Hilbert modular forms and algebraic models for Teichmuller curves
- Abstract: We will present an explicit algebraic curve C of genus one and an algebraic immersion f: C ---> M₂ into the moduli space of genus two RIemann surfaces with the amazing property that f is a local isometry for the hyperbolic metric on C and the Teichmuller metric on M₂. Such an immersion is called a Teichmuller curve. The curve C is naturally embedded in a Hilbert modular surface and our equations for C were obtained, in collaboration with A. Kumar, by studying a particular Hilbert modular form that vanishes along C.
- Expository
- Tuesday May 7 at 3pm in Eck 308
- Dave Witte Morris, University of Lethbridge
- Lattices in SL(3,R) --1
- Abstract: SL(3,Z) is the easiest example of an arithmetic lattice in SL(3,R), because SL(3,Q) is the obvious Q-form of SL(3,R). We will see how to construct all of the other lattices, by finding all of the possible Q-forms. They come from either division algebras or unitary groups, or a combination of the two.
- Expository
- Wednesday May 8 at 3:30pm in Ry 358
- Dave Witte Morris, University of Lethbridge
- Lattices in SL(3,R) --2
- Abstract: SL(3,Z) is the easiest example of an arithmetic lattice in SL(3,R), because SL(3,Q) is the obvious Q-form of SL(3,R). We will see how to construct all of the other lattices, by finding all of the possible Q-forms. They come from either division algebras or unitary groups, or a combination of the two.
- Thursday May 9 at 3pm in Eck 308
- Dave Witte Morris, University of Lethbridge
- SL(n,Q) has no volume-preserving actions on (n-1)-dimensional compact manifolds
- Abstract: As part of the "Zimmer program," numerous authors have studied volume-preserving actions of the group SL(n,A) on compact manifolds, where A is either the ring Z of integers or the field R of real numbers. On the other hand, very little seems to be known about the intermediate case where A is the field Q of rational numbers. As a first step in this direction, we show that SL(n,Q) has no nontrivial, C-infinity, volume-preserving action on any compact manifold of dimension strictly less than n. The proof has two main ingredients: a theorem of Zimmer tells us that the action of any "S-arithmetic" subgroup must extend (a.e.) to a measurable action of its profinite completion, and the Congruence Subgroup Property provides a very nice description of this profinite completion. This is joint work with Robert J. Zimmer of the University of Chicago.
- Tuesday May 14 at 3pm in Eck 308
- Sa'ar Hersonsky, University of Georgia
- From tiling and packing to uniformization II
- Abstract: The celebrated Riemann mapping theorem asserts that a non-empty simply connected open subset of the complex plane (which is not the whole of it) is conformally equivalent to the open unit disk in the complex plane. Conformally equivalent means that there exists an angle-preserving map from this subset to the open unit disk which is one to one and onto, and that map has an inverse with the same properties. Such a map preserves the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it. Can one visualize this map? We will start with a remarkable conjecture made by Thurston which was first proved by Rodin-Sullivan, continue with related work of Dehn, Schramm and Cannon-Floyd-Parry that involves tiling by squares, and end with current research directions aim at addressing more general questions such as: Given a surface with some combinatorics defined on it, such as a cellular decomposition, can one use the combinatorics to obtain an effective version of uniformization theorems?
- Thursday May 16 at 3pm in Eck 308
- Ben Mcreynolds, Purdue University
- Decision problems, complexity, and representations
- Abstract: In this talk, we will investigate the conjugacy problem on finitely presented groups. I will discuss a few different methods for solving the conjugacy problem that are geometrically motivated. I will then connect the complexity of the conjugacy problem with linear representations of the group. This connection has geometric ties. The talk will focus largely on surface groups and should be easy to follow. This is joint work with Sean Lawton and Lars Louder (possibly some of the results are separately joint with Khalid Bou-Rabee and Priyam Patel).
- Thursday May 23 at 3pm in Eck 308
- Egor Shelukhin, University of Montreal
- Braids and L^p-norms of area-preserving diffeomorphisms
- Abstract: We survey results on the large-scale metric properties of groups area-preserving diffeomorphisms of surfaces endowed with the hydrodynamic L²-metric (or more generally the L^p-metric). The first such property is the unboundedness of the metric, which we establish for the last unknown case among compact surfaces - the two-sphere. Our methods involve quasimorphisms on braid groups and differential forms on configuration spaces. This talk is based on a joint work with Michael Brandenbursky.
- Thursday May 30 at 3pm in Eck 308
- Nir Avni, Northwestern University
- Representation count and the space of local systems
- Abstract: I will talk about a connection between the asymptotic representation theory of arithmetic lattices and the singularities and volumes of the moduli spaces of local systems. In particular, I will talk about the following theorems:
  1. For d>2, the number of irreducible n-dimensional representations of SL(d,Z) is less than n²³.
  2. If g>11, the moduli space of SL(d)-local systems on an algebraic curve of genus g has rational singularities.
  This is a joint work with Rami Aizenbud.
- Monday September 16 at 1-2pm in Eck 308
- Peter Lee, Toronto
- The Pure Virtual Braid Groups and Representation Stability
- Abstract: The virtual braid group is an extension of the classical braid group by the symmetric group, introduced by L. Kauffman, and forms a natural setting in which to explore the concept of representation stability proposed by T. Church and B. Farb. We will review the definition of the virtual braid group vB_n and its subgroup PvB_n of pure virtual braids, the relation of these groups to the classical braid groups and the groups of string motions, and the action of the symmetric group S_n on the cohomology modules H^k(PvB_n,Q). We then give a description of H^k(PvB_n,Q) as sums of S_n-modules induced from certain one-dimensional representations of specific subgroups of S_n. We will recall Church and Farb's notion of representation stability and show how the above description allows us to conclude that H^k(PvB_n,Q) is uniformly representation stable. Time permitting, we will discuss what is known about the decomposition of H^k(PvB_n,Q) into irreducible S_n modules, including the multiplicity of the trivial and alternating representations, and some of the techniques available to uncover this decomposition.
- Wednesday October 9 at 3-4pm (tentative) in Eck 203
- Martin Scharlemann, UCSB
- TBD
- Abstract: TBD
- Thursday October 10 at 3-4pm (tentative) in Eck 308
- Martin Scharlemann, UCSB
- TBD
- Abstract: TBD
- Thursday October 17 at 3-4pm in Eck 308
- Jenya Sapir, Stanford University
- TBD
- Abstract: TBD

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