Geometry/Topology Seminar

Spring 2018

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

- Thursday October 11 at 3-4pm in Ry 358
- Rita Gitik, University of Michigan
- On Tame Subgroups of Finitely Presented Groups
- Abstract: We describe several examples of tame subgroups of finitely presented groups and prove that the fundamental groups of certain finite graphs of groups are locally tame.
- Thursday October 18 at 3-4pm in Ry 358
- Daniel Ramras, IUPUI
- Homological stability for spaces of representations
- Abstract: I will discuss recent work with Mentor Stafa on homological stability for various spaces built from commuting elements in Lie groups. These results depend on rational models for these spaces due to T. Baird, and utilize J. Wilson's theory of FI_W-modules. A key aspect of this work is the relationship between stability in ordinary and in equivariant (co)homology. Time permitting, I'll also discuss what is known regarding stability for character varieties of free groups and surface groups.
- Thursday October 25 at 3-4pm in Ry 358
- Daniel Studenmund, Notre Dame
- Commensurability growth of nilpotent groups
- Abstract: A classical area of study in geometric group theory is subgroup growth, which counts the number of subgroups of a given group Gamma as a function their index. We will study a richer function, the commensurability growth, associated to a subgroup Gamma in an ambient group G. The main result of this talk concerns the case that Gamma is an arithmetic subgroup of a unipotent group G, starting with the simplest example of the integers in the real line. This is joint work with Khalid Bou-Rabee.
- Thursday November 01 at 3-4pm in Ry 358
- Yulan Qing, University of Toronto
- Loops with Large Twist Get Short Along Quasi-geodesics in Out(Fn)
- Abstract: We study the behaviour of quasi-geodesics in Out(Fn) equipped with word metric. Given an element \phi of Out(Fn), there are several natural paths connecting the origin to \phi in Out(Fn). We show that these paths are, in general, not quasi-geodesics in Out(Fn). In fact, we clear up the current misunderstanding about distance estimating in Out(Fn) by showing that there exists points in Out(Fn) where all quasi-geodesics between them backtracks in all of the current Out(Fn) complexes .
- Thursday November 08 at 3-4pm in Ry 358
- Kevin Schreve, University of Chicago
- Action Dimension of Simple Complexes of Groups
- Abstract: The action dimension of a discrete group G is the minimal dimension of contractible manifold that admits a properly discontinuous G action. I will compute the action dimension of several examples,including Artin groups, graph products, and hyperplane arrangements.
- Thursday November 15 at 3-4pm in Ry 358
- Trevor Hyde, University of Michigan
- Moduli of multivariate irreducible polynomials and liminal reciprocity
- Abstract: I will share some recent results on the moduli space of multivariate polynomials which are irreducible over a field K, including a surprising connection between univariate polynomials and the limiting space of irreducibles in infinitely many variables.
- Thursday November 29 at 3-4pm in Ry 358
- Fedor Manin, Ohio State University
- Geometric shadows of rational homotopy theory
- Abstract: Given compact simplicial complexes or Riemannian manifolds X and Y, what is the least Lipschitz constant of a nullhomotopy of an L-Lipschitz map X \to Y, as a function of L? If X is the circle, then this is certainly bounded below by the square root of the Dehn function of \pi₁(Y), which for certain Y grows faster than any computable function. On the other hand, if Y is simply connected, then for any X whatsoever the answer is O(L²) (this is almost sharp in some cases.) This, among other results, follows from a geometric upgrade to a fundamental correspondence in algebraic topology due to Sullivan, which I will attempt to describe.
- 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.
- Thursday April 4 at 2:30-3:30pm in Ry 358
- Weiyan Chen, Minnesota
- Cohomology of the space of complex irreducible polynomials in several variables
- Abstract: We will show that the space of complex irreducible polynomials of degree d in n variables satisfies two forms of homological stability: first, its cohomology stabilizes as d increases, and second, its compactly supported cohomology stabilizes as n increases. Our topological results are inspired by counting results over finite fields due to Carlitz and Hyde.
- Friday April 12 at 4-5pm in Ry 251
- Vladimir Markovic, California Institute of Technology
- Namboodiri Lecture 1: The Surface Subgroup Problem
- Abstract: The surface subgroup problem asks whether a given group contains a subgroup that is isomorphic to the fundamental group of a closed surface. In this talk I will survey the role the surface subgroup problem plays in some important solved and unsolved problems in 3-manifold topology, the theory of arithmetic manifolds and geometric group theory.
- Monday April 15 at 4-5pm in Ry 251
- Vladimir Markovic, California Institute of Technology
- Namboodiri Lecture 2: Rigidity and geometry of Harmonic Maps
- Abstract: Harmonic maps play a prominent role in geometry. I will explain some of these applications including Siu's rigidity of negatively curved Kahler manifolds and the Corlette-Gromov-Schoen rigidity of representations theorem, as well as the very recent results of Markovic and Benoist-Hulin about the existence and uniqueness of harmonic maps between rank-1 symmetric spaces.
- Tuesday April 16 at 4-5pm in E 206
- Vladimir Markovic, California Institute of Technology
- Namboodiri Lecture 3: Teichmueller flow and complex geometry of Moduli Spaces
- Abstract: I will explain why in general the Caratheodory and Teichmueller metrics do not agree on Teichmueller spaces and why this yields a proof of the convexity conjecture of Siu. Moreover, I will illustrate how deep theorems in Teichmueller dynamics play an important role in classifying Teichmueller discs where the two metrics agree.
- Thursday April 18 at 2:30-3:30pm in Ry 358
- Sam Nariman, Northwestern
- Dynamical and cohomological obstruction to extending group actions
- Abstract: For any 3-manifold M with torus boundary, we find finitely generated subgroups of \Diff₀(\partial M) whose actions do not extend to actions on M; in many cases, there is even no action by homeomorphisms. The obstructions are both dynamical and cohomological in nature. We also show that, if \partial M = S², there is no section of the map \Diff₀(M) \to \Diff₀(\partial M). This is joint work with Kathryn Mann.
- Thursday May 9 at 2:30-3:30pm in Ry 358
- Alex Duncan, South Carolina
- TBA
- Abstract: TBA
- Thursday May 16 at 2:30-3:30pm in Ry 358
- Kathryn Mann, Brown
- TBA
- Abstract: TBA
- Monday May 27 at 2:30-3:30pm in Ry 358
- Corey Bregman, Brandeis
- TBA
- Abstract: TBA

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