|
|
There will be three workshops that run in parallel sessions on the morning and afternoon of May 8th and the morning of May 9th
Parallel 1 (A). Hyperbolic Structures and Kleinian Groups
Organizers: Canary, Minsky, Gilman and Masur
Parallel 2 (B) . Geometric and Algebraic Structures
Organizers: Wolf, Wolpert, Gilman, Goldman, Loftin and Reid
Parallel 3 (C). Conformal, Quasiconformal Geometry and Dynamics
Organizers: Saric, Loftin, and Markovic | Date and Time | Parallel 1 (A) | Parallel 2 (B) | Parallel 3 (C) | | Thursday, May 8, AM * | Englehard 203 | Englehard 213 | Englehard 201 | 9:30-11:45 coffee and REGISTRATION 1st Floor Hall Englehard | | | | | Talk # 1 10 - 10:30 | Kent | Freixas | Koch | | Talk # 2 10:40 -11:10 | Storm | Mclntyre | Hubbard | Coffee 11:10-11:30 1st Floor Hall Englehard | | | | | Talk # 3 11:30 -12:00 | Duchin | Mondello | Kim | | Talk # 4 12:10 -12:40 | Magid | Guo | Snipes | | Lunch 12:24-2:30 | | | | | Thursday, May 8, PM | Englehard 215 | Englehard 213 | Englehard 201 | Registration continues 2:00-3:00 1st Floor Hall Englehard | | | | | Talk #5 2:30-3:00 | Lecuire | Huang | Matsuzaki | | Talk #6 3:10 - 3:40 | Min | Kapovich | Fletcher | | Coffee 3:40 - 4:10 | | | | | Talk #7 4:10 - 4:40 | Adeboye | Piggott | Hakobyan | | Talk #8 4:50 - 5:20 | Walker | Schleimer | Bowman | Pizza and Beer and Wine immediately after last talk in PRCC room 224** | TIME-5:30 to 7:30PM | | | | Friday, May 9 AM | Englehard 201 | Englehard 213 | Englehard 215 | Registration and coffee 8:30-11:45 1st Floor Hall Englehard | | | | | Talk #9 9:00 - 9:30 | Biringer | Loftin | Zhan | Talk #10 9:40 - 10:10 | Mangahas | Charette | Mitra | Talk #11 10:20 - 10:50 | Tao | Dumas | Handel | Coffee 10:50-11:20 1st Floor Hall Englehard | | | | | Talk #12 11:20 - 11:50 | Namazi | Bernstein | Kahn | | Talk #13 12:00 - 12:30 | Rafi | Lee | Saric |
*) Thursday AM: Shuttle bus will run continual repeat loop from Hilton, Penn Station, Robert Treat to campus from 9-10 AM. Last pick up loop at 10 AM. *) Thursday Evening: Shuttle bus continual loop returning from campus (Outside Robeson on MLK) to Station, Hilton and Robert Treat from 8:00-9:15. Last campus pick up 9:15. *) Friday AM: Shuttle bus loop: To campus from Robert Treat, Penn Station and Campus 8-9 AM Last Pick up at 9 AM. Titles and Abstracts for the Workshops On Volumes of Hyperbolic 4-orbifolds By Ilesanmi Adeboye (adeboye@usc.edu)
Abstract: We will construct an explicit lower bound for the volume of a hyperbolic orbifold dependent on dimension and the maximal order of torsion in the orbifolds' fundamental group. We will then discuss progress in developing a sharp bound in dimension 4.
Helicoid-Like Minimal Disks By Jacob Bernstein (jbern@math.mit.edu)
Abstract: Colding and Minicozzi have shown that if an embedded minimal disk in $B_R\subset\Real^3$ has large curvature then in a smaller ball, with radius proportional to R, it looks roughly like a piece of a helicoid. In this talk, we will see that near points whose curvature is large (relative to nearby points) the description can be made more precise. That is, the region of the surface in a ball, centered at a such a point and with radius inversely proportional the curvature at the point, is bi-Lipschitz to a piece of a helicoid. Moreover, as R goes to $\infty$, the Lipschitz constant goes to 1. This follows from Meeks and Rosenberg's result on the uniqueness of the helicoid of which, time permitting, we will discuss a proof. Joint work with C. Breiner.
Sequences of Hyperbolic 3-Manifolds with Unfaithful Markings By Ian Biringer (biringer@math.uchicago.edu)
Abstract: I will present some new results linking the pointwise convergence of a sequence of representations of a fixed group into Isom(H^3) with Gromov-Hausdorff convergence of the corresponding quotient manifolds. A detailed analysis already exists for sequences of faithful representations; I will give examples that illustrate the failure of these theorems in the unfaithful setting, and offer some useful replacements. Joint work with Juan Souto.
Applications of Delaunay triangulations to Teichmüller Theory By Joshua Bowman (joshua.bowman@gmail.com)
Abstract: I will define Delaunay triangulations of flat surfaces and prove several properties that make them useful in the realm of Teichmüller theory. These include a tessellation of the cotangent bundle to Teichmüller space Teich(g,n) that is Mod(g,n)-equivariant (as first shown by Veech), simple combinatorial descriptions of flat surfaces, and other tools that uncover geometric and dynamical properties of individual surfaces and moduli spaces of flat surfaces. These tools will be applied to the study of several examples, especially the genus 3 Arnoux-Yoccoz surface.
Affine Deformations of the Holonomy of a Three-holed Sphere By Virginie Charette (Virginie.Charette@USherbrooke.ca)
Abstract: Let T be a complete hyperbolic surface homeomorphic to a three- holed sphere and let G denote the image under the holonomy representation of its fundamental group. Identifying the group of hyperbolic isometries with an appropriate component of the group of isometries of Minkowski spacetime, we may consider affine deformations of G; we may ask, when does this affine deformation act properly discontinuously on $R^3$? An important invariant for affine isometries with non-elliptic linear part is the Margulis invariant, which is a measure of signed Lorentzian displacement. We show that an affine deformation of G acts properly discontinuously if and only if the Margulis invariant is positive for each of the three isometries corresponding to the pant holes of T. More precisely, we show that such an affine deformation admits a fundamental domain. This is joint work with Drumm and Goldman.
Lengths on Translation Surfaces By Moon Duchin (mduchin@gmail.com)
Abstact: We show that Flat(S), the space of marked flat metrics on a surface S, embeds into the (projectivized) length functions on simple closed
curves, just as for marked hyperbolic metrics in the classical Thurston theory. This allows us to consider questions such as identifying the
boundary in that projective space of Flat(S), analogous to the Thurston compactification of Teichmüeller space. This is joint work with Chris
Leininger and Kasra Rafi.
Grafting and the Teichmüller Metric By David Dumas (ddumas@math.brown.edu)
Abstact: We present joint work with Young-Eun Choi and Kasra Rafi on the geometric properties of grafting maps with respect to the Teichmüller metric on Teichmüller space. In particular, we show that grafting maps are Lipschitz, and that grafting rays are stable quasi-geodesics.
Local Bi-Lipschitz Equivalence of Teichmüeller Spaces By Alastair Fletcher (Alastair.Fletcher@nottingham.ac.uk)
Abstract: The Bers embedding is a holomorphic mapping of Teichmüeller space into a Banach space of holomorphic quadratic differentials. Whenever M is a Riemann surface of infinite analytic type, this Banach space Q(M) is isomorphic to the sequence space l^{\infty}. Using these two results, we will see that any two Riemann surfaces of infinite analytic type have locally bi-Lipschitz equivalent Teichmüller spaces. An analogous result holds for asymptotic Teichmüeller spaces.
Hermitian Tautological Line Bundles on the Moduli Space of Pointed Stable Curves By Gerard Freixas (Gerard.Freixas@math.u-psud.fr) Abstract: Following ideas in Arakelov geometry in the spirit of the works of Bismut, Deligne, Gillet and Soule, we construct natural hermitian line bundles on the moduli space of curves and establish a Riemann-Roch type isometry between them. This isometry links a number of results due to S. Wolpert, concerning the Selberg zeta function, the family hyperbolic metric and the Takhtajan-Zograf Kahler form. As an application, we compute special values of Selberg zeta functions for open modular curves. Global Fixed Points for Centralizers and Morita's Theorem By Michael Handel (MICHAEL.HANDEL@lehman.cuny.edu)
Abstract: We prove that the mapping class group of a closed surface S does not lift to the diffeomorphism group $\Diff(S)$ of S if the genus of S is greater than or equal to 3. The proof makes uses of the Thurston stability theorem and a fixed point theorem for certain subgroups of $\Homeo(D^2)$ (joint work with John Franks).
Parametrizations of the Teichmüller Space of Surfaces with Boundary By Ren Guo (renguo@math.rutgers.edu)
Abstract: The Teichmüller space of a surface with boundary is the space of all isotopy classes of hyperbolic metrics with totally geodesic boundary. Using the cosine law of a hyperbolc right-angled hexagon, F. Luo constructed a family of new coordinates of the Teichmüller space. Under each of the new coordinates, the Teichmüller space is an open convex polytope. Conformal Dimension of Products By Hrant Hakobyan (hhakob@math.toronto.edu)
Abstract: Bishop and Tyson asked for a charachterization of the subsets E of the line which have the property that ExY is minimal for conformal dimension for every compact Y. We show that E satisfies this property if and only if E is minimal for conformal dimension itself.
On Curvatures along a Weil-Petersson Geodesic in Teichmüller Space by Zeno Huang (huang@gauss.math.csi.cuny.edu)
Abstract: I will report what one finds about the shape of a Weil-Petersson geodesic if travels along it.
Monodromy of Horseshoe Hénon Mappings and Automorphisms of the Two-shift By John Hamal Hubbard (hubbard@math.cornell.edu)
Abstract: Quadratic Hénon mappings are written
Ha,c : (x, y)→(x^2+ c − ay, x).
For a ≠0 these are polynomial diffeomorphisms of C^2. Among these, in particular for |c| sufficiently large, one finds maps conjugate to a “Smale horseshoe”. Let L \subset C^2 be the set of parameters for which the corresponding map is a horseshoe. For (a, c) \in L, we consider the set Ka,c \subset C^2 of points (x, y) \in C^2 with bounded forward and backward orbits. The action of Ha,c on Ha,c is conjugate to the full shift on two symbols. If (a0, c0) \in L is a basepoint, then there is a monodromy action
r : π1(L, ((a0, c0))) →Aut(Ka0,c0),
giving rise to automorphisms of the full two-shift. The group of automorphisms of the full two-shift is a pretty mysterious group. We will try to explain why certain loops give rise to certain automorphisms
The Ehrenpreis Conjecture for Punctured Surfaces
By Jeremy Kahn (kahn@math.sunysb.edu) Abstract: The Ehrenpreis conjecture states that any two Riemann surface of the same general type have finite covers that are arbitrarily close in the Teichmüller metric.
Let S be a non-compact finite-area hyperbolic Riemann surface, and let $\epsilon > 0$. We normalize the Weil-Peterssen metric on hyperbolic surfaces by dividing the standard inner product by the area of the surface; this makes it invariant under passing to a finite cover and strictly smaller than the Teichmüller metric.
We prove that there is a finite degree cover $\hat S$ of S such that the normalized Weil-Peterssen distance between $\hat S$ and a cover of the modular surface is less than $\epsilon$. This is joint work with Vladamir Markovic.
Harmonic Functions and Ends of Groups
By Michael Kapovich (kapovich@math.ucdavis.edu) Abstract: I will sketch a proof of a compactness theorem for a certain family of harmonic functions on noncompact manifolds and its application to a proof of Stallings' theorem on groups with infinitely many ends. Skinning Maps
By Richard Kent (rpkfour@gmail.com) Abstract: The skinning map is a holomorphic self map of the Teichmüller space that arises naturally in Thurston's proof of Geometrization for Haken manifolds. Thurston's Bounded Image Theorem says that the skinning map of a hyperbolic manifold with totally geodesic boundary has bounded image. Yair Minsky has asked if bounds on the diameter may be obtained given topological information about the manifold. I'll discuss "sharp" upper and lower bounds that only depend on the volume of the metric with totally geodesic boundary. These follow from: a filling theorem, which says that skinning maps converge uniformly as higher Dehn fillings are performed; the Bounded Image Theorem, together with a finiteness theorem of Jorgensen; and a theorem (joint with D. Dumas) that skinning maps are never constant. The theorem is sharp in the sense that the upper and lower bounds tend to infinity and zero, respectively, as the volume grows (the fact there there is no universal lower bound to the diameter is joint with K. Bromberg). Quasiconformal non-stability in Hyperbolic 4-space
By Youngju Kim (ykim@gc.cuny.edu) Abstract: An n-dimensional Mobius group is said to be quasiconformally stable if its sufficiently small deformations into Isom+(H^n), the group of orientation-preserving isometries acting on hyperbolic n-space, are all quasiconformally conjugate to its identity deformation. Our main result states that there is a Mobius group acting on hyperbolic 4-space which is geometrically finite, but is not quasiconformally stable. This is in contrast to lower dimensions, where any geometrically finite Mobius group is quasiconformally stable. A New Link between Teichmüller Theory and Complex Dynamics
By Sarah C. Koch (kochs@math.cornell.edu) Abstract: Inspired by Thurston's theorem of the characterization of rational maps, J. H. Hubbard posed the "twisted rabbit" problem. This problem was recently solved by L. Bartholdi and V. Nekrashevych using original techniques involving iterated monodromy groups. A key part of their solution contains the construction of a map on a certain moduli space. We discuss Thurston's theorem and present the "twisted rabbit" problem. We then generalize the construction of these Bartholdi-Nekrashevych maps and discuss their dynamical significance. Quasi-Fuchsian Manifolds with Particles
By Cyril Lecuire (cyril.lecuire@polytechnique.org) Abstract: We consider hyperbolic structures with vertical cone singularities on a an I-bundle over a closed surface. We show that, as long as the singular angles are less than pi, results known for Quasi-Fuchsian manifolds hold for those "Quasi-Fuchsian manifolds with particles". Namely, we have Ahlfors-Bers coordinates and we can characterise the bending measured geodesic laminations of the convex core. Fundamental Domains of Convex Projective Structures
By Jaejeong Lee (zlee@math.ucdavis.edu)
Abstract: Convex (or properly convex) projective structures on manifolds share many common features with non-positively curved metrics. The lack of invariant metrics, however, makes it harder to study them. For example, some of the well-known facts about fundamental domains in the case of constant curvature geometries are no longer obvious in projective geometry. In my talk, I will show that every properly convex projective structure admits a convex fundamental polyhedron, which is the Dirichlet domain with respect to a certain distance-like function. The proof makes an essential use of the solution (by Cheng and Yau) of Calabi's conjecture on complete hyperbolic affine spheres and the duality relation between them.
Real Projective Surfaces and Holomorphic Cubic Differentials By John Loftin (loftin@andromeda.rutgers.edu)
Abstract: A convex real projective structure on a closed surface S is given by S=Omega/Gamma, where Omega is a convex domain in $R^2\subset RP^2$, and Gamma is a discrete subgroup of PGL(3,R). There are many such structures: the Klein model of the hyperbolic plane shows that every hyperbolic structure on S induces a convex real projective structure.
There is a canonical identification of a convex real projective structure on an orientable surface S of genus g>1 and a pair consisting of a conformal structure $\Sigma$ together with a holomorphic cubic differential U on the surface. $(\Sigma,U)$ can be used to explicitly calculate the $RP^2$ holonomy along loops on S in various limitingcases: neck pinches (the Deligne-Mumford compactication of the moduli space of curves), and the case that U homothetically goes to $\infty$. The proofs use affine differential geometry results of Cheng-Yau and C.P. Wang, and PDE estimates.
Deformation Spaces of Hyperbolic 3-Manifolds that are Not Locally Connected By Aaron Magid (magid@umich.edu)
Abstract: For a compact 3-manifold, M, and a disjoint collection, P, of annuli and tori in the boundary of M, we will define the relative deformation space, AH(M,P), of all hyperbolic 3-manifolds homotopy equivalent to M with cusps associated to P. After reviewing some classical results concerning the topology of the interior of AH(M,P), we will survey some of the progress that has been made in bumponomics (i.e., the pathological topological behavior near the boundary of AH(M,P)). We find infinitely many pared 3-manifolds, (M,P), for which the relative deformation space, AH(M,P), is not locally connected. This generalizes Bromberg's result that the space of Kleinian punctured torus groups is not locally connected.
Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group By Johanna Mangahas (mangahas@umich.edu)
Abstract: Like linear groups and hyperbolic groups, subgroups of the mapping class group with exponential growth have uniform exponential growth; furthermore, their minimal growth rates have a lower bound which depends on the surface, not the particular subgroup. I'll describe the proof, for which one plays ping-pong on sets of simple closed curves on the surface.
Self-covering of Hyperbolic Surfaces By Katsuhiko Matsuzaki (matsuzak0608@mocha.ocn.ne.jp)
Abstract: First, we give a necessary condition for a hyperbolic Riemann surface to admit a (non-injective) holomorphic self-cover in terms of the
corresponding Fuchsian group. Namely, if the Fuchsian group is of divergence type at the critical exponent of its Poincare series, then
the Riemann surface has no self-covers. The proof uses uniqueness of the Patterson-Sullivan measure and can be extended to higher
dimensional cases. A holomorphic self-cover of a Riemann surface induces a non-surjective holomorphic self-embedding of its Teichmüeller space. Next, we investigate the dynamics of such a self-embedding and examine the distribution of isometric tangent vectors over the Teichmüeller space. We also extend our observation to quasiregular self-covers of Riemann surfaces.
Determinants and Zeta Functions for Kleinian Groups By Andrew Mcintyre (amcintyre@bennington.edu)
Abstract: I will discuss the determinant of the Laplacian operator on Riemann surfaces corresponding to geometrically finite Kleinian groups. In particular, I will discuss connections to the geometry of the associated open hyperbolic 3-manifold, and analogies with number theory.
Hyperbolic Graph of Surface Groups By Honglin Min (hmin@andromeda.rutgers.edu)
Abstract: We find conditions under which the fundamental groups of the graphs of surface groups are hyperbolic. And we construct an example of such a group which is hyperbolic but is not commensurate to any surface-by-free group.
Metric Properties of some Generalized Teichmüller Spaces By Sudeb Mitra (sudeb.mitra@qc.cuny.edu) Abstract: Let X be a hyperbolic Riemann surface, and let E be a closed subset of X. We will discuss some metric properties of the Teichmüller space of X rel E, denoted by T(X ,E). In particular, we will talk about Teichmüller contraction for T(X ,E) and holomorphic maps from the open unit disk into T(X , E). There is a "generalized" Teichmüller curve V(X,E) over T(X,E) -- at the end of the talk, we will discuss some properties of this curve, and how they may be related with holomorphic motions of the closed set $E$ in a holomorphic family of Riemann surfaces. Natural Triangulations of Riemann Surfaces with Boundary and Weil-Petersson Poisson Structure By Gabriele Mondello (g.mondello@imperial.ac.uk)
Abstract: Natural cellularizations of the Teichmüller space via ribbon graphs have been invented using decorated hyperbolic surfaces or Jenkins - Strebel differentials. We show that these two constructions are interpolated by infinitely grafted hyperbolic surfaces with boundary. We also discuss the behavior of the Weil-Petersson symplectic structure in the flat limit.
Splittings and Hyperbolic Geometry
By Hossein Namazi (hossein@Math.Princeton.EDU)
Abstract: We discuss how Heegaard splittings can be used to study the hyperbolic metric on a closed 3-manifold. We try to explore various results and examples where the combinatorics of the splitting detects the geometric structure.
Primitive Elements in the Free Group of Rank Two
By Adam Piggot (adam.piggott@tufts.edu)
Abstract: Let F denote the free group with basis {x, y}. An element w of F is a palindrome if the unique reduced word for w reads the same when read from left-to-right and right-to-left, and primitive if there exists an automorphism f in Aut F such that f(x) = w. Some primitive elements are also palindromes, and the rest are products of two palindromes. We will explain why.
Convexity of Lengths along a Teichmüeller Geodesic
By Kasra Rafi (kasra.rafi@gmail.com)
Abstract: We will show that both the extremal length and the hyperbolic length of a simple closed curve along a Teichmüeller geodesic are quasi-convex functions of time. We conclude that a round ball in Teichmüeller space is quasi-convex. It is notable that these lengths are not in general convex functions. (Joint work with Lenzhen.)
Realization of the Mapping Class Group by Homeomorphisms
By Dragomir Saric (dragomir.saric@qc.cuny.edu)
Abstract: Let S be a closed surface of genus g>1. Let Pr be a natural projection from the group of homeomorphism Homeo(S) of S onto the mapping class group MC(S). Morita showed that Pr has no homomorphic section E from MC(S) into Diffeo(S) such that Pr(E)=id when g>4. Markovic showed that there is no section E from MC(S) into Homeo(S) when g>5. Recently, Franks and Handel showed that there is no section E from MC(S) into Diffeo(S) when g>2.
We show that there is no section E from MC(S) into Homeo(S) when g>1. That also settles the case of diffeomorphisms for g=2. Joint work with V. Markovic.
Compressed Words in Hyperbolic Groups
By Saul Schleimer (saulsch@math.rutgers.edu)
Abstract: Suppose that G is a Gromov hyperbolic group. Then the "compressed word problem" in G has a polynomial-time solution. (When G is a free group this result is due to Markus Lohrey.) As a consequence, the word problems in Aut(G) and Out(G) are also polynomial-time. Since surface groups are hyperbolic, this gives a new solution to the word problem in the mapping class group.
Flat Forms in Banach Spaces
By Marie Snipes (marie_snipes@sbcglobal.net) Abstract: The flat forms of Whitney have been useful in solving many problems in geometric analysis and elsewhere. A classical theorem of Wolfe states that the space of flat forms is in fact the dual to the space of flat chains in Euclidean space. Recent work by T. Adams generalized flat chains to Banach spaces. We will define a flat differential form in a Banach space and discuss the generalization of Wolfe's theorem to this setting. Maps between Spheres of Minimal Stretchs
By Peter Storm (peterastorm@gmail.com) Abstract: Given a homotopy class of maps between spheres, curiosity leads some to wonder about the existence of a map with minimal Lipschitzconstant, i.e. minimal stretch. I'll present a couple results in this relatively uncharted topic focusing on Hopf maps. The talk will be more general geometric topology than hyperbolic geometry. This is joint work with Herman Gluck. Linear Bound for the Length of a Conjugating Element in the Mapping Class Group
By Jing Tao (jingtao@math.uic.edu) Abstract: Given two conjugate mapping classes f and g, we produce a conjugating element w such that | w | ≤ K ( | f | + | g | ), where | * | denotes the word length with respect to a fixed generating set, and K is a constant depending only on the generating set. Masur and Minsky previously showed this in the case when f and g are pseudo-Anosovs. This is a preliminary report. Fundamental Groups of Moduli Spaces of Quadratic Differentials
By Katharine Walker (kaceyw@umich.edu) Abstract: I will use facts about configuration spaces of points on Riemann surfaces and their relationship to sections of line bundles, as well as some properties of hyperelliptic Riemann surfaces, to construct a quotient group of the fundamental group for certain strata of the moduli space of quadratic differentials over Teichmüller space. Reversibility and Duality of SLE
By Dapeng Zhan (dapeng.zhan@yale.edu) Abstract: I will explain the proof of the reversibility of chordal SLE$_\kappa$ for $\kappa\in (0,4]$, and the proof of some versions of Duplantier's duality conjecture about SLE.
|
|
|
|