Geometry/Topology Seminar

Spring 2008

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

- Thursday April 3 at 2pm in Ry358
- Daniel Groves, University of Illinois at Chicago
- Homomorphisms to mapping class groups
- Abstract: Let S be an orientable surface of finite type, and Mod(S) its mapping class group. If B is a compact space, and G is its fundamental group then the set of isomorphism classes of S-bundles over B is naturally parametrized by
  X_S(G) = Hom(G,Mod(S))/~
  where the equivalence relation is post-composition by conjugation in Mod(S).
  I'll present some results about the structure of the set X_S(G), for an arbitrary finitely presented group G. Morally, X_S(G) being infinite implies that G admits a splitting as a graph of groups, and the structure of the set X_S(G) can (often) be understood via studying splittings of G.
- Thursday April 17 at 2pm in Ry358
- Kasra Rafi, University of Chicago
- Convexity of lengths along a Teichmuller geodesic
- Abstract: We will show that both the extremal length and the hyperbolic length of a simple closed curve along a Teichmuller geodescic are quasi-convex functions of time. We conclude that a round ball in Teichmuller space is quasi-convex. It is notable that these lengths are not in general strictly convex functions. (Joint work with Lenzhen.)
- Thursday April 24 at 2pm in Ry358
- David Dumas, Brown University
- Grafting and the Teichmuller metric
- Abstract: We present joint work with Young-Eun Choi and Kasra Rafi on the geometric properties of grafting maps with respect to the Teichmuller metric on Teichmuller space. In particular, we show that grafting maps are Lipschitz, and that grafting rays are stable quasi-geodesics.
- Thursday May 1 at 2pm in Ry358
- Enrico Leuzinger, Universitaet Karlsruhe
- Reduction theory for arithmetic and mapping class groups
- Abstract: Reduction theory is concerned with the construction of (coarse) fundamental domains for groups acting properly discontinuously. I will first outline this theory for arithmetic groups and then describe a remarkable analogy for mapping class groups. As an application I will also discuss asymptotic cones for locally symmetric spaces and moduli spaces.
- Special day and room
- Wednesday May 7 at 1:30pm in E202
- Eriko Hironaka, Florida State University
- Mapping classes from graphs
- Abstract: Mapping classes of compact surfaces can be constructed in a natural way from finite graphs. We describe the effect of subdividing edges of graphs on the geometry of the corresponding mapping class, and use this to find bounds on dilatations of pseudo-Anosov maps.
- Thursday May 8 at 2pm in Ry358
- Daniel Allcock, University of Texas at Austin
- The metric completion of a nonpositively curved space
- Abstract: If X is an noncomplete nonpositively curved space (precisely: it is locally CAT(0)) then its completion may easily fail to be. But in some interesting cases the completion is CAT(0), for example Teichmuller space and the many branched covers of Riemannian manifolds over totally geodesic submanifolds. We have found a simple criterion which forces X to be CAT(0). Its key hypothesis is a statement about X *near* the boundary but not at the boundary.
- Thursday May 15 at 2pm in Ry358
- Jianguo Cao, University of Notre Dame
- An extension of Perelman's soul theorem for singular spaces
- Abstract: The study of Alexandrov spaces played an important role in the recent work of Perelman on Poincare-Thurston conjecture. In particular, the critical point theory of distance functions lead to Perelman's stability theorem and collapsing theorem for 3-manifolds with lower curvature bound.
  In this lecture, using the moving half-space method and the one-sided gradient flow of distance functions on Alexandrov spaces, we establish an extension of Perelman's soul theorem for singular spaces:
  “Let M be a non-compact, complete, n-dimensional Alexandrov space of non-negative curvature. Suppose that M has no boundary and M has positive curvature on a non-empty open subset. Then M must be contractible".
  This is a joint work with B. Dai and J. Mei.
- Special day and time
- Tuesday May 27 at 12pm in E206
- Larry Guth, Stanford University
- Area-contracting maps and Steenrod squares
- Abstract: The k-dilation of a map measures how much the map stretches k-dimensional areas. For example, consider the linear map from R³ to itself which multiplies the x-coordinate by 10⁶ but divides the y and z coordinates by 10⁶. This map does not increase the 2-dimensional area of any surface in R³ -- it has 2-dilation equal to 1. We fix two Riemannian manifolds (Mⁿ, g) and (Nⁿ, h) and consider the smallest k-dilation of any degree 1 map from M to N. This minimal dilation is hard to estimate even for simple examples such as when M and N are each the n-sphere and the metrics g and h correspond to n-dimensional ellipses in Rⁿ⁺¹. The natural-seeming linear maps have much larger k-dilation than some other rather twisty maps. We will discuss upper and lower bounds for this problem, and the lower bounds are related to Steenrod squares.
- Thursday May 29 at 2pm in Ry358
- Svetlana Katok, Penn State University
- Reduction theory and coding of geodesics
- Abstract: I will discuss a method of coding of geodesics on surfaces of constant negative curvature using boundary expansions of the endpoints. For the modular surface this is related to a new 2-parameter family of continued fractions, some of which can be used for coding. I will also give a dynamical interpretation of the “reduction theory” which underlines these constructions and its relation to the attractor of a certain associated natural extension map. This is a joint project with Ilie Ugarcovici and Don Zagier.
- Thursday June 5 at 2pm in Ry358
- Ben McReynolds, University of Chicago
- Manifold covers of arithmetic orbifolds
- Abstract: I will discuss some geometric applications of a generalization of Selberg's lemma for arithmetic lattices that permit one to control some geometric properties of manifold covers of arithmetic orbifolds. The talk will be fairly elementary.
- Wednesday July 9 at 1pm in TBA
- Danny Calegari, Caltech
- Stable commutator length is rational in free groups
- Abstract: For any group G and any g in [G,G] the commutator length cl(g) is the smallest number of commutators in G whose product is equal to g. The stable commutator length scl(g) is the limit lim cl(gⁿ)/n. We show scl is rational in free groups. More generally, scl extends to a pseudo norm on group 1-boundaries B₁(G). We show in a free group, and some closely related groups, that this pseudonorm is piecewise rational linear. The method of proof has a number of surprising corollaries, some of which we discuss, including applications to constructing closed surface groups in certain graphs of free groups amalgamated over cyclic subgroups. We also present a conjectural picture of scl in free groups and some other groups.
- Thursday July 10 at 2pm in E308
- Danny Calegari, Caltech
- Combable functions, quasimorphisms and the central limit theorem
- Abstract: A function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left and right invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include (i) homomorphisms to Z (ii) word length with respect to a finite generating set (iii) most known explicit constructions of quasimorphisms (e.g. the Epstein-Fujiwara counting quasimorphisms) We show that bicombable functions on word-hyperbolic groups satisfy a central limit theorem: if \bar{\phi}_n is the value of \phi on a random element of word length n (in a certain sense), there are E and \sigma for which there is convergence in the sense of distribution n^-1/2(\bar{\phi}_n - nE) \to N(0,\sigma), where N(0,\sigma) denotes the normal distribution with standard deviation \sigma. As a corollary, we show that if S₁ and S₂ are any two finite generating sets for G, there is an algebraic number \lambda_1,2 depending on S₁ and S₂ such that almost every word of length n in the S₁ metric has word length n\lambda_1,2 in the S₂ metric, with error of size O(\sqrt{n}).
- Wednesday July 23 at 11am in Tea Room
- Roozbeh Hazrat, Queen's University Belfast
- Reduced K-theory for Azumaya Algebras, An overview
- Abstract: The theory of Azumaya algebras developed parallel to the theory of central simple algebras. However the latter are algebras over fields whereas the former are algebras over rings. One wonders how the K-theory of these objects compare to each other. We look at higher K-theory and reduced K-theory of these objects. We ask nice questions!
- Thursday August 7 at 2pm in E308
- Martin Moeller, Max Planck Institute
- Teichmueller curves and uniformizing differential equations
- Abstract: The talks starts with an overview of the known Teichmueller curves and their origin via flat surfaces. We then focus on the fields of definition of the Teichmueller curves. For particular cases of Teichmueller curves we can actually write down their equation as families of algebraic cruves. This will yield recursion relations with surprising integrality properties, as in Apery's proof of the irrationality of \zeta(3).

For questions, contact

Benson Farb, farb at math.uchicago.edu
Alex Eskin, eskin at math.uchicago.edu
Shmuel Weinberger, shmuel at math.uchicago.edu
Nathan Broaddus, broaddus at math.uchicago.edu