Department of Mathematics

Details

Curves on surfaces with large topology

by Hugo Parlier  (www) | University of Toronto
Time: 16:10  (Monday, Mar. 22, 2010)
Location: HU1018, 215 Huron St
Abstract:
Surfaces of large genus are intriguing objects. Their geometry has been studied by finding geometric properties that hold for all surfaces of the same genus, and by finding families of surfaces with unexpected or extremely geometric behavior. A classical example of this is the size of systoles where on the one hand Gromov showed that there exists a universal constant $C$ such that any (orientable) surface of genus $g$ with area normalized to $g$ has a homotopically non-trivial loop (a systole) of length less than $C log(g)$. On the other hand, Buser and Sarnak constructed a family of hyperbolic surfaces where the systole roughly grows like $log(g)$. Another important example, in particular for the study of hyperbolic surfaces and the related study of Teichmüller spaces, is the study of short pants decompositions, first studied by Bers. The talk will discuss two ideas on how to further the understanding of surfaces of large genus. The first part will be about joint results with F. Balacheff and S. Sabourau on upper bounds on the sums of lengths of pants decompositions and related questions. In particular we investigate how to find short pants decompositions on punctured spheres, and how to find families of homologically independent short curves. The second part, joint with L. Guth and R. Young, will be about how to construct surfaces with large pants decompositions using random constructions.

Dates in this series

· Monday, Mar. 08, 2010: The fundamental group of random 2-complexes (Matthew Kahle)

· Monday, Mar. 15, 2010: Three Dimensional Manifolds All of Whose Geodesics Are Closed (John Olsen)

· Monday, Mar. 22, 2010: Curves on surfaces with large topology (Hugo Parlier)

· Friday, Mar. 26, 2010: New invariants in singularity theory (Rustam Sadykov)

· Friday, Mar. 26, 2010: The hidden smoothness of the space of stable maps (Jonathan Wise)

· Monday, Apr. 05, 2010: Moduli spaces of conics in curves (Chris Judge)

· Monday, Sep. 20, 2010: Total Q-curvature for CR manifolds (Kengo Hirachi)

· Monday, Sep. 27, 2010: Group completions of moduli spaces (Rustam Sadykov)

· Tuesday, Oct. 12, 2010: Pontryagin invariants and integral formulas for Milnor's triple linking number (Herman Gluck)

· Monday, Oct. 18, 2010: Positively curved polar manifolds (Fuquan Fang)

· Monday, Nov. 01, 2010: Rational formality and special holonomy (Manuel Amann)

· Monday, Nov. 08, 2010: Isoperimetric Inequalities -- methods and applications (Emanuel Milman)

· Monday, Nov. 15, 2010: From the sandwich to the waist (Alfredo Hubard)

· Monday, Nov. 29, 2010: TBA (Keith Burns)

· Monday, Nov. 29, 2010: Conjugate points and non-injectivity of the exponential map (Keith Burns)

· Monday, Jan. 17, 2011: Loop products and resonance for loop homology on spheres (Nancy Hingston)

· Monday, Jan. 31, 2011: A Variational Characterization of the Catenoid (Jacob Bernstein)

· Monday, Apr. 18, 2011: On the structure of conformally covariant powers of the Laplacian (Andreas Juhl)

· Monday, May. 09, 2011: Causality, Low conjecture and globally hyperbolic spacetimes (Vladimir Chernov)

· Monday, Sep. 26, 2011: Mapping degrees of self-maps of simply-connected manifolds (Manuel Amann)

· Monday, Oct. 17, 2011: Formality of Positive Quaternion Kähler manifolds (Vitali Kapovitch)

· Friday, Oct. 21, 2011: Kahler-Einstein metrics singular along a divisor (Yanir Rubinstein)

· Monday, Oct. 24, 2011: Diastolic inequality on a 2-sphere (Yevgeny Liokumovich)

· Monday, Nov. 14, 2011: TBA (Ian Hambleton)

· Monday, Nov. 14, 2011: Coarse geometry and group actions (Ian Hambleton)

· Monday, Nov. 21, 2011: Fundamental groups of manifolds with lower Ricci curvature bounds (Vitali Kapovitch)

· Monday, Nov. 28, 2011: Fundamental groups of manifolds with lower Ricci curvature bounds II (Vitali Kapovitch)

· Tuesday, Nov. 29, 2011: Asymptotic geometry, finite generation of fundamental groups, and harmonic functions (Bruce Kleiner)

· Friday, Dec. 09, 2011: Topological Complexity and H-Spaces (Gregory Lupton)

· Tuesday, Jan. 24, 2012: A new construction of compact $G_2$ manifolds by glueing families of Eguchi-Hanson spaces (Spiro Karigiannis)

· Thursday, Jan. 26, 2012: Coarse Differentiation and the rank of Teichmüller space (Kasra Rafi)

· Thursday, Feb. 16, 2012: Manifolds and spaces with Ricci curvature bounds (Toby Colding)

· Monday, Feb. 27, 2012: Homological and homotopical filling functions (Robert Young)

· Monday, Mar. 12, 2012: Algorithmic decidability of Thurston equivalence and geometrization of branched coverings (Michael Yampolsky)

· Monday, Mar. 26, 2012: On some problems in the geometry of surfaces in normed spaces (Dmitri Burago)

· Monday, Apr. 02, 2012: CANCELLED -- Scattering rigidity with trapped geodesics; the two dimensional case (Chris Croke)

· Monday, Sep. 10, 2012: Realisation and dismantlability (Piotr Przytycki)

· Friday, Oct. 05, 2012: The metric completion of Outer Space (Yael Algom Kfir)

· Monday, Oct. 15, 2012: Complexes which require large filling distortion in Euclidean space (Dominic Dotterrer)

· Tuesday, Nov. 06, 2012: A theorem of existence for Asymptotically Conical Calabi-Yau manifolds (Ronan Conlon)

· Monday, Nov. 19, 2012: Ricci Solitons of Cohomogeneity One (McKenzie Wang)

· Monday, Nov. 26, 2012: Complete affine 3-manifolds, Lorentzian and hyperbolic geometry (Bill Goldman)

· Monday, Dec. 03, 2012: Hyperbolic Group Boundaries and Cannon-Thurston Maps (Owen Baker)