Geometry/Topology Seminar

Winter 2019

Thursdays (and sometimes Tuesdays) 2:30-3:30pm, in Ryerson 358

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• 1/10/19
• 1/24/19
• 2/7/19
• 2/14/19
• 03/05/19
• 3/07/19
• 03/12/19
• 3/28/19

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• contacts

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• previous quarters
• F2018
• S2018
• W2018
• F2017
• S2017
• W2017
• F2016
• S2016
• W2016
• F2015
• S2015
• W2015
• F2014
• S2014
• W2014
• F2013
• S2013
• W2013
• F2012
• S2012
• W2012
• F2011
• S2011
• W2011
• F2010
• S2010
• W2010
• F2009
• S2009
• W2009
• F2008
• S2008
• W2008
• F2007
• S2007
• W2007
• F2006
• S2006
• W2006

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• • Thursday January 10 at 3:30-4:30pm in Ry 358
  • Izzet Coskun, UIC
  • The stable cohomology of moduli spaces of sheaves on surfaces
  • Abstract: Moduli spaces of Gieseker semistable sheaves on surfaces play a central role in mathematics and have many applications to cycles and linear systems on surfaces, Donaldson's 4-manifold invariants and mathematical physics. In this talk, I will describe a conjecture with Matthew Woolf on the cohomology of these moduli spaces. We conjecture that the Betti numbers of these moduli spaces stabilize as the discriminant tends to infinity and that the stable numbers are independent of the rank and the first Chern class. In particular, calculations of Gottsche determine the stable numbers. I will give some evidence for the conjecture. This is joint work with Matthew Woolf.
• • Thursday January 24 at 3:30-4:30pm in Ry 358
  • Nate Harman, Chicago
  • Effective and infinite rank superrigidity for special linear groups
  • Abstract: Superrigidity for SL_n(Z) tells us that any finite dimensional complex representation virtually extends to (i.e. agrees along a finite index subgroup with) an algebraic representation of SL_n(C). We look at effective improvements of this statement, and explore the interaction of superrigidity with representation stability. This leads to a formulation of superrigidity for the infinite rank special linear group -- despite the fact that it has no nontrivial finite dimensional representations or finite index subgroups.
• • Thursday February 7 at 3:30-4:30pm in Ry 358
  • Kasia Jankiewicz, Chicago
  • Groups acting and not acting on CAT(0) cube complexes
  • Abstract: CAT(0) cube complexes are built by gluing Euclidean cubes of various dimensions together, such that a certain combinatorial condition is satisfied, which guarantees that they are as least as nonpositively curved as the Euclidean space. The combinatorial nature of CAT(0) cube complexes gives them a feel of simplicial trees, yet the family of groups admitting actions on CAT(0) cube complexes is rich and various. There are strong relations between actions on CAT(0) cube complexes and algebraic properties of groups. Constructing group actions on CAT(0) complexes is usually done by a standard construction, and obstructing them is in general harder. In my talk I will survey properties and examples of groups acting nicely on CAT(0) cube complexes. I will also discuss some examples of groups that do not admit such actions, including the 4-strand braid group (joint work with J. Huang and P. Przytycki).
• • Thursday February 14 at 3:30-4:30pm in Ry 358
  • Jingyin Huang, Ohio State University
  • Commensurability and virtually specialness of some uniform cubical lattices
  • Abstract: A classical result by Bieberbach says that uniform lattices acting on Euclidean spaces are virtually free abelian. On the other hand, uniform lattices acting on trees are virtually free. This motivates the study of commensurability classification of uniform lattices acting on RAAG complexes, which are cube complexes that "interpolate" between Euclidean spaces and trees. We show that the tree times tree obstruction is the only obstruction for commmensurability of label-preserving lattices acting on RAAG complexes. Some connections of this problem with Haglund and Wise's work on special cube complexes will also be discussed.
• • Tuesday March 05 at 3:30-4:30pm in Ry 358
  • Christin Bibby, University of Michigan
  • Supersolvable posets and arrangements
  • Abstract: The structure of a supersolvable geometric lattice has proven to be fruitful in the theory of hyperplane arrangements, where it arises as the intersection poset of a fiber-type arrangement. A nice partition of the atoms in the poset determines the roots of the characteristic polynomial, thus giving a factorization of the Poincare polynomial of the arrangement complement. The cohomology ring (the Orlik-Solomon algebra) is a Koszul algebra, which allows one to extract information about the rational homotopy theory of the complement. We explore these ideas for toric and elliptic arrangements, where the analogue of the intersection poset is not even a semilattice but a notion of supersolvability can still be applied. The main motivating example is an analogue of reflection arrangements, where the complement is an orbit configuration space and the poset is a generalization of partition lattices. Based on joint work with Emanuele Delucchi.
• • Thursday March 07 at 3:30-4:30pm in Ry 358
  • Sebastian Hensel, LMU Munchen
  • Virtual homology representations of mapping class groups and topology
  • Abstract: Given a mapping class f of a surface S (or an automorphism of a free group), one can extract basic information about f from the action on the first homology of S. While this representation is very well understood, it is also very coarse -- most interesting topological or group theoretic properties of f are not determined by the homology action. Somewhat surprisingly, this picture changes drastically if one is willing to consider the homology of finite covers as well. We will discuss theorems that give examples of this behaviour, in particular concering the question when mapping classes extend over handlebodies.
• • Tuesday March 12 at 3:30-4:30pm in Ry 358
  • John Wiltshire-Gordon, Wisconsin
  • Configuration space in a product
  • Abstract: Write Conf(n,X) for the space of injections {1,...,n} -----> X. For example, the space Conf(n, R) is homotopy equivalent to a discrete space with cardinality n!. In contrast, the space Conf(n, R x R) seems much more interesting and complicated. In this talk, we explain how to compute the homotopy type of Conf(n, X x Y) using only information about configurations in each factor. As an application, we show that configuration space distinguishes the two real line bundles on a circle, and find a homological stability result for trivial complex vector bundles of high rank.
• • Thursday March 28 at 3:30-4:30pm in Ry 358
  • Wolfgang Lueck, Bonn
  • Universal L²-torsion, L²-Euler characteristic, Thurston norm and polytopes (joint with S. Friedl)
  • Abstract: Given an L²-acyclic connected finite CW-complex, we define its universal L²-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group \Wh^w(G). We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead group \Wh^w(G) to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal L²-torsion can be identified with many invariants such as the L²-torsion, the L²-torsion function, twisted L²-Euler characteristics and, in the case of a 3-manifold, the Thurston norm and the (dual) Thurston polytope.

Due to the high number of requests, we are no longer accepting speakers via self-invitations.

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
• Maxime Bergeron, mbergeron at math.uchicago.edu
• Nate Harman, nharman at math.uchicago.edu
• Kasia Jankiewicz, kasia at math.uchicago.edu
• Kevin Schreve, kschreve at math.uchicago.edu